PARAMETER ESTIMATION OF GROUND
THERMAL PROPERTIES
by
NAGENDRA KUMAR JAIN
Bachelor of Science
Indian Institute of Technology
Bombay, India
1997
Submitted to the Faculty of the
Graduate College of the
Oklahoma State University
in partial fulfillment of
the requirements for
the degree of
MASTER of SCIENCE
July, 1999
PARAMETER ESTIMATION OF GROUND
THERMAL PROPERTIES
Thesis Approved:

II
b
ACKNOWLEDGEMENTS
I would like to thank my parents and my brother for their constant upport during my graduate tudie.
Without their support, this work wOllld not have been possible.
I would like to extend my reverence to Dr. Jeffrey D. Spitler for his constant guidance and valuable input
f
during the course of this project. You have shown me a way to my future and I hope to adopt your
principles in my futllfe life.
I would like to thank Dr. Rees Simon for his expert guidance and comments during lhi project I learned a
lot from you and I respect you for all yOllr support.
I would like to extend my thanks and appreciation to the following people:
The members of my advisory committee for their willingness to provide me input and carve thi work in a
better way.
Cenk Yavuzturk for providing me his valuable support, input and working late hour' with me together in
the lab.
Andrew Chaisson for working together on the project which was useful to me in my thesis.
Ravikant Sharma for his support during pessimistic days.
Rajesh Chakravarthy for being supportive during strange situation and saving my time by making a way
through them.
Naren for being quite helpful out of the way.
1lI
Chapter
TABLE OF CONTENTS
Page
1. Introduction 1
1.1. Overview 1
1.2. Literature Review 3
1.3. Experimental Apparatus 4
1.3.1. Power Supply System 5
1.3.2. Data Acquisition System 5
1.3.3. Temperature measurement 6
1.4. The Numerical Model 6
1.5. Objective function 7
2. Optimization Domain y
2.1. Exhaustive Search 9
3. Optimization Methods II
3.1. Neider Mead Simplex 11
3.1.1. Reflection 11
3.1.2 Expansion 1.2
3.1.3 Contraction 13
3.2. O'Nei.ll 's implementation of Neider Mead Simplex 15
3.3. Box's Complex method 16
3.4. Hooke and Jeeves' method 17
3.4.1. Algorithm 18
3.5. Powell's method 20
3.5.1. Methodology 22
3.6. BroydenFJetcherGoldfarbShanno (BFGS) method 23
3.6.1. Background 23
IV
3.6.2. AJgorithm 24
3.7. Genetic AJgorithms 26
3.8. Quadratic fit method 27
3.9. O'Neill's implementation of NeIder Mead Simplex with exploratory search 28
3.10. Summary of all the methods 31
4. Offline Estimation Results and Discussion 33
4.1. Results from NeIder Mead Simplex 34
4.2. Results from O'Neill's implementation of NeIder Mead Simplex 37
4.3. Results from Box's Complex method 41
4.4. Results from Hooke and Jeeves method .45
4.5. Results from Powell's method 46
4.6. Results from BFGS method 49
4.7. Results from Genetic AJgorithms 51
4.8. Results from Quadratic fit method 52
4.9. Results from O'Neill's implementation of NeIder Mead Simplex
method with exploratory search 53
4.10. Comparison of results 55
5. Online Parameter Estimation 58
5.1. Nonlinear Recursive Estimator 59
5.2. Application of Nonlinear optimization method 60
5.3. Results from Salsbury's method 61
5.4. Results from application of optimization method 62
5.5. Determination of the .length of the experimental test.. 71
6. Conclusions and Recommendations 73
6.1. Offline parameter estimation 73
6.2. Online parameter estimation 75
v
6.3. Recommendations 77
References 78
VI
Figure
LIST OF FIGURES
Page
11. A schematic diagram of trailer.. 5
12. Domain for numerical model 7
13. A typical plot of comparison between Experimental and Estimated
temperatures by numerical model 8
21. Domain of objective function 10
31. Reflection Process 12
32. Expansion Process 13
33. Contraction Process 14
34. Contraction towards minimum 14
35. Complex in 2D space 17
36. Steps taken in Powell's method 23
37. Quadratic Curve fitting in a valley 28
38. Line source model 29
41. Simplex in 2D domain 34
42. Movement of Simplex towards the minimum for SiteA5_93 37
43. Small excursions in both the coordinate directions 38
44. Estimated Standard Deviation vs. Exponent of errOL 41
45. Reflection of the worst point for reflection coefficient of 0.8 .44
46. Steps taken by Hooke and Jeeves method for SiteA 1_98 .46
47. Steps taken by Powell's method for SiteA2_170 .48
48. Steps taken by BFGS method for SiteA6_240 50
51. Variation of updated parameters for a fixed interval of optimization time
as the test for SiteAI_98 progresses 65
VII
52. Variation of estimated parameters for minimization at each data set for
SiteA 1_98 68
53. Variation of estimated parameters for minimization at each data set for
SiteA2_114 68
54. Variation of estimated parameters for minimization at each data set for
SiteA2_170 69
55. Variation of estimated parameters for minimization at each data set for
SiteA5_93 69
56. Variation of estimated parameters for minimization at each data et for
SiteA6_240 70
57. Variation of estimated parameters for minimization at each data set for
Chickasha data 70
58. Variation of estimated parameters for minimization at each data set for
Weatherford data 71
Vlll
Table
LIST OF TABLES
Page
31. A summary of characteristics of all the methods 32
41. Summary of Experimental tests used for parameter estimation 34
42. A Summary of results from NeIder Mead Simplex Algorithm for
50 hours of data 36
43. Summary of results for O'Neill's Simplex Algorithm for 50 hours of data 38
44. Summary of results for O'Neill's Simplex algorithm for step size=0.05
in both Ksoil and Kgrout directions 39
45. Summary of results from O'Neill's Simplex Algorithm for the sum of
the fourth power of the error for all the test sites .40
46. Comparison of results from BoxI s Complex method for SiteA 1_98 with
different parameter values 42
47. Results from Box's Complex method for all the sites with the best et of
parameter values 43
48. Results from Box's Complex method for reflection coefficient of I .3 44
49. Results from Hooke and Jeeves method for all the sites .46
410. Results from Powell's method for all the Sites 48
411. A summary of results from BFGS method for all the sites 49
412. A summary of results from Genetic Algorithms for all the site for
maximum of 100 generations 52
413. Results of all the sites using Quadratic fit method 53
4 I4. Results of exploratory search for all the sites 54
4 15. Results of O'Neill Simplex method with exploratory search for all the sites 55
416. Comparison of methods based on average number of objective
function evaluations 56
417. Comparison of results with exploratory search 57
IX
51. Coinparison of results for six testsites for updating the parameter after every
collected data point. 63
52. Comparison of results for six testsites for updating the parameter after every 15
collected data points 64
53. Comparison of results for updating parameters during a fixed amount of time 65
54. Comparison of results obtained from minimization at each step and
offline NeIder Mead Simplex method 67
55. A summary of length of experiment for all the test sites 72
x
1. Introduction
1.1. Overview
Ground source heat pump systems (GSHP) offer a great advantage over
conventional heat pump systems. GSHPs have high efficiency, low maintenance costs
and low overall operating costs. The heat is absorbed and rejected through water, which
is a more desirable heat transfer medium than air as used in conventional systems. The
heat is absorbed from the ground during heating operation and rejected to the ground
during cooling operation. Since the ground is nearly at a constant temperature, the heat is
absorbed and rejected at a more constant temperature than in air source heat pump
systems, which leads to high coefficient of perfonnance (COP) of these systems.
In GSHPs, there are three loops in the system. The first loop is the water/air loop
depending on whether the heated/cooled space requires water or air to heat/cool the
space. The second loop is the refrigerant loop which exchanges heat with both source and
sink loops. The third loop is the ground loop in which water exchanges heat with the
ground and refrigerant. This loop might be dosed loop or open loop depending on the
GSHP system.
For commercial buildings the ground loop typically consists of a series of vertical
heat exchangers, made up of three main components, the highdensity polyethylene pipe
(HDPE), material around the pipe called grout and the soil around grout. The pipe, which
has a shape of Vtube, is inserted in a vertical borehole and the borehole is fil1ed with
grout around the pipe. The length of the borehole varies depending on the insitu test.
Designing ofGSHP systems requires accurate estimation of the thennal properties
of the ground. Hence, to accurately estimate the ground thennal properties, a trailer was
designed by Austin (1998, 2000) to experimentally measure power and temperature
response. The power and temperature response can be used to inversely find the ground
thermal conductivity.
Using power measurements, borehole geometry and a guess of ground thermal
properties, a twodimensional numerical model developed by Yavuzturk et al (1999)
simulates the borehole by finite difference methods and temperatures are computed at the
appropriate location of the borehole. The difference between the experimental and
numerical temperature profiles, at each time step is calculated, squared and added. The
thennal properties of the ground are adjusted such that this sum of the square of the error
(SSQERR) is minimized. An optimization algorithm is required to systematically vary
the properties to reach the minimum of SSQERR. This inverse problem of finding the
parameters using an experimental test and a numerical model is called a parameter
estimation problem. If the parameter estimation is perfonned after all the experimental
data has been collected, it is called offline parameter estimation.
To minimize SSQERR, severa] optimization methods have been applied so as to
reduce the time taken in estimating the thennal properties of the ground. Since, it takes
about three minutes of computer CPU time on a 233 MHz Pentium II processor to get the
temperature profile at each set of ground thennal properties, it is extremely critical to
reduce the number of times the model computes the temperature profile. Hence, the main
aim of the project is to reduce the number of objective function evaluations (getting the
SSQERR value from the model).
Only two parameters, soil and grout thennal conductivity, are considered for the
parameter estimation results. Additional parameters are probably redundant and add
unwarranted complexity to the problem. For more details on choosing the parameters, see
the thesis by Austin (1998) and research paper Austin et al (2000).
It is also desirable to be able to estimate the ground thermal properties while the
insitu test is perfonned. This reduces the overhead of taking the experimental data and
performing the offline estimation of parameters. By performing online parameter
estimation, we can obtain a plot of parameters as test is performed which helps 111
deciding the questions like the length of the test and final parameter values.
2
This thesis focuses mainJy on the application of different optimization algorithms
and finding the best algorithm to expedite the process of estimating the ground thennal
properties. Online parameter estimation is also investigated.
1.2. Literature Review
Since the parameters for this problem are nonlinearly related to the objective
function and are unbounded, unconstrained nonlinear optimization methods are
considered. Optimization methods are divided into two major classes:
• Detenninistic Methods.
• Direct Search Methods.
• lndirect Search (Gradient) Methods.
• Stochastic Methods.
Detenninistic methods find the mInlJTIum by searching the mImmum In a
particular direction with finite number of steps. Stochastic methods use random search to
find the minimum. Hence, the minimum found with stochastic methods is a minimum
only in probabilistic sense. Most of the methods considered in this thesis are detenninistic
methods.
Detenninistic methods are further divided into two classes, direct search methods
and indirect search methods. Direct search methods require only the value of objective
function (SSQERR value) to be calculated at a particular set of parameters. Indirect
search methods use gradient of objective function to minimize.
The optimization methods used in this thesis are as follows:
• Exhaustive Search.
• Neider Mead Simplex Algorithm, NeIder and Mead(1965)
• O'Neill's implementation of NeIder Mead Simplex, R. O'Neill (1971)
• Box's Complex method, M. J. Box (1972)
3
• Hooke and Jeeves' method, Hooke and Jeeves ( 1972)
• Powell's method, M. J. D. Powell (1964)
• BFGS method, S. S. Rao (1996)
• Genetic Algorithms, D. E. Goldberg (1989)
• Quadratic fit method.
• O'Neill's implementation of Neider Mead Simplex with exploratory search.
Exhaustive search is just used to see how the optimization domain looks like. This
method of optimization is not feasible since the time taken to estimate the thermal
properties is enormous.
NeIder Mead Simplex and O'Neill's Simplex are methods that use only the
evaluation of objective function value at points that are obtained using reflection,
expansion and contraction of a simplex (a n+1 dimensional figure in n dimensional
space). Box's Complex method is similar to NeIder Mead Simplex except that it uses a
2*n+ 1 dimensional figure called a "Complex". Powell's method is generates conjugate
directions and searches for the minimum along those directions. BFGS method is another
gradientbased method. Genetic algorithms are stochastic method, which use random
number generators to find the minimum in a domain. The Quadratic fit method is based
on line minimization and fitting a quadratic polynomial. Exploratory search is useful in
getting a better starting guess of parameters.
1.3. Experimental Apparatus
The experimental apparatus is contained within an enclosed trailer, constructed by
Austin (1998), that contains all the components required to perform the insitu test. The
main components in the trailer are water heating elements, water supply/purge tank,
pumps, valves, SCR power controller and two 7000 watts power generators. The
instrumentation and data acquisition equipment are flow meter, two thermistor probes,
watt transducer, thermocouple and a data logger. Since this trailer must be capable of
testing even in undeveloped areas, it contains generator and water tank to supply power
4
and water. The trailer is divided into subsystems namely water supply, power supply,
water heating, flow sensing/control equipment, pipe insulation, temperature measurement
and data acquisition. Power Supply System, Data Acquisition System and Temperature
measurement are described here. Figure 11 shows the schematic diagram of the trailer.
f4 Purge Pumps
Heating elements
~ It
~ ~
Circulating Pumps
t
Trailer Wall I
Water Supply
Tank
Into the borehole Out of the borehole
Figure 11. A schematic diagram 0 f trai ler
1.3.1. Power Supply System
Three heater elements are used to heat the circulating water. The heat input can be
varied in the range of 0 kW to 4.5 kW using SCR power controller which can vary the
power between 0 kW to 2 kW with other two heater elements of 1 kW and 1.5 kW on or
off. The total heat input is the energy input from the three heater elements and power
input to the circulating pumps. This energy input is measured with watt transducer.
1.3.2. Data Acquisition System
The output of watt transducer and digital display is measured by a Fluke Hydra
Data Logger. Digital display displays the DC voltage on the scale of 010 Volts for each
measurement. Three readings are recorded by the data logger is:
5
• Outlet temperature ofwater (leaving the trailer)
• Inlet temperature of water (returning the trailer)
• Flow rate
All of the measurements are in DC Volts. These measurements are converted
from DC volts to the actual units by taking the raw voltage and fitting a linear curve.
Inside and outside temperatures of the trailer are also recorded using thennocouples.
These data are stored in the data logger's memory. If the data logger is connected to a
computer, the data can be loaded on to the computer at any time by scanning the data
logger's memory.
1.3.3. Temperature measurement
The experimental apparatus uses three thennistor probes. Two probes measure the
temperature of water at the inlet ana the outlet of the borehole. Third probe is used to
measure the temperature of the pipe wall.
1.4. The Numerical Model
Yavuzturk et al (1999) developed the numerical model, which simulates the
borehole with the Utube, grout and soil. A sketch of the numerical domain is shown in
Figure 12. The geometry of Utube has been approximated by a "piesector". Only one
half of the numerical domain has been simulated due to symmetry of the domain. The
power is superimposed at each time step and is considered constant during a single time
step. Radial conduction heat transfer equations have been solved over the geometry and
temperatures have been calculated. The initial condition is that the temperature is
constant and equal to far field temperature everywhere. The boundary conditions are far
field temperature at the outer boundary and a constant heat flux at the inner boundary
during a small time step.
Input parameters to the numerical model are borehole geometry (borehole radius,
Utube length, pipe diameter, distance between the legs of Vtube, thickness of pipe, heat
capacity of pipe, heat capacity of grout and soil, number of hours for which the
6
experimental data has been collected, time step, far field temperature). All of thes
parameters are fixed due to the experimental setup. Other input parameters are initial
guesses of soil and grout conductivity. Experimental temperatures are calculated by
averaging the inlet and the outlet water temperature to the borehole. The subroutine
ConductSing/eSim gives the temperature values for the whole length of the test at a
particular Ksoi1 and Kgrout.
UTube
1.5. Objective function
Borehole
Figure] 2. Domain for numerical model
The objective function is the sum of the square of the errors between experimental
temperatures and numerically estimated temperatures, specifically:
N
SSQERR =I (Texperim<mal,n  Tnwnerical.l1P
n=1
(11)
Where, N is the total number of temperature measurements,
Texperimental.n is the experimental temperature at the nth time step,
Tnumerical,n is the average fluid temperature obtained by numerical model at the nth
time step.
7
A typical plot of experimentally obtained temperatures and numerically estimated
temperatures is shown in Figure 13.
95,,
90
Numerical
/
85
~
OJ
Ql e. 80
~ .a
~
~ 75
E
Ql
I
'"Experimental
70
65
30 40 50 60
Time (Hours)
10 20
60 t.,,,.ri
o
Figure 13. A typical plot of comparison between Experimental and Estimated
temperatures by numerical model
8
2. Optimization Domain
2.1. Exhaustive Search
Exhaustive search has been performed just to view how the objective function
surface looks like. The main aim oftms search is to fmd out whether there are any ripples
and local minima exist in the valley or the valley is smooth with no ups and downs. This
is helpful because the minimum found by different optimization methods will be the
global minimum if the method converges.
To apply exhaustive search to the optimization domain, we divide the domain of
ground thermal properties (namely, Ksoil and Kgrout) into a grid of small step sizes in both
the directions. At each grid node, a value of objective function is calculated from the
SSQERR function. In the current application of this method, we have applied this to our
SiteA1 insitu test. The domain is divided into 100 X 80 grid in the direction of ~oil and
Kgrout respectively. The range of Ksoil is 1.0 Btu/hrftF to 1.8 BtuIhrftF. The range of
Kgrout is 0.2 BtuJhrftF to 1.0 Btu/hrftF. These ranges are chosen based on some
previous experience about the properties. This leads to 8000 objective function
evaluations, which takes around 20 days to run on a Pentium II 233 MHz computer.
After computing all the objective function values, we find the minimum of all the
values, which is the true global minimum. This minimum is not quite accurate because of
the finite size of the grid. This method has a drawback that it requires a lot of objective
function evaluations and uses an enormous amount of computer time. Hence, other
optimization methods have been sought to expedite the process of estimating the ground
thermal properties.
Figure 21 shows that the objective function surface is a deep turning valley.
From the figure it can be observed that there are no local minimum in the valley and the
floor of the valley is quite flat near the minimum. It is not possible to see the absolute
minimum point with naked eyes.
9
Figure 21. Domain of objective function
10
3. Optimization Methods
3.1. Neider Mead Simplex
This algorithm, first proposed by Spendley et all (1962) and later developed by
NeIder and Mead (1965), is a multidimensional minimization method, that is, finds the
minimum of a function ofmore than one independent variable. This method requires only
objective function evaluations, not derivatives. This is useful since it is quite costly to
calculate derivatives for our problem. By costly, we mean that each objective function
evaluation takes a lot of time, approximately three minutes on a Pentium II 233 MHz
computer.
This algorithm requires a geometric figure of n+1 vertices in n dimensional space,
called a "Simplex". In two dimensions, this figure becomes a triangle. The process of
minimization requires different steps called "reflection", "expansion" and "contraction".
For our case of twodimensional minimization (Ksoil and KgrOUl), the code starts with a
starting simplex and objective function values at each vertex of the simplex. The simplex
has to be formed in such a way as to ensure that all the three points are not collinear
(actually a right angle triangle is constructed). Then the actual process of moving the
simplex towards the minimum starts. This process uses reflection, expansion and
contraction of the simplex. The amoeba code, implemented for this method, is available
in Numerical Recipes (Press et. ai, 1986).
3.1.1. Reflection
If Yh is the vertex at which the objective function value is largest, then a point Yr
is obtained by reflecting this vertex across the centroid of the remaining vertices (Yo) and
a lesser function value is expected at this vertex. Mathematically the reflection process is
given by:
(31)
11
Where a is the reflection coefficient and Yh is the vertex corresponding to the maximum
obj ective function value:
(32)
j=l,n+l
and Yo is the centroid of all vertices except j = h:
1 n+1
Yo = 'LYj
n j=1
j ..h
and a is defined as
(33)
(34)
Distance between Yr and Yo
a =''
Distance between Yh and Yo
It is not entirely possible to reach a minimum by only reflection process due to
certain difficulties. If the objective function value at reflected vertex turns out to be
greater than or equal to the value at reflecting vertex, then we have to stop since this does
not lead to a new simplex or we will have to reflect the second worst vertex. Using this,
further improvements towards the minimum are made. To absolutely find the minimum
we have to consider other operations like expansion and contraction.
Kgrout
Y.r.'o
.....
Ksoil
Figure 31. Reflection Process
3.1.2 Expansion
If the objective function value at the reflected vertex is minimum of all the
vertices in the simplex, it is expected that a further improvement is possible in the same
direction. Hence, expansion process is introduced. The expansion can be shown as:
12
Ye= Yo +y (Yr  yo)
Where y is the expansion coefficient defined as:
Distance between Ye and Yo
y=
Distance between Yr and Yo
(35)
(36)
The value of this coefficient is greater than 1. If f(Ye) < f(Yr), then Yr is replaced
by Ye and the reflection process is restarted. If f(Ye) > f(Yr) , which means expansion
process is not successful, Ye is rejected and Yh is replaced by Yr and reflection process is
started again.
.... .....
Kgrout
Ksoil
Figure 32. Expansion Process
3.1.3 Contraction
If the reflection produces a vertex with the highest objective function value, a
contraction is tried. First Yh is made equal to Yh or Yr (the reflected point) depending
upon the lower objective function value at Yh and Yr, or Yh = min (Yh, Yr). Now, a
process of contraction is tried using Yhand Yo (the centroid) according to:
(37)
13
(38)
Where Pis contraction ratio defined as
Distance between Y and YO
~= c
Distance between Yh and YO
This contraction ratio is always less than 1. If the objective function value at Yc is
!:,rreater than the value at Yh then a point with lower objective function value could not be
produced using reflection and contraction. Hence, a contraction about the vertex of
minimum objective function value is perfonned as:
Yi = (Vi + Y1ow)/2 for i = 1...ndim+1 (39)
KgrOUI
Y....r....o
.....
Kgrout
Ksoil
Figure 33. Contraction Process
Figure 34. Contraction towards minimum
The reflection process is started again using this new simplex.
The convergence criteria to stop is:
14
2·(f(Yh ) f(Y)ow»
(f(Y,,) + f(YloW» < £ (310)
Where, f is the function value at the co.rresponding point, € is the specified tolerance and
f(Y1ow)is defined as:
(311)
3.2. O'Neill's implementation of Neider Mead Simplex
R. O'Neill (1971) modified NeIder Mead Simplex algorithm and presented it in a
different form. The advanced features of this algorithm are:
• Instead of passing an already constructed simplex to the optimization subroutine, this
implementation requires that a starting vertex and step size in all the coordinate
directions be provided. Then, the subroutine itself constructs a feasible simplex in
such a manner that all the vertices of the simplex are not collinear. It computes the
objective function values also at the vertices of the simplex.
• The convergence criterion has been changed. In this implementation, standard
deviation is calculated using the centroid of the simplex. If the standard deviation is
less than € (a userspecified tolerance) then the process is stopped otherwise
continued until the convergence criterion is met.
1/2
n + I [f(Y.)  Centroid]2
Standard deviation ={L J } ~ E
1=I n +1
Where Centroid is calculated as:
1 n+1
Centroid = Lf(Y;)
n+ 1 ;=1
(312)
(313)
• To ensure that the algorithm did not stop at the local minimum, a small number equal
to 8*step size is added/subtracted to the current minimum in all the coordinate
directions and objective function value is calculated at those vertices. If the objective
15
function values obtained are greater than the current muumum then th global
minimum has presumably been reached. If not, then the algorithm takes tha vertex as
a starting vertex and restarts itself with the step sizes set to original step size tim s a
factor 8 which is decided by the user.
The code for this method is available as Algorithm AS 47 in. Applied Statistics
(1971).
3.3. Box's Complex method
Box's complex method of optimization is a constraint optimization method and
our problem domain is unbounded. B~t, fortunately we can restrict our attention to a
range of h<=Ksoil. Kgrout <=g, where h=O.1 BtuJhrftF and g=2.0 BtulhrftF (from
experience) and eliminate the rest of the problem domain.
This method starts with a geometric figure of 2·n+1 points (or vertices) called
"Complex" rather than the n+1 points in simplex method. A starting vertex is provided
which satisfies the constraints and other vertices (2·0.) are generated randomly using the
random number generators in the range of (0, 1) and the range provided as follows:
Yi = h + r • ( g  h )
Where, r is pseudorandall} number in the range (0,1),
g is higher limit on the variable,
h is lower limit on the variable.
(314)
At each vertex, the value of the objective function is calculated. The vertex
corresponding to the highest objective function value is reflected across the centroid of
the vertices other than the vertex with highest objective function value. The amount of
reflection can be varied using a parameter a, called reflection coefficient. The definition
of reflection coefficient a is given by equation (34) in which the number n, is replaced
with 2·n.
16
If the reflection process produces a vertex with objective function value lower
than the highest objective function value of the complex, the vertex is accepted and the
reflection process is started again using the vertex having the highest objective function
value in the new complex. Otherwise, the value of reflection coefficient, <x, is r duced by
half and the point is reflected again. This makes the reflected point come closer to the
centroid of the complex. This process is repeated until the value of <X becomes less than a
minimum called <Xlow. If this still does not produce an objective function value less than
the highest, the vertex corresponding to the second highest objective function valu is
reflected. This process is repeated until all the vertices are converged within some userspecified
tolerance. Figure 35 shows the complex with 2n+ I vertices and the point with
highest objective function value is reflected. The algorithm was available from Box M. J.
(1965) and we developed the code for it.
Highest Point
~ Centroid
Reflected Point
/ '''0
Figure 35. Complex in 2D space
3.4. Hooke and Jeeves' method
Hooke and Jeeves (1972) devised a method for staying on the crest of the valley,
while searching for an optimum. This pattern search method is based on the conjecture
that the coordinate directions are worth trying again and again to search for an optimum
along a ridge. This method starts with a small perturbation in each coordinate direction
from the starting point and the step size increases as the success is achieved in the
subsequent direction. A failure in all the directions would mean that the step size should
be decreased.
There are two types of moves that are the main feature ofthis algorithm:
17
• Exploratory moves.
• Pattern moves.
In an Exploratory move, a small step in one coordinate direction is taken from the
base point and a point corresponding to lower value of objective function is considered a
temporary vertex. Then, next exploratory move is taken in the next coordinate direction
(depending on the number of independent parameters) from this temporary vertex. This is
repeated until all the coordinate directions are searched like this. The final point is
designated as second base point.
In Pattern moves, a pattern direction is set with the two base points detennined in
the exploratory move. It is assumed that the same direction will result in a better point
and the step size is doubled. This point is designated the temporary base point. Now,
exploratory moves arc carried out about this point and the point that gives a favorable
objective function value is compared with the second base point. If this point turns out to
be better than the second base point, another pattern direction is obtained by joining
second base point and the current temporary vertex. This is repeated until no further base
points could be obtained and exploratory steps are started again.
Step size is decreased when no further favorable steps can be taken in all the
coordinate directions. This method has the advantage that it takes larger and larger steps
in the favorable direction. The pattern direction turns as the valley turns and stays on the
crest of the valley.
The algorithm was available from Rao S. S. (1996) and we developed the code for
it.
3.4.1. Algorithm
• The algorithm starts with an initial guess XI = (XI, X2, X3 ....Xn)T. This is tenned as
starting base point.
18
• A step size Axj is selected by the user in each independent unit coordinate direction Uj
(i=1, 2.... n), where n is the number of parameters and Uj has ith element as one.
• Evaluate objective function value at the starting point Xk, f (Xk), where k =1.
• Set Yk,O = Xk. Start the Exploratory step.
• Exploratory step: This step is performed to get a new temporary base point. Evaluate
the objective function value at Yk,iI+ LlXj· Uj. If this new point is better than Yk.i.1 then,
call Yk,j\+ Llxj· Uj the temporary base point Yki. IfYk,j_\+ Llxi· Uj is not better than Yk,il
then Yk,i\ Llxi • Uj is tried. If this is better than Yk.iI then, this is made temporary head
otherwise Yk,iI itself is made temporary base point. In summary, for minimization,
Yk,i =
Yk.il+ Axj· Ui if f(Yk,il+ ~Xj· Uj) < f(xl)
Yk,il iff(Yk,i_l) < min (f(Yk,i\+ Axj· Ui), f(Yk,jl Axi· Uj» (315)
Yk,il  LlXi • Ui if f(Yk,il ~Xi • Uj) < f(Yk,il)
This process is repeated with respect to the new base point found until all the
coordinate directions are perturbed and final base point is designated as Yk,n. If the new
base point remains same as Xk, then the step sizes Llxj are reduced (by factor of 2) and the
exploratory step is repeated again, otherwise, this new base point is
(316)
• With Xk and Xk+l, a pattern direction S is established,
and a new point is found as
Yk+l,O = Xk+1 + AS
19
(317)
(318)
Ais assumed to be one in our optimization algorithm. Alternately, it can be found using
onedimensional minimization method in the pattern direction S. In that case, A will be
replaced with A·, the optimum step length.
• Increment to the next step and set k = k+1. Get the objective function value at Yk,O,
and repeat the exploratory move about this base point. If at the end of exploratory
move, objective function value at the temporary base point is better than Xk, then this
point is assigned as new base point Xk+1=Yk,n and new pattern direction is established
again using this new base point and the old base point obtained in previous step. If the
function value at Yk,O is greater than the function value at the old base point, then
reduce the step size .:lXi, set k = k+1 and repeat the Exploratory step about this current
base point.
• The process is assumed to have converged whenever the maximum step length falls
below a user defined tolerance E. Thus, the process is terminated when
Max (~Xi) < E (319)
i = l,n
3.5. Powell's method
This method is a wellknown pattern search method. In multidimensions this
method consists of a sequence of line minimization. The main feature of these types of
algorithms is to compute the next search direction starting with the coordinate directions.
The method could be simpler if we just have to minimize in coordinate directions only.
When the minimization is performed only in coordinate directions, the function does
reach the minimum but it is extremely inefficient in the cases where a long turning valley
exists at an angle to the coordinate directions. This is because the method has to take
small steps in the coordinate directions to reach the minimum. Hence, for a long, narrow
and twisted valley, like in our case, this method is extremely inefficient.
20
Hence, an algorithm, which computes a better set of directions, is needed than just
using the coordinate directions. Some of these directions can reach very far in the narrow
valley near the minimum. The concept of "noninterfering" directions (also call d
conjugate directions) helps the method reaching the minimum faster than with just the set
ofcoordinate directions.
Two direction vectors, u and v are said to be conjugate with respect to function f
if they satisfy the following criteria:
(320)
Where the components of A matrix are second partial derivative of function f and A is
called Hessian matrix.
(321)
For twodimensional case, the matrix becomes:
(322)
p
According to Powell, a quadratic function will converge in one cycle of line
minimization if all the directions are mutually conjugate directions. Hence, Powell's
method tries to generate N (number of dimensions) mutually conjugate directions. Powell
states that for a quadratic function it will take N cycle of minimization to generate N
mutually conjugate directions. Hence, a total of N·(N+1) number of line minimization
will take us to the correct minimum. But this is only true when:
• The function is exactly quadratic.
• Each line minimization produces absolute minimum in that direction.
In reality, for any objective function like our problem this may not be true. Hence, it will
take more number of line minimization than N·(N+1).
21
More heuristic schemes were suggested to Powell's basic method of quadratic
convergence. The idea behind the modified Powell's method which is implemented h re
is to take PN  Po (last point after N line minimization  starting point) as the new search
direction for the next cycle of minimization and discard the direction in whi h there was
a maximum decrease in the objective function value. This is because this was the best
direction in the previous cycle of minimization and can playa major role in the new
direction (P  Po). Hence, by discarding this direction, the problem of linear dependence
of directions can be reduced and this helps in generating the conjugate directions that are
"linearly independent".
Both the algorithm and the code are available in Numerical Recipes (Press et. aI,
1986).
3.5.1. Methodology
This method starts with coordinate directions as the starting search directions for
the minimum starting with the first coordinate direction as shown in Figure 36. Then, it
generates a pattern direction by taking the direction that is obtained after N univariate
steps from the starting point. A univariate step is one in which the objective function is
minimized along a particular direction. In each direction, the objective function is
minimized using a one dimensional minimization routine. Our implementation of the
method uses Brent's method to minimize in a particular direction. After this cycle of
minimization, the new search direction is accepted or rejected based on the maximum
decrease criterion and one of the old search directions is replaced with this new direction.
Now, we have only N search directions. The new cycle of minimization is started again
with these N search directions. This procedure is repeated until the desired minimum is
reached.
The convergence criteria used is:
2 . (f(X 0)  feX N»
(f(X 0) + f(X »
22
< e (323)
Where f(Xo) is the objective function value at the beginning of a cycle of line
minimization and f(XN) is the objective function value at the end of a whole cycle of
minimization consisting of N line minimization. E is the tolerance value chosen by the
user.
Figure 36. Steps taken in Powell's method.
3.6. BroydenF1etcberGoldfarbSbanno (BFGS) method
3.6.1. Background
This method is one of the indirect search (gradient based) methods. Sometimes it
is also called QuasiNewton method. Different methods under this category differ only in
their updating of inverse of Hessian matrix. Instead of directly taking the inverse of
Hessian matrix, an iterative updating technique is used. The basic equation describing
Newton's method is:
23
(324)
Where, the Hessian matrix [Jj ] is composed of second partial derivatives of f and Xi is the
vector in previous iteration. In quasiNewton methods the idea is to approximate the
inverse of Hessian matrix [Jjr l by some other matrix [Bi]. Hence, the approximated
equation becomes:
(325)
Where, "A/ is the optimal stcp length in the direction
(326)
The matrix [Bj] is approximated as:
(327)
Where, Zl and Z2 are two independent ncomponent vectors and Cl, C2 are constants. n is
the number of dimensions. This is also called a rank 2 update of inverse of H ssian
matrix. Finally, the BroydenFletcherGoldfarbShanno (BFGS) formula is given by
djg[[Bd
d;gj
(328)
The algorithm is available from Rao S. S. (1996) and we developed the code for
it.
3.6.2. Algorithm
1. Algorithm starts with an initial guess Xi.
24
2. Initialize the inverse of Hessian matrix [Bd with a positive definite symmetric matrix
called identity matrix [1].
3. Calculate the gradient of objective function (Vf) at the point Xi using fOlWard finite
difference fonnula.
(329)
4. Compute the direction vector Si = [Bd Vf (Xi).
5. Minimize the function in the direction Sj and find the optimum length 1./. In our
implementation, minimization is done using Golden Section Search method.
6. Compute the new vector X i+\ =Xi + Aj· Sj.
7. Calculate the gradient ofobjective function (V£+I) at the point Xi+1
8. If IIVti + 1115 E, where E is a user defined tolerance, then we assume that the minimum
of the objective function is reached and stop the process. OthelWise we update the
inverse ofHessian matrix.
9. Update of inverse ofHessian is done by
d.d: [ gT[Bj]g J [D. ]g.d:
[Bi +1 ] =[Bj ] +_L_' 1+ 1 T 1  I T I I
dTg. d j gj d j gj
l I
Where
25
djg;r[BJ
dTgj
(330)
(331)
(332)
10. After updating the inverse of Hessian, we restart th whole process from st p 2 until
the condition in step 8 is satisfied.
3.7. Genetic Algorithms
Genetic algorithms (GA's) refer to a class of stochasticbased optimization/search
techniques that rely on the theories of natural selection (i.e. "survival of the fittest"). As
such, the purpose of a GA is to optimize a particular problem by simulating or imitating
the evolution of life in the natural world.
According to Goldberg (1989), GA's owe their robustness over other optimization
and search procedures to the following four attributes:
1. GA's work with a binary coding of the parameter set not the parameters
themselves.
2. GA's search from a population of points, not a single point.
3. GA's use objective function information, not derivatives or other auxiliary
knowledge.
4. GA's use probabilistic transition rules, not deterministic rules.
As an example of how GA's work, consider a problem where n parameters are to
be optimized. The user selects a population size of 5, for example, meaning that at each
generation, th.ere are 5 points at which the objective function will be evaluated. At the
start of the problem, all 5 parameter sets are generated randomly within a given range.
For each parameter set, a binary string is formed which is made up of the binary
equivalents of each individual parameter in the set. The result is that 5 binary strings (or
5 "parents") are fanned. Each bit making up the parent is called a "gene". Parents are
then selected for "reproduction" based on their function evaluation or "fitness". Parents
with higher fitness are allowed to reproduce more times than those with a lower fitness.
Reproduction is accomplished in GA's by "crossovers", which refer to a swapping of
binary digits. Crossovers by "genetic mutations" can also be simulated by specifying a
26
probability at which a mutation may occur. Crossovers resulting from simulated
biological "niching" can also occur. The results of binary digit swapping are the
"children" which form the next generation of parents.
Computer Algorithm:
Select number of parameters (nparam) to be optimized, the population size to work
with (npopsize), and the number of generations to evaluate (maxgen).
Initialize a random population ofindividuals at time =0
For i = 1 to nparam
For j =1 to npopsize
Parent (ij) =random(parameter)
Nextj
Next i
Evaluate the objective function for each set of parameters (i.e. determine fitness)
For k = 1 to maxgen
Test for termination criteria (no. of generations, etc.)
Select a subpopulation of parents for reproduction based on fitness
Code parameters ofP(rowj) into one binary string and store in P'(row), where row is
the selected parameter set
Recombine "genes" of selected parents to form "children"
Perturb the mated population stochastically
Evaluate new fitness
Nextk
End
3.8. Quadratic fit method
This method, developed for this proj ect, is based on some experience from the
exhaustive search method. From the exhaustive search we found that the SSQERR
27
surface in the domain of Ksoil and Kgrout is nearly quadratic and th re exists a st p vall y
which is turning. Figure 37 shows the turning valley in twodimensional domain. First
optimization in the Kgrout direction is perfonned using Brent's Quadratic Optimization
method at three guessed Ksoil values. After the three vertices are obtained along the floor
of the valley, a quadratic curve is fit through these three vertices and minimization is
performed along the quadratic by one dimensional line minimization routine called
"Golden Section Search". Brent's method was used in Kgrout direction since this method
is based on fitting a quadratic curve that reduces the number of objective function
evaluations in reaching the floor of the valley. This procedure leads the optimum point
very close to the true global minimum of the valley in some cases. The idea of fitting the
quadratic curve and optimizing along the quadratic comes from the fact that the valley of
SSQERR surface is nearly quadratic and turning. We developed the code for this method.
\
\
\
\
\
\
\
\
\
\
..............................................~..
\
\
'~,,,
..............................._ ~~ ..
' , ...
.........._ _ ~.>::::::e"'.;;::::::::: _.~ ~.~ ..
Kgrout
Ksoil
Figure 37. Quadratic Curve fitting in a valley
3.9. O'Neill's implementation of Neider Mead Simplex with exploratory search
To apply O'Neill's Simplex or any other search method for parameter estimation,
an initial guess of parameters is required to start with. If the initial guess of parameters is
28
close to the actual minimum then the search algorithms will find the minimum in less
number of objective function evaluations than the initial guess being far away. Hence, a
method that quickly gives an initial estimate of parameter values close to the minimum is
desired.
Kelvin's line source theory and the cylinder source method presented by Ingersoll
(1948, 1954) are the two wellknown analytical methods to solve radial earthcoupled
heat transfer problems using a series solution. Figure 38 shows the line source model.
According to Hart and Couvillion (1986), line source theory is applicable for pipes also
with modified interpr~tation of terms in equations (334) and (335). The line source
method is modified to include a film of constant thickness around the equivalent diameter
of the pipe and the resistance of this film is made equal to the grout resistance. The ratio
of equivalent diameter to one pipe diameter, which is based on empirical results, can be
changed based on our own experience. Our implementation uses the value of ratio to be
1.1 as opposed to J2 because the estimates of grout conductivity are better with this
ratio.
Line source Film Thickness
Figure 38. Line source model
Line source model is applicable for both constant heat flux per unit length of the
source and constant temperature of the pipe. Since the power input varies with time
during the experiment, the timevarying heat flux is decomposed into a series of constant
heat flux during a time step and superposition principle is applied to get the temperature
29
response at any time. The expression for transient temperature distribution around the
heat source/sink may be written as follows for our implementation:
(333)
Where Teo is the farfield temperature. 6Tsoil is the temperature difference due to
soil and 6Tgrout is the temperature difference due to grout. Temperature difference due to
soil is given by:
q [ r", 4r 2 (_I)N+1 (4r 2IN]
6Tsoil = Tsoil  T", =. In  0.9837 +2 • , 2 (334)
2 1t k soil r 2r", 2N N. r",
Where.
r",=4~ (335)
Where a is thermal diffusivity of ground. ksoil is thermal conductivity of soil. roX> is
the radius in which the rejected heat is completely absorbed in the ground. t is the time. N
is the number of tenns chosen by the user.
For equations (334) and (335) to be applicable for pipes. r", ~ 15 has to be
ro
satisfied and the line source heat flux q has to be same as the heat flux at the inner radius
of the pipe. This condition is achieved approximately after 45 minutes of experimental
data for our test cases. Generally. two or three terms are sufficient in the equation (334).
We have taken six terms in our implementation.
To estimate the temperature difference due to grout, the resistance of the film
thickness around the equivalent diameter ofthe pipe is calculated as:
1 ( RAD BH )
og~
R grout =""=2
·n . kgrout
30
(336)
Where RAD_BH is the radius of the borehole, DIS is the thickness of the grout
around the pipe and kgrout is the thermal conductivity of the grout material. Temperature
difference due to grout is then calculated as:
q
~Tgrout =Tgrout  Too =. RgroUI
L 100p
Where L\oop is the length of the borehole and q is the power imposed.
(337)
O'Neill's Simplex algorithm is applied after the starting guess is obtained by this
method which we hope is closer to the actual minimum. Step sizes of 0.2, smaller than
the O'Neill step sizes of 0.5, are chosen in both the directions. The vertex obtained from
exploratory search and the step sizes in both the directions are passed to the O'Neill
Simplex method and optimum parameter values are obtained. We developed the code for
exploratory search method.
3.10. Summary of all the methods
A summary of all the methods and their characteristics is shown in Table 31.
Most of the methods are deterministic and nongradient based.
31
Table 31. A summary ofcharacteristics of all the methods
Method
Deterministicl Gradient/Non
Pattern Search
Stochastic Gradient
Neider Mead Simplex Deterministic NonGradient No
O'Neill Simplex Deterministic NonGradient No
Box Complex Deterministic NonGradient No
Powell Deterministic NonGradient Yes
Hooke and Jeeves' Deterministic NonGradient Yes
BFGS Deterministic Gradient No
Genetic A1Qorithm Stochastic NonGradient No
Quadratic fit Deterministic NonGradient No
O'Neill's implementation with
Deterministic NonGradient No
exploratory search
32
4. Offline Estimation Results and Discussion
The results presented here are for 7 test sites. Table 41 shows the summary of
experimental tests used for parameter estimation results. Insitu experiments were
conducted using the trailer designed by Austin (1998) and temperature and power
consumption data were collected. The twodimensional numerical model that simulates
the borehole with Utube and grout uses this power data and the borehole parameters
including the radius of the borehole, pipe conductivity, soil volumetric specific heat,
grout volumetric specific heat, fluid viscosity, fluid density and fluid conductivity. This
model generates the finite difference grid around the borehole and calculates the average
fluid temperature, which is then compared with experimental data and sum of the square
ofthe difference is calculated. This difference is minimized using the search methods.
It has already been established that 50 hours of experimental data are required to
correctly estimate the soil and grout thermal conductivity with twoparameter estimation
procedure. Hence, most of the parameter estimation results here are presented for 50
hours of data unless otherwise specified. For more details on the duration of experimental
test, see Austin (1998).
Results obtained from Nelder Mead Simplex/O'Neill Simplex algorithms are
considered to be "true minimum" because they provide the lowest objective function
value (i.e. sum of the square of the error value) when compared to the other methods.
Other method results are considered converged to true minimum if the percentage
difference in Ksoil is within ± 2.5%.
Results from all the methods are characterized for goodness of fit by the
Estimated Standard Deviation (Scheaffer et aI.1995). The Estimated Standard Deviation
value is calculated as:
Estimated Standard Deviation =
33
N
L(Error)~
i=1
N2
(41)
This provides a unifonn comparison, regardless of the 1 ngth of the xperim ntal
test.
Table 41. Summary ofExperimental tests used for parameter estimation
~ Location Description CodeNll:re 1batia(1Ts)
fi2J97 StiIMEtEJ', 0<SiteA
#1  3 1/Z' Bcrehde, 244' deep, gwta::l wlh 3)%
SlteA.1JIB ~
sdids 8ertaite. FUteed ~ elEdric lire.
f:l2Pjg7 StiIMEter,O<SiteA
#2 3 112" Ixrehde, 2SZ deep, gwta::l wlh~
SiteA2_114 114
QwI: 85. PoMre:l ~ elEdric line. ,
1f:i97 StiIMEter,O<SiteA
#2 3 1/Z' bcretde, 2SZ deep, gwta::l wlh lherrra
SlteA2_170 170
QaJ85. PoMre:l ~ elaDic line.
4/21/97 StiIMEter. 0<SiteA
#5 3 1/Z' bcretde, 2SZ deep, gwta::l wlh
SlteA5_93 93
8erlsea. Po.\erej ~elEdricline.
11/19'00 StiIMEter, 0<SiteA
#6 4.88"1xretde, 2fJ8 deep,~1.
~_240 240
FbMred '"elEdric lire.
9/2£/97 Qli~ 3 1/Z' bcretde, 2ffJ 00ep, gaia::l wlh :u'/o sdkS
O1ickasha 99
Bentmite. FtMe'OO uy ~ It:! calYS.
317/97 'MlEitefud
4 ~4" tx:retde, 248 deep, gaia::l wlh 8a'lsE9 EZ
V61IheIfad f()
MD. FbMred uy ~ ICl C1UI;:'.
4.1. Results from NeIder Mead Simplex
The initial guess for the parameters (Ksoi \' Kgrout) is taken to be (1.5 BtuJhrftF,
0.575 Btu/hrftF) which is assumed to be close to minimum. A simplex, which is a right
angle triangle in this case, is constructed using 0.5 Btu/hrftF as the step size. The
triangle is shown in Figure 41.
(2.0,1.075)
KgrOUI
(0.0,0.0)
(1.5,0.575)
K..oil
(2.0,0.575)
Figure 41. Simplex in 2D domain
34
The value of the objective function is calculat d at th s thr v rtices and this
infonnation is passed to the subroutine "amoeba" since the subroutine requires an already
constructed simplex. This algorithm then does all the processes of reflection, expansion
and contraction of the simplex to give the optimum values of th two parameters, for
which estimated temperature profile fits the best to the experimental temperature profile.
The algorithm gives back the simplex whose first vertex contains the pararnet rs
corresponding to the minimum objective function value.
We do not have a local minimum in the valley as established by the exhaustive
search. The nature of NeIder Mead Simplex search algorithm is such that it searches the
objective function only at discrete points or computes the objective function value at
certain points. When the three vertices of the simplex have their objective function values
within the specified tolerance, 1% in our case, the algorithm stops. If the algorithm is
started with a different starting guess, it is likely that the simplex will roll down the
valley taking a different path. The chances are high that another simplex might satisfy the
same tolerance criteria and the minimum vertex, thus found, has a lower objective
function value than the previous one. Making tolerance smaller than 1%, which is used
currently, will also not cure the problem because we can still find a vertex, which will
have slightly lesser objective function value and will lie within the flnal small simplex.
This will increase the number of objective function evaluations only.
Hence, it is expected that a restart ofthe method sometimes give a lower objective
function value than that obtained from the first run but certainly not worse than the
previous one. Because the method is restarted with one vertex to be the same as the
minimum found and the rest are constructed again with lesser step sizes since we are
close to the minimum now. This gives us more confidence in the optimum parameter
values. Sometimes a change of 11.5% in the value of soil conductivity is obtained. Table
42 gives the values of estimated Ksoil , Kgrout, Estimated Standard Deviation in the
temperature and the number of objective function evaluations.
35
Table 42. A summary of results from NeIder Mead Simplex algorithm for 50 hours of
data
Estimated
Function
Site Ksoil Kgrout Standard
Evaluations
Deviation
SiteA1 98 1.367 0.560 0.25762 73
SiteA2 114 1.460 0.858 0.22395 64
SiteA2 170 1.427 11.157 0.17603 72
SiteA5 93 1.435 0.581 0.15723 72
SiteA6 240 1.319 0.783 0.09842 83
Chickasha 1.490 0.758 0.15566 76
Weatherford 1.638 1.187 0.12046 109
Figure 42 shows the movement of simplex towards the minimum in NeIder Mead
Simplex algorithm for SiteA5_93. The initial simplex is constructed with three
(1.5,0.575), (2.0,0.575) and (1.5,.1.075) noncollinear vertices forming a right angle
triangle. The highest objective function value is at (1.5,0.575), not apparent from the
fi.gure. This vertex is reflected and then contracted to find a favorable vertex (1.625,
0.825) in terms of objective function value. The new simplex now has vertices (1.625,
0.825), (2.0,0.575) and (1.5,1.075). The highest objective function value in the new
simplex is at vertex (1.625, 0.825) which is reflected against the centroid of remaining
vertices and then contracted to find a favorable vertex (1.688, 0.7). This process of
reflection and contraction continues until the minimum of the objective function is
reached. Only the objective function value is calculated at each vertex and compared with
the other vertices in every simplex. The minimum point could not be shown clearly due
to clustering of small simplexes near the minimum.
36
1.2r,
1.1
1.5, 1.075)
Starting Simplex
/'
U; 0.9
~
.c ::;
~ 08
:::l eC> :.::
0.7
0.6
rm~~:::::::===:==========~=_::_~""'a«2.00.0.575)
0.5
16 1.7 1.8 1.9 2 2.1
Ksoil BluJhrftF
1.4 1.5
0.4 >~~____.____.____.___,.__._l
1.3
Figure 42. Movement of Simplex towards the minimum for SiteA5_93
4.2. Results from O'Neill's implementation of Neider Mead Simplex
As mentioned earlier, this method is same as NeIder Mead Simplex except that
the convergence criterion is different. Also, this algorithm restarts with a smaller step size
if the small excursions at the converged point in the coordinate directions show a lower
objective function value than the current minimum. A summary of results from this
method is shown in Table 43.
In some cases, the algorithm does not restart since the small excursions don't
provide the objective function value less than the previously found minimum. Figure 43
shows the small excursions in the coordinate directions.
37
Minimum found
Kg.,UI
Ksoil
0.woUI
Figure 43. Small excursions in both the coordinate directions
Table 43. Summary of results for O'Neill's Simplex Algorithm for 50 hours ofdata
Estimated
Function
Site Ksoil Kgrout Standard
Evaluations
Deviation
SiteA1 98 1.368 0.562 0.25764 54
SiteA2 114 1.450 0.866 0.22356 59
SiteA2 170 1.432 1.158 0.17610 83
SiteA5 93 1.426 0.585 0.15725 47
SiteA6 240 1.317 0.782 0.09832 74
Chickasha 1.478 0.764 0.15523 58
I
Weatherford 1.634 1.193 0.12062 55
This algorithm was started with small simplex (with step size = 0.05) in both the
Ksoil and Kgrout directions to see the effects of different starting simplexes. The summary
of results is shown in Table 44. It is observed that the number of objective function
evaluations is, in some cases, less than the 0.5 step size case and more in some other
cases without any specific order.
When this algorithm runs, a log file is created which keeps track of the Simplex
vertices. When these vertices are plotted, it was observed that since the valley of the
objective function is flat, the simplex rolls down the valley and lot of objective function
38
values are evaluated in the flat part. Hence, to make the valley deeper, the urn of the
fourth power of the error values was calculated and minimized, or
ERROR =L (Texperiment aI, n  Tnumerical, n) ~
0=1
(42)
Table 45 shows a summary of the results thus obtained. It is observed that soil
and grout conductivity values converge to a different number other than their true
minimum values.
Table 44. Summary of results for O'Neill's Simplex algotithm for step size=O.05 in both
K soil and Kgrout directions
, Estimated
Site Ksoil Kgrout Standard
Function
Deviation
Evaluations
SiteA1 98 1.371 0.561 0.25762 73
SiteA2 114 1.459 0.863 0.22390 56
SiteA2 170 1.416 1.177 0.17654 69
SiteA5 93 1.443 0.577 0.15877 38
SiteA6 240 1.312 0.782 0.09844 66
Chickasha 1.496 0.751 0.15575 45
Weatherford 1.640 1.191 0.12064 49
39
Table 45. Summary of results from O'Neill's Simplex Algorithm for the sum of the
fourth power of the error for all the test sites
% Ksoil
Estimated
Site Ksoil
Difference
Kgrout Standard
Function
from True Evaluations
Minimum
Deviation
SiteA1 98 1.251 8.557 0.609 0.48034 84
SiteA2 114 I 1.335 7.903 0.978 0.37408 50
SiteA2 170 1.282 10.471 1.420 0.42543 64
SiteA5 93 1.385 2.859 0.604 0.19956 98
SiteA6 240 1.316 0.081 0.781 0.13809 43
Chickasha 1.547 4.639 0.713 0.34211 40
Weatherford 1.596 2.342 1.214 0.24472 46
Other exponents of the error, 0.8, 1.5, 1.8, were also tried and the results did not
converge to the true parameter values. For comparison purposes, Estimated Standard
Deviation was calculated in each case and then compared with true minimum. Figure 44
shows the exponent of the error with respect to the Estimated Standard Deviations for
four sites. It can be concluded that the Estimated Standard Deviation is minimum for the
case of exponent of 2. Other cases provide higher Estimated Standard Deviation than the
mll1lmum,
40
0.35.,,
0.3
~0.25
CD o
c: o~
.:; 0.2
~
E
Ol
"0
~ 0.15
rJ)
"0
!!
ell
E:g 0.1
w
0.05
• II • •
SiteA1_98
SiteA2_114
SiteA2_170
SiteA593
2 2.5 3 3.5 4 4.5
Exponent of Error
0.5 1.5
Ol...r,,.,..........,....,.,1
o
Figure 44. Estimated Standard Deviation vs. Exponent of error
4.3. Results from Box's Complex metbod
The algorithm of Box's method is similar to NeIder Mead Simplex except that it
has 2·ndim+1 vertices instead ofndim+l vertices. Box's method combines the expansion
and reflection steps into one by reflecting the vertex by an amount a. greater than 1. [t
then contracts if the objective function value comes out to be more than the maximum
objective function value of the complex. The parameters that can be varied are the value
of a., the value of a.low and the number of vertices which can be higher or lower than
2·ndim+1. Table 46 shows the comparison of results for SiteA1_98, for 50 hours of data
with different values of parameters.
It is observed that if the value of a.low is changed from 0.000001 to 0.001, the
number of function evaluations get reduced by about 6%, without effecting the results up
to 3 significant digits in the conductivity values and SSQERR value. This is because
when the contraction about the centroid is tried, 0.001 times the distance between the
41

reflected point and the centroid is considered suffici nlly close. Going very clos to the
centroid of the complex is not that significant. Hence, a value of alow = 0.001 is
sufficient. As the value of a is changed, for some values of a, parameter values did not
converge to the right numbers (especially where a values were less than I) for six
vertices in the complex. When the number of vertices in the complex were r duced from
6 to 5, the number of function evaluations decreased by 36% for other parameters b ing
constant.
Table 46. Comparison ofresults from Box's Complex method for SiteAI_98 with
different parameter values
~stimated
Function No of
Alpha Alphalow Ksoil Kgrout Standard
Evaluations Vertices
Deviation
0.6 0.001 1.560 0.518 0.36987 74 6
0.8 0.001 1.389 0.561 0.27417 85 5
0.8 0.001 1.463 0.529 0.34738 83 4
0.9 0.001 1.378 0.568 0.27265 128 6
1 0.000001 1.359 0.573 0.27341 167 6
1.1 0.001 1.384 0.567 0.27315 125 6
1.3 0.000001 1.374 0.568 0.27254 192 6
1.3 0.001 1.374 0.568 0.27254 182 6
1.3 0.001 1.381 0.570 0.27317 123 5
1.6 0.000001 1.381 0.567 0.27286 241 6
Table 47 shows the results for other sites for the best set of parameters from Table 46
(a=O.8, alow=O.OOI, generation=5).
42

Table 47. Results from Box's Complex method for all the sites with the best s t of
parameter values
% Ksoil
Estimated Converged
. Difference Function
Site Ksoil
from True
Kgrout Standard
Evaluations
to True
Minimum
Deviation Minimum
SiteA1 98 1.389 1.535 0.561 0.27417 85 Yes
SiteA2 114 1.886 30.067 0.621 0.73046 140 No
SiteA2 170 2.000 39.665 0.760 0.46847 74 No
SlteA5 93 1.463 2.574 0.569 0.16151 297 Yes
SiteA6 240 1.909 44.925 0.616 0.50817 353 No
Chickasha 1.789 21.009 0.598 0.29678 392 No
Weatherford 2.000 22.399 0.974 0.27314 158 No
This method does not always converge to the true parameter values for reflection
coefficient less than 1. Different reflection coefficients were tried and Table 48 shows
that the value of the parameters converged in all cases for reflection coefficient 1.3 which
is greater than 1. In fact, it can be concluded from Table 46 that the complex always
converges for the values of reflection coefficient greater than or equal to one. Hence, the
reflection coefficient with a value greater than 1 is required to guarantee that the complex
will converge to true parameter values, although the complex might oonverge for lower
value of reflection coefficient too.
Figure 45 shows the reflection process for reflection coefficient of 0.8. The
vertex corresponding to the worst objective function value in the complex ABCDE is B,
which is reflected to produce vertex B'. The new complex is ACDEB'. The size of the
complex ACDEB' is less than the size of the complex ABCDE because vertex B is
reflected with reflection coefficient of 0.8 which is less than 1. If this process is
continued further, the size of the complex will keep reducing. There is a possibility that
the fmal complex eventually becomes so small that the objective function values at all the
43
....
vertices fall within the userspecified tolerance without converging to the true minimum.
Hence, in some cases the complex may not converge to the true minimum for reflection
coefficient less than one.
2.5 ,,
2 A Centroid
1.5
0.5
...... ..:::'.":: ..
B' E
o
• B
1.6 1.8 2 2.2
Ksoll
1.2 1.4
0+,....,....,.........,1
1
Figure 45. Reflection of the worst point for reflection coefficient of 0.8
Table 48. Results from Box's Complex method for reflection coefficient of 1.3
% Ksoil
Estimated Converged
Difference Function
Site Ksoil
from True
Kgrout Standard
Evaluations
to True
Minimum
Deviation Minimum
SiteA1 98 1.381 0.924 0.570 0.27317 123 Yes
SiteA2 114 1.455 0.313 0.865 0.22158 116 Yes
SiteA2 170 1.437 0.346 1.132 0.17631 154 Yes
SiteA5 93 1.432 0.454 0.584 0.15720 168 Yes
SiteA6 240 1.324 0.525 0.783 0.10153 155 Yes
Chickasha 1.476 0.166 0.766 0.15530 170 Yes
Weatherford 1.635 0.087 1.194 0.12068 174 Yes
44
4.4. Results from Hooke and Jeeves method
Hooke and Jeeves method is well known for problems where the domain is a
steep, turning valley. The advantage of this method is that it takes larger steps in the
favorable direction called pattern moves and if the valley is turning, the direction also
keeps tuming due to the exploratory steps.
Since our problem domain is also a deep turning valley, we implemented this
method to see if we can reduce the time taken to minimize the objective function or the
number of objective function evaluations. Table 49 shows the results for all the sites
using this method. Figure 46 shows the steps taken by this method for SiteAl_98.
Results for all the sites converge to their true minimum values. But ifwe compare it with
O'Neill's Simplex method, this method reaches the minimum by evaluating more number
of objective function values than O'Neill's Simplex method for all the test sites.
The reason behind the higher number of objective function evaluations is the
small excursions, which do not result in a favorable objective function value. Another
source might be evaluating the objective function values when step size is reduced in all
the coordinate directions. All 2ondim+1 objective function evaluations of the previous
step are wasted because no favorable vertex is obtained during this step.
45

Table 49. Results from Hooke and Jeeves method for all the sit s
0/. Ksoil
Estimated Converged
Difference Function
Site Ksoil
from True
Kgrout Standard
Evaluations
to True
Minimum
Deviation Minimum
SiteA1 98 1.402 2.454 0.550 0.25978 113 Yes
SiteA2 114 1.453 0.215 0.866 0.22368 150 Yes
SiteA2 170 1.451 1.298 1.125 0.17746 126 Yes
SiteA5 93 1.457 2.204 0.576 0.15952 79 Yes
SiteA6 240 1.326 0.667 0.777 0.09910 154 Yes
Chickasha 1.489 0.748 0.755 0.15535 113 Yes
Weatherford 1.624 0.623 1.198 0.12116 149 Yes
0.58.,,
Slarting Point
0.57
0.58
CL=~~
0.55
'5eCl
~
End Point
0.54
053
1.44 1.46 1.48 1.5 1.52
Ksoil (BtulhrftF)
1.4 1.42
0.52 !.,........,..I
1.38
Figure 46. Steps taken by Hooke and Jeeves method for SiteAl_98
4.5. Results from Powell's method
Powell's method is supposed to he very efficient in minimizing the sum of the
square of the error. This method consists of line minimization in the coordinate directions
46
in the beginning and then generates directions depending upon the maximum decrease in
the objective function value along a particular direction. This process is repeated until the
minimum of the valley of the objective function is reached. Results from this method
have been summarized in Table 410.
It can be observed from Table 410 that the parameter values do not necessarily
reach the true minimum for all the test cases. For some cases the number of objective
function evaluations are quite large (e.g. SiteA2_170, 561 objective function evaluations)
and still the method does not reach the true minimwn. Hence, it is not guaranteed with
this method, that the minimum reached will be the true minimum after one pass of the
method, even after large number of objective function evaluations. The reason for not
converging to true minimum can be attributed to stopping prematurely at a point where
the convergence criteria is satisfied. Since the convergence criteria can not be improved,
the only possibility of obtaining a converged value is by restarting the method with
previous best point. This works for some cases and we obtained a converged value of
parameters close to the optimum. But the number of objective function evaluations is
unusually high hence a restart was not performed for all the cases. According to Powell,
it takes more than n*(n+1) line minimization for a nonquadratic objective function to
reach minimum, where n is the number of coordinate directions. That is why this method
takes very large number of objective function evaluations.
47
Table 410. Results from Powell's method for all the Sites
%Ksoit Estimated Converged
Site Difference Function Ksoil
from True
Kgrout' Standard
Evaluations
to True
Minimum
Deviation Minimum
SiteA1 98 1.334 2.489 0.574 0.26255 124 Yes
SiteA2 114 2.186 50.730 0.583 0.70103 159 No
SiteA2 170 1.576 10.039 0.961 0.24088 561 No
SiteAS 93 1.458 2.234 0.575 0.15944 123 Yes
SiteA6 240 1.553 17.911 0.684 0.27855 319 No
Chickasha 1.494 1.070 0.752 0.15563 462 Yes
Weatherford 4.506 175.781 0.608 0.93958 226 No
Figure 47 shows the steps taken by Powell's method for the test site SiteA2_170.
End Point
0.95
0.9
0.85
~ 0.8
f.c :3 0.75
~
'5e 0.7
OJ
~
0.65
0.6 Starting Point
0.55
0.5
1.4 1.6 1.8 2 2.2
Ksoil (Blu/hrftF)
2.4 2.6 2.8 3
Figure 47. Steps taken by Powell's method for SiteA2_170
48
4.6. Results from BFGS method
BFGS method is a wellknown gradient based method for minimization. The
inverse of Hessian matrix is updated with some approximation rather than evaluating the
Hessian matrix whose components are second partial derivative of objective function.
Since, it is already difficult to calculate the gradients due to uncertainty in step size, it
will even be more difficult to calculate the second partial derivatives. Hence, this method
is better than some other QuasiNewton methods that require accurate computation of
inverse of Hessian matrix. Once the direction is computed in which the objective function
has to be minimized, golden section search is used to minimize the objective function in
that particular direction. Table 411 summarizes the results for aU the sites for this
method.
Table 411. A summary of results from BFGS method for all the sites
% Ksoil
Estimated Close to
Site Ksoil Difference Kgrout Standard
Function
True
from True
Deviation
Evaluations*
Minimum
Minimum
SiteA1 98 1.466 7.147 0.531 0.29286 672 No
SiteA2 114 1.484 2.349 0.841 0.22771 419 Yes
SiteA2 170 1.434 0.153 1.149 0.17753 988 Yes
SiteA5 93 1.438 0.809 0.580 0.15949 412 Yes
SiteA6 240 1.443 9.560 0.736 0.16581 326 No
Chickasha 1.759 19.021 0.600 0.27369 498 No
Weatherford 2.551 56.147 0.797 0.52859 220 No
* Program was stopped and the convergence criteria was not satisfied
From Table 411, it is observed that this method does not always converge to the
true minimum even after a large number of objective function evaluations. For the cases
this method provide parameter values close to the minimum but the program did not stop
49
due to convergence and was stopped by us, it took a very large number of objective
function evaluations as compared to other methods like Neider Mead Simplex and
O'Neill Simplex.
The reason for not converging to true minimum can be attributed to inaccurate
computation of the gradient of the objective function due to numerical errors.
Susceptibility to divergence is caused if step lengths during line minimization are not
found accurately. Also the inverse of Hessian matrix looses its positive definiteness after
a lot of updates which is only approximate. Figure 48 shows the steps taken by this
method for SiteA6_240. The initial direction takes the starting point directly into the
valley but the future steps just hover around the minimum without satisfying the
convergence criteria.
0.8...,
0.75
0.7
Final P.o.int 
_0.65
u..
¢:.
~
~
§. 0.6
"5 eOl
:'::0.55
0.5
Starting Point
0.45
1.46 1.47 1.48 1.49 1.5 1.51
Ksoll (Btu/hrf\F)
1.44 1.45
0.4 +,r.,...,r.i
1.43
Figure 48. Steps taken by BFGS method for SiteA6_240
50

4.7. Results from Geneti.c AJgorithms
This stochastic based method is a global search method in a domain. Mutation and
Crossover are the two main features of these classes of algorithms. This search uses
random number generation. David L. Carroll (1996), Ph.D. at the University of Illinois,
UrbanaChampaign, developed the FORTRAN code for this algorithm. Table 412 shows
the results for all the sites using genetic algorithms for our problem.
From Table 412 it can be concluded that for maximum of 100 generations,
genetic algorithms does not always reach true minimum but it is close to the minimum.
So, it is expected that if we run the program for more than 100 generations, it might reach
the true minimum. This has been verified for some test cases. We chose 100 generations
due to the time constraint because there is no specific convergence criteria in these kinds
of algorithms that use random numbers. Hence, the minimum obtained is optimum only
in probabilistic sense. These methods are suitable only where the objective function
evaluations are not the performance criteria and obtaining the global optimum parameters
are more important.
Hence this method is not recommended for this type of problem. Deterministic
methods will be more appropriate.
51

Table 412. A summary of results from Genetic Algorithms for all the sites for maximum
of 100 generations
Of. Ksoil
Estimated Converged
Difference Function
Site Ksoil
from True
Kgrout Standard
Evaluations
to true
Minimum
Deviation minimum
SiteA1 98 1.363 0.388 0.563 0.26040 500 Yes
SiteA2 114 1.478 1.940 0.850 0.22655 500 Yes
SiteA2 170 1.480 3.355 1.084 0.18363 500 No
SiteA5 93 1.431 0.352 0.584 0.15983 500 Yes
SiteA6 240 1.411 7.117 0.739 0.14521 500 No
Chickasha 1.359 8.079 0.899 0.20967 500 No
Weatherford 1.749 7.019 1.095 0.15066 500 No
4.8. Results from Quadratic fit method
This method is based on line minimization and minimization along a quadratic
curve. The line minimization, which is perfonned by fixing KsoiJ. produces a point that
has minimum objective function value along the Kgrout direction. Threeline minimization
at three different Ksoil values produce three different points on the floor of the valley. A
quadratic polynomial is fit using those three points and the objective function is
minimized along the quadratic. Table 413 shows the results for all the sites using this
method. The main reason of not converging to true optimum parameter values is that the
fitting of quadratic may not pass through the optimum point for all the test cases and
valley is not truly quadratic. It can provide only a set of parameters that are close to the
optimum. Hence, the errors encountered range from less than 1% to as high as 13% in the
soil conductivity values. Number of objective function evaluations present a great
advantage of this method and this method might give a set of parameters close to
optimum if the valley turns out to be close to quadratic for any test case. A further
investigation into this method is required to fully realize the advantages of this method
52
and variations of this method. No recommendation on whether to use this method or not
can be made at this stage.
Table 413. Results of all the sites using Quadratic fit method
% Ksoil
Estimated Converged
Difference Function
Site Ksoil
from True
Kgrout Standard
Evaluations
to True
Minimum
Deviation Minimum
Site A1 98 1.416 3.539 0.548 0.27136 26 No
Site A2 114 1.444 0.395 0.841 0.33753 24 Yes
Site A2 170 1.489 4.005 I 1.074 0.18727 22 No
Site AS 93 1.556 9.097 0.550 0.20628 27 No
Site A6 240 1.500 13.895 0.720 0.21934 25 No
Chickasha 1.427 3.447 0.779 0.23929 23 No
Weatherford 1.496 8.450 1.357 0.17969 22 No
4.9. Results from O'Neill's implementation of Neider Mead Simplex method with
exploratory search
O'Neill's Simplex method provides good results in the sense that it converges for
all initial guesses and even for small starting simplexes. But if we can start with an initial
guess that is close to the minimum, the objective function evaluations can be reduced to
some extent. Hence, an exploratory search is perfonned which is based on the line source
method. This exploratory search gives a better starting point than just any arbitrary guess.
Since we have a better guess of parameters, we can start with a smaller simplex that we
hope will reduce the number of objective function evaluations and which in turn reduce
the time taken to minimize the objective function. Table 414 shows the results ofjust the
exploratory search for all the sites.
53
From Table 414, it can be seen that the values ofK oil and Kgrout are closer to the
optimum values than our starting guess of Ksoil =1.5 Btu/hrftF and Kgrout = 0.575 Btu/hrft
F. It is also observed that the average deviation from the optimum values is between
0.10.15 Btu/hrftF except for one case. Hence, we can take a step size of 0.1 in both the
parameter directions to construct the simplex.
Table 414. Results of exploratory search for all the sites
% Ksoil
Estimated Exploratory
Difference
Site Ksoil
from True
Kgrout Standard I Function
Minimum
Deviation Evaluations
SiteA1 98 1.127 17.642 0.651 0.67879 69
SlteA2 114 1.354 6.655 0.797 0.40956 59
SiteA2 170 1.352 5.613 0.941 0.27989 45
SiteA5 93 1.264 11.370 0.632 0.37550 47
SiteA6 240 2.182 65.674 0.625 1.12559 107
Chickasha 1.351 8.578 0.879 0.15427 67
Weatherford 2.057 25.862 0.918 0.61358 114
Table 415 provides the results for all the sites by O'Neill's Simplex method
using exploratory search results as the starting guesses. The average number of objective
function evaluations without exploratory search and with exploratory search is 62 and 49
respectively. The average is calculated based on 7 sites used to compare the results.
Hence, we can say that there is an improvement of about 20% in the number of objective
function evaluations.
54
Table 415. Results of O'Neill Simplex method with exploratory search for all th sit s
% Ksoil
Estimated Converged
Difference Function
Site Ksoil
from True
Kgrout Standard
Evaluations
to True
Minimum
Deviation Minimum
SiteA1 98 1.363 0.357 0.561 0.25731 60 Yes
SiteA2 114 1.469 1.343 0.853 0.22592 36 Yes
SiteA2 170 1.441 0.658 1.147 0.17574 52 Yes
SIteA5 93 1.430 0.256 0.585 0.15728 40 Yes
SiteA6 240 1.309 0.618 0.786 0.09875 49 Yes
Chickasha 1.485 0.466 0.759 0.15520 57 Yes
Weatherford 1.636 0.130 1.190 0.12083 49 Yes
4.10. Comparison of results
The basis of comparing the results from all the methods is the number of objective
function evaluations. This is because the number of objective function evaluations is
directly proportional to the time taken to obtain the optimum parameter values. All the
methods are compared against Neider Mead Simplex / O'Neill Simplex method, which
consistently provides the best solution. Another fonn of comparison is made between
using an exploratory search and not using an exploratory search for the same method.
Genetic Algorithms do not require a starting guess, hence exploratory search can not be
applied for this method.
Table 416 shows the average number of objective function evaluations for all the
methods used. For the cases in which the method does not converge to the true minimum
it is assumed that a restart of the method might allow it to converge. However, the
number of objective function evaluations will only add to the current average value. It
can also be concluded from Table 416 that the O'Neill Simplex method takes the
minimum average number of objective function evaluations. Although the Quadratic fit
method takes less objective function evaluations compared to O'Neill Simplex method, it
55
does not converge to true minimum in aU cases. Hence, the O'Neill Simplex method is
considered the best optimization method for our problem of steep, turning valley.
Table 4162
. Comparison of methods based on average number of objective function
evaluations
Method
Average Function
Evaluations
Neider Mead Simplex 79
O'Neill Simplex 62
Box Complex 152
Powell* > 292
Hooke and Jeeves. 127
BFGS* > 505
Genetic A1gorithm* > 500
Quadratic Fit* 25
NeIder Mead Simplex method and Hooke and Jeeves method are the two potential
methods that might be further considered. Hence, these methods are rerun with
exploratory search so that they can start with a better initial guess. The results are shown
in Table 417.
2 • Indicates that the method does not converge to the true minimum in all the test cases.
56
Table 417. Comparison of results with exploratory search
Method
Average Function
Evaluations
Neider Mead Simplex 68
O'Neill Simplex 49
Hooke and Jeeves 138
From Table 416 and Table 417 it can be concluded that O'Neill Simplex method
with exploratory search produces the best results and minimizes the objective function in
an average of49 objective function evaluations.
Although the Quadratic fit method does not converge to true optimum parameter
values, this method may have the potential to reduce the number of objective function
evaluations significantly if we use exploratory search to obtain an initial guess of
parameters. This option was realized towards the latter part of the project and could not
be considered in a greater detail. Hence, a further investigation will establish whether this
method has the potential to reduce the number of objective function evaluations or not.
57
5. Online Parameter Estimation
Offline parameter estimation has the disadvantage that the temperature and
power data is first collected in the field and then taken to the computer to estimate ground
thennal properties. This estimation takes some computer time before we can get ground
thennal property values. Another disadvantage of offline parameter estimation is that if
there is some error while collecting the data in the field, may it be experimental error, it
can not be detected.
Disadvantages of offline parameter estimation can be overcome by some other
means and one of them is online parameter estimation. Online parameter estimation, as
the name implies, is a parameter estimation method by which the values of ground
thermal properties can be estimated as the data is collected in the field. A graph of
parameters can be plotted to see where the values of parameters might be converging.
This is very advantageous because if the value of parameters are not changing much (if
they are within some tolerance), then the experiment can be stopped. Hence, this provides
a better measurement of the length ofthe experimental test.
Online parameter estimation is also called Sequential NonLinear Estimation in
some literature (Sorenson, 1980). The term sequential refers to updating the previous
parameter estimate using the newly collected data sequentially. Two approaches to
estimate the parameters are considered here:
• Nonlinear Recursive Estimator.
• Application of nonlinear optimization method.
Sorenson (1980) and Salsbury (1996) presented Nonlinear recursive algorithm
for general optimization problems. lang et al (1986) presented extended Kalman filter
approach to account for general nonlinear systems and standard optimization method
approach which they call Horizon approach. They compared both the approaches for
their problem of chemical process.
58
5.1. Nonlinear Recursive Estimator
Salsbury (1996) applied the nonlinear recursIve estimator equations for the
problem of fault detection in HVAC systems. The nonlinear recursive estimator quations
presented are in the following fonn:
Ck =f k(9k_.)Yk (51)
\II k =V'fk(9kJ ) (52) r r Pk = L'Vi'VT . (53)
.=1
~9k =9le  9k1 (54)
Where, ~ is the vector containing parameters at k1h step, \Ilk is the gradient of
objective function at kth step, P contains curvature information and Yk is the experimental
value of temperature.
The complete algorithm is stated as:
9 k =9k_1 Lot '€k
L k =Pkl'Vk(I + \jI~Pkl'Vk)
Pk =(I L.t",r)Pk••
(55)
(56)
(57)
Lk is called gain vector, P matrix is the inverse of Hessian matrix and \II is the
direction of steepest descent according to Salsbury.
The algorithm starts with a positive definite symmetric matrix Po of ndim x ndim
dimensions. It is taken as Identity matrix for the first step. 90 contains an initial guess of
parameters. \Ilk 1S calculated for the next step from equation (52) and Lk is computed
from equation (56). 9k is then updated using equation (55) and Pk is then updated for the
next step. This process is repeated until the parameters are converged.
59
5.2. Application of Nonlinear optimization method
The basic idea behind online parameter estimation method is that if we compute
the parameter values for M points, can this value be utilized for M+N points where N is
the number of additional points collected within the time period of estimation of
parameters for M points. This value will certainly be closer to the optimum value instead
of any initial guess for the next set of points. It has already been established from offline
parameter estimation that the number of objective function evaluations is reduced if we
have an initial guess of parameters which is closer to optimum values of those
parameters.
Another issue that is worth addressing is how much information or how many
hours of data are needed before updating the parameters. Should we update the
parameters for every collected data point or should parameters be updated after a set of
data points have been collected?
It mainly depends on the information that is coming from the experiment. If the
experimental data is changing slowly with time, probably we do not need to update the
parameters for each collected data point. But if the incoming information is changing
rapidly with time, we need to update the parameters for each collected point.
An extensive analysis of the data collected from the seven test sites has been done
and online parameter estimation has been performed using different strategies. NeIder
Mead Simplex algorithm is used to perform the online parameter estimation. Four
different alternatives were investigated.
• Construct a starting simplex for a fixed number of data points, increase the number of
points by one at each reflection, contraction and expansion, evaluate the average
SSQERR and find the best set of parameters at every step.
60
P""'""
• Construct a starting simplex for a fixed number ofdata points increase the number of
points by a fixed number at each reflection, contraction and expansion, evaluate the
average SSQERR and find the best set of parameters at every step.
• Start with a fixed number of data points. Get the best set of parameter values by
optimizing for a fixed time. Increase the number of points by a fixed number,
optimize for a fixed amount of time using the previous set of parameters as initial
guess to this set.
• Start with a fixed number of points. Get the optimum parameter values. Then take this
optimum value as initial guess for the next set of points collected during that time
period. Construct the initial simplex and minimize. In other words, minimize the
average SSQERR for a variable number of experimental data points.
5.3. Results from Salsbury's method
Salsbury's method of recursive parameter estimation was applied for our nonlinear
problem of parameter estimation. This approach yields parameter values that are
quite off from the true optimum parameter values obtained from offline parameter
estimation. The main reason for this method to fail to converge appear to be:
• This method requires an initial guess of parameter values to begin with. It is desirable
to choose a guess of parameters, which is closer to the true values. This depends upon
the experience of the problem at hand. The performance of the algorithm is strongly
influenced by the initial guess. Inability to start with a better initial guess might be a
reason for the divergence ofthe method.
• To compute the gradient of SSQERR function, a finite difference approach is used.
Since the problem is highly nonlinear, the uncertainty in step size in both the
parameter directions can cause an inaccurate first derivative of the objective function.
The objective function is linearized using Taylor series given in equation (58) and
61
this linear approximation is used to derive the recursive form of quations, which
introduces a significant amount of error for highly nonlinear problems like ours.
(58)
5.4. Results from application of optimization method
The four variations in application of offline optimization method applied to
online parameter estimation presented above are analyzed for the seven test sites. For the
case of updating the parameters after every data point, since one objective function
evaluation takes somewhere between one to three minutes depending on the number of
points, the program has to wait till the next data point is collected. In our case, since we
already had the data, we assumed that each data point is coming only after 2.5 minutes
and ran the simulation for 50 hours of data. Table 51 shows the results for all the test
sites for this case.
This process of updating the parameters at each collected point does not always
lead to the optimum value of parameters as seen by Table 51. The errors in the
conductivity values are as high as 33%, as compared to the values obtained from offline
NeIder Mead Simplex method. The reason might be that if the starting guess is far away
from the minimum point, then the chances are more that it will converge at some point
other than the optimum.
62
I•"
.,
, I
Table 51. Comparison of results for six testsites for updating the paramet r after every
collected data point
% Ksoil
Estimated
Site Ksolil Kgrout
Difference
Standard
f.ro.m. True Deviation
SiteA1 98 0.921 0.695 32.643 0.35612
SiteA2 114 1.289 0.977 11.712 0.19322
SiteA2 170 1.427 1.156 0.005 0.17539
  SiteA5 93 1.168 0.650 18.587 0.08935
SiteA6 240 1.279 0.795 3.003 0.06387
Chickasha 1.843 0.667 23.657 0.28906

Weatherford 1.310 1.393 20.028 0.13172 
Taking a fixed number of points for updating the parameters at each reflection,
contraction and expansion process also has the same problem as updating the parameters
at each collected points has. Table 52 shows the comparison of results for this case
where the number of collected points is equal to 15. Table 52 also shows the percent
difference of Ksoil and Kgroul from offline parameter estimation. It is observed that the
percentage error can be as high as 1012% in both soil conductivity and grout
conductivity after 50 hours of test. Hence, the method does not lead to the true optimum
point in some cases.
63
"
I'
l~
Table 52. Comparison of results for six testsites for updating the paramet r after every
15 collected data points
°/. Ksoil
Estimated
Site Ksoil
Difference
Kgrout ' Standard
from True
Minimum
Deviation
SiteA1 98 1.336 2.269 0.571 0.26544
 
SiteA2 114 1.305 10.662 0.962 0.19972

SiteA2 170 1.411 1.093 1.184 0.17671
SiteA5 93 1.329 7.363 0.612 0.12200
SiteA6 240 1.300 1.386 0.785 0.08352
Chickasha 1.484 0.415 0.759 0.15644
Weatherford 1.550 5.368 1.254 0.13788
The previous two methods don't necessarily lead to the minimum point. Hence, a
better method is needed, which can provide a better update of the values of the
parameters at each step. A method, which takes a fixed number of collected data points
and minimizes with those set of points till we receive the next set of data points will be
more suitable. In our case, we tried 15 experimental data points to be the fixed number of
points for which the minimization process continues. The time for the process is fixed
since we have fixed the number of data points and each data point is collected after 2.5
minutes. Hence, total time will be 15*2.5=37.5 minutes before the minimization starts for
the next set of points. This gives us a better estimate of parameters since at each step we
get a value of parameters which is closer to optimum for those set of points. A new
simplex is constructed after every 37.5 minutes and minimization is done during that time
starting with the best guess from the previous step. Table 53 gives a comparison of the
results obtained from this method at the end of 50 hours of test length. Figure 51 shows
how the parameters vary as the experimental data is collected for SiteA1_98.
64
l,.
Table 53. Comparison ofresults for updating parameters during a fix d amount of time
% Ksoil
Estimated
Site Ksoil
Difference
Kgrout Standard
from True
Minimum
Deviation
SiteA1 98 1.364 0.164 0.562 0.26117
SiteA2 114 1.450 0.728 0.868 0.22820
SiteA2 170 1.424 0.189 1.161 0.17764
SiteA5 93 1.431 0.259 0.584 0.16118
SiteA6 240 1.318 0.034 0.781 0.10373
Chickasha 1.476 0.980 0.769 0.15442
1.6 r,
1.4 .............,... .
1.2
~ ......
=
0.4
0.2
+Ksol
..... Kgroul
30 40 50 60
Hours
10 20
O.J.,r,.i
o
Figure 51. Variation of updated parameters for a fixed interval of optimization time
as the test for SiteAl_98 progresses
65
If we want to find out whether the experiment should be stopped or continu d to
collect more data, we have to look at the plot which shows the variation of the
parameters. If the parameters were not changing significantly then we would like to stop
the experiment. The method which updates parameters for a fixed interval of time will
not give us optimum value at each step since the optimization process may not have
reached the true optimum within that intervaL With this we can not find out whether we
should stop the experiment or not since we are not dealing with optimum parameter
values.
To avoid this and get an optimum value at each step, we start with 5 hours of data.
Construct the simplex, minimize using Neider Mead Simplex and get the optimum set of
parameters. This process takes some time and during that time we must have collected
some data from the experiment. We take those data points, add them in the previous set
and again minimize for that set of data points starting with the optimum value of the
parameters from the previous step. Table 54 shows the comparison of results obtained
from this method and offline Neider Mead Simplex method. The difference between the
Neider Mead Simplex values and values obtained from online parameter estimation are
within 1% for 50 hours of data. The variation of parameters is shown in Figure 52 for
SiteAl_98, Figure 53 for SiteA2_114, Figure 54 for SiteA2_170. Figure 55 for
SiteA5_93, Figure 56 for SiteA6_240, Figure 57 for Chickasha, Figure 58 for
Weatherford.
66
Table 54. Comparison of results obtained from minimization at each step and offline
NeIder Mead Simplex method
Ksoil e;. Ksoil
Site Ksoil
Obtained Difference
Kgrout
from NM from True
SimDlex Minimum
SiteA1 98 1.373 1.367 0.472 0.558
SiteA2 114 1.453 1.460 0.476 0.865
SiteA2 170 1.432 1.427 0.355 1.153
SiteA5 93 1.435 1.435 0.027 0.584
SiteA6 240 1.323 1.319 0.319 0.779
Chickasha 1.485 1.490 0.364 0.760
Weatherford 1.634 1.638 0.238 1.194
67
1.6 ~ ,
1.4
1.2
U, 1
iii'
~
~ 08
U
:> g
8 0.6 ~_ __ ..~..._. ......
0.4
0.2
30 40 50 60
Hours
10 20
Ol,__r___,_....,....~
o
Figure 52. Variation of estimated parameters for minimization at each data set for
SiteAl 98
1.6.,,
1.4
1.2
~N~
~ 08
~
"0 c: 8 0.6
0.4
• L 
•
0.2
30 40 50 60
Hours
10 20
O+~__r___,___r___,r_____l
o
Figure 53. Variation of estimated parameters for minimization at each data set for
SiteA2_114
68
~
2, ,
1.8
1.6
LL
"" 1.4
~
iii 1.2
&:
~U
::l
Uc8
08
+Ksoil
0.6
Kgroul
0.4
0.2
25 30 35 40 45 50 55
Hours
5 10 15 20
Ol_...,.,r..,..,.~l
o
Figure 54. Variation of estimated parameters for minimization at each data set for
SiteA2 170
1.6,,
........... II It ••• ' ••
1.4
1.2
u.
"~"
iii
.~> 0.8
tl ::l
U
25 0.6
U .. •••••••••
04
0.2
25 30 35 40 45 50 55
Hours
5 10 15 20
O+..,._~_._____.~,__..,._~~_l
o
Figure 55. Variation of estimated parameters for minimization at each data set for
SiteA5_93
69
1.4,., ............... .
........ ....fI'H ..... • • •• II' •• •• I' ••
1.2
.............  ..  .
0.4
0.2
25 30 35 40 45 50 55
Hours
5 10 15 20
O+.,.,.,.,....,...,...,...,,,i
o
Figure 56. Variation of estimated parameters for minimization at each data set for
SiteA6 240
2.5 ,,
2
lJ..
i<
~ 1.5
CD
~.;;
TI
:J
"0
Co
U
\.~ ~...... .., ,
.",,. __.  ··_••M _••
0.5
25 30 35 40 45 50 55
Hours
5 15 20
O+.,....rrrrr~
o
Figure 57. Variation of estimated parameters for minimization at each data set for
Chickasha data
70
1.8.,
30 40 50 60
Hours
10 20
0+·
o
02
1.4
0.4
1.6
u.. 1.2
~
.c
~ 1
[0
~
'S:
i$ 0.8
'" cco
() 06
Figure 58. Variation of estimated parameters for ~ation at each data set for
Weatherford data
5.5. Determination of the length of the experimental test
As the experiment is performed and parameters are updated, one is interested in
knowing how long the test has to be run to get the set of parameter values close to the
true minimum within some error band. The analysis of online parameter estimation
results obtained from optimization algorithm for variable point has been perfonned for
the seven test sites. The convergence criterion is if there is a variation of less than X% in
the parameter values for Y hours of collected data, then the experiment is stopped and the
length of the experiment obtained is sufficient for practical purposes. The variation is
calculated as:
X=
N
100 (59)
71
Several convergence c.riteria are applied to get an optimum length of the
experimental tests by changing the X and Y values. Some of the criteria did not converge
for all the test sites, hence are not discussed here. The criteria were applied only within
50 hours of experimental test. The possibility that a particular criterion will converge
after 50 hours of experimental data can not be avoided. The best set of values are X =
0.5% and Y = 5 hours, which are obtained by trial and error. Table 55 shows a summary
of the length of the experimental test for all the seven test sites.
Table 55. A summary of length of experiment for all the test sites
Ksoil %
Max Time
Site Ksoil from Difference Kgrout
Taken(Hrs)
O'Neill Ksoil
SiteA1 98 1.290 1.367 5.624 0.581 26.46
SiteA2 114 1.453 1.450 0.202 0.865 48.21
SiteA2 170 1.393 1.432 2.670 1.185 34.50
, SiteA5 93 1.334 1.426 6.467 0.612 28.38
SiteA6 240 1.256 1.317 4.672 0.800 10.08 !
Chickasha 1.420 1.478 3.890 0.794 14.79
Weatherford 1.513 1.634 7.403 1.270 21.88
From Table 55, it can be observed that in the worst case, soil conductivity is off
by 7.5% from the true minimum obtained from O'Neill Simplex method. The length of
the experimental test can be reduced as low as 10 hours in some cases with only 4.6% of
error in soil conductivity. The average length of the test turns out to be approximately 27
hours.
72
6. Conclusions and Recommendations
The main objective of this project, as stated earlier, is to investigate the best
parameter optimization algorithm which will reduce the time taken to estimate ground
thermal properties using the numerical model which best fits the experimental data.
Investigation of online parameter estimation method is also performed which will
eliminate the need for offline parameter estimation and help in determining the length of
the experimental test so that unnecessary collection of data can be avoided.
6.1. Offline parameter estimation
Several parameter optimization methods have been applied to see whether the
number of objective function evaluations can be reduced because each objective function
evaluation takes about 3 minutes on a Pentium II 233 MHz computer for 50 hours of
experimental data. Hence, a lot of computer time is required if number of function
evaluations are quite large.
Both gradient and nongradient based, deterministic and stochastic methods have
been applied for 50 hours of experimental data. Nongradient and deterministic methods
perform the best if comparison is made in terms of number of objective function
evaluations.
Both Neider Mead Simplex and O'Neill simplex algorithm perform in a similar
manner except that convergence criterion is different. O'Neill simplex method has
advantages of builtin construction of simplex and restart of the algorithm. There are
always slight differences in the optimum parameter values depending upon the starting
guess because of the basic nature of search being at discrete points and no absolute
convergence criteria. Hence differences of about 1% in the optimum parameter values are
ignored for comparison purposes and results from other optimization methods are
compared with O'Neill's Simplex method. These two implementations minimize the
SSQERR in the least number of function evaluations. Neider Mead Simplex takes about
79 and O'Neill's implementation requires about 62 objective function evaluations.
73
Box's Complex method is also a "simplex" method with larg f number of
vertices, 2*n+l, as opposed to n+] in Neider Mead Simplex. This method has been
investigated for various values of reflection coefficient greater than 1 as well as less than
]. For some cases of reflection coefficient, the parameters do not converge to true
minimum. Results obtained for an appropriate reflection coefficient show that the average
number of objective function evaluations is 152, higher than NeIder Mead simplex. This
can be attributed to multiple contractions of the reflected point in the case when reflection
is not successful. Contraction is tried until a favorable point is obtained which involves
waste of objective function evaluations.
Hooke and Jeeves' method performs next best to NeIder Mead Simplex in terms
of number of objective function evaluations. This method has the advantage of increasing
the step length and turning of pattern direction as the favorable points are obtained.
Higher number of function evaluations is due to an unsuccessful exploratory step, which
causes at least 5 objective function evaluations per unsuccessful exploratory step. The
average number of objective function evaluations for this case is 127, a bit higher than
NeIder Mead simplex but less than all the other methods.
Powell's method is considered very efficient in minimizing sum of the square of
the error for quadratic functions. But due to nonquadratic nature of the objective
function surface, this method requires very large number of objective function
evaluations. Each line minimization requires on an average of 15 objective function
evaluations, which means 30 function evaluations for each cycle of minimization for two
dimensions. According to Powell, more than n*(n+ 1), 6 in our case, line minimization are
required for nonquadratic functions. Hence, this method becomes very inefficient for our
purposes. The average number of objective function evaluations for this case is 292,
considerably higher than NeIder Mead Simplex.
BFGS method, which is based on calculating the gradient of objective function, is
tried because if successful, the gradient method can minimize the objective function
quickly. For our problem, this method could not converge for several test sites. This can
be attributed to uncertainties in computing the gradient of objective function. This
74
method is susceptible to divergence if step lengths during line minimization is not found
accurately. The inverse of Hessian matrix looses its positive definiteness after a lot of
updates. For the cases where this method converged, it took about 505 objective function
evaluations.
Genetic algorithms are stochastic based optimization methods, which requIre
random number generation, manipulating the bits of a parent by mutation and crossovers
and evaluation of objective function at those points. The optimum values found from this
method are probabilistic. Even after 100 generations, 500 objective function evaluations
the minimum is not reached for some test sites. But it is observed that the optimum for
those sites can be reached if we run the simulation for more number of generations.
Overall this method presents a hopeless case for our problem.
The Quadratic fit method, which is based on experiences from exhaustive search,
gives parameters values close to optimum for some cases but it is off as high as 14% for
the other test cases. The only advantage is number of objective function evaluations,
which are not useful at the cost of accuracy.
Exploratory search is applied to obtain a better set of initial guess of parameters
because several methods require a good initial guess. The line source method is
implemented and a good starting guess is obtained for both the parameters. Then
O'Neill's Simplex method is applied to get the optimum value. The number of objective
function evaluations reduce to about 49 as opposed to 62 without exploratory search.
This saves about (6249)*3.0=39 minutes of computer time which is about 21% of actual
time. Hence, the O'Neill's implementation of Neider Mead Simplex method with
exploratory search is the best method for our problem ofturning valley.
6.2. Online parameter estimation
Online parameter estimation was performed to get the parameter values, as the
experiment commences which saves time to estimate the parameters after the test.
Optimum length of the test can be obtained by applying the convergence criteria on the
estimated parameters with some error. Two conventional estimation methods are applied
75
for our case. Salsbury's method diverges due to its approximation of nonlinear objective
function as linear. Another method, which is actually application of an offline
optimization method, is successful in predicting the optimum parameters and is useful in
detennining the length of the test. NeIder Mead Simplex optimization algorithm is
applied to get the optimum parameters for variable number of data points. The
convergence criterion for online parameter estimation, obtained by trial and error, is 0.5%
variance in conductivity values applied on 5 hours ofestimated parameter values.
The convergence criteria when applied to all the test cases, provides that the
required length of the test varies from 10 hours to 48 hours depending upon the specific
test case. But the average length is 27 hours. The maximum error in the predicted soil
conductivity values for all the test cases is 7.5%.
76
6.3. Recommendations
• Apply more optimization algorithms that are gradient based and are more robust than
BFGS method to investigate how gradientbased methods fair for this problem.
• A more robust online parameter estimation technique should be applied rather than
the offline optimization method applied for online parameter estimation, which
updates the parameters in a manner that a true optimum can be obtained successfully
at a particular data set. This will help in determining the length of the experimental
test with further confidence.
• Numerical model should be modified in such a way so that the time taken for each
objective function evaluation is reduced.
• More test sites should be investigated to gain confidence in the average number of
objective function evaluations for offline parameter estimation case.
• A plot of parameters can be obtained which will help in observing where the
parameter values are converging and when to stop the experimental test.
• Quadratic search method has to be refined to get closer estimates of parameter values
because this method has the potential to reduce number of objective function
evaluations.
• Implement combined data acquisition and online parameter estimation on site.
77
References
Austin Trey 1998, Development of an In Situ System for Measuring Ground Thennal
Properties, Masters Thesis, Oklahoma State University, Stillwater Oklahoma. (Also
available at http://www.mae.okstate.edu/Faculty/spitler/Austin_thesis.pdf.)
Austin W. A., Yavuzturk C. and Spitler J. D. 2000, Development of an In Situ System for
Measuring Ground Thermal Properties, Draft submitted to ASHRAE TraTlsactions.
Box M. J. 1965, A New Method of Constrained Optimization and a Comparison with
Other Methods, Computer Journal, Vol. 8, No.1, pp. 4252.
Broyden G. G. 1970, The Convergence of a Class of DoubleRank Minimization
Algorithms, Parts I and IT, Journal of the Institute of Mathematics and Its Applications,
Vol. 6, pp. 7690,222231
Carroll D. L. 1996, Chemical Laser Modeling with Genetic Algorithms, AIAA Journal,
Vol. 34, 2, pp. 338346.
Carroll D. L. 1999, FORTRAN Genetic Algorithm (GA) Driver, Online. Available:
http://www.staff.uiuc.edu/can·olJ/ga.htmlll March 1999.
Fletcher R. 1970, A New Approach to Variable Metric Algorithms, Computer Journal,
Vol. 13, pp. 317322.
Goldfarb D. 1970, A Family of Variable Metric Methods Derived by Variational Means,
Mathematics ofComputation, Vol. 24, pp. 2326.
Goldberg, D.E. 1989, Genetic Algorithms in Search, Optimization, and Machine
Learning, AddisonWesley Publishing Company, Inc., pp. 412.
78
Hart D. P. and Couvillion R. C, Earthcoupled heat transfer, National Water Well
Association, 1986.
Hooke R. and T. A. Jeeves 1961, Direct Search Solution of Numerical and Statistical
Problems, Journal ofthe ACM, Vol. 8, No.2, pp. 212229.
lang S. S., Joseph B. and Mukai H. 1986, Comparison of Two Approaches to OnLine
and State Estimation of Nonlinear Systems, Ind. Eng. Chern. Process Des. Dev. Vol. 25,
pp. 809814.
Neider 1. A. and Mead R. 1965, A Simplex Method for Function Minimization,
Computer Journal, Vol. 7, pp. 308.
O'Neill R. 1971, Function Minimization using a Simplex Algorithm, J. R. Statist. Soc. C
Vol. 20, No.3, pp. 33845.
Powell M. 1. D. 1964, An Efficient Method for Finding the Minimum of a Function of
Several Variables Without Calculating Derivatives, Computer Journal, Vol. 7, No.4, p.
303307.
Rao, S. S. 1996, Engineering Optimization, Theory and Practice, Third Edition, John
Wiley & Sons, Inc.
Salsbury T. 1., Fault Detection and Diagnosis in HVAC Systems USlOg Analytical
Models, Ph.D. thesis 1996, Loughborough University, UK.
Scheaffer R. L. and McClave J. T. 1995, Probability and Statistics for Engineers, 4th
Edition, Duxbury Press.
Shanno, D. F. 1970, Conditioning of QuasiNewton Methods for Function Minimization,
Mathematics ofComputation, Vol. 24, pp. 647656.
79
Spendley W., Hext G. R. and Himsworth F. R 1962, Sequential Application of Simplex
Designs in Optimization and Evolutionary Operation, Technometrics, Vol. 4, pp. 441.
Yavuzturk c., Spitler J. D. and Rees S.l. 1999, A Transient TwoDimensional Finite
Volume Model for the Simulation of Vertical VTube Ground Heat Exchangers, to be
published in ASHRAE Transactions.
80
VITA
Nagendra Kumar Jain
Candidate for the Degree of
Master of Science
Thesis: PARAMETER ESTI1v1ATION OF GROUND THERMAL PROPERTIES
Major Field: Mechanical Engineering
Biographical:
Education: Graduated from Indian Institute of Technology, Bombay India ill
May 1997 with a Bachelor of Technology Degree. Completed the
requirements for Master of Science degree with a major in Mechanical
Engineering from Oklahoma State University, Stillwater OK in July 1999.
Experience:
Worked for a manufacturing plant on the project "Inventory Analysis".
Employed as Teaching Assistant and Research Assistant at Oklahoma Stale
University from January 1998 to current.
Professional Membership:
American Society of Heating, Refrigeration and AirConditioning
Engineers (ASHRAE).