Velten K. Mathematical Modeling and Simulation: Introduction for Scientists and Engineers
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4.11 Other Mechanistic Modeling Approaches 315
system parameters (the thermal conductivity K). Now suppose that the cube is a
black box to us in the sense that nothing is known about its internal structure except
for the fact that it is made up of two materials A and B. Suppose that all we can do
to explore its internal structure is to perform experiments of the following kind:
•
Fixed temperatures are imposed at some of the cube surfaces.
•
The ‘‘answer’’ (output) of the system is determined in terms of
the temperatures that are observed at the remaining cube
surfaces. In addition to this, we may also have data from a few
temperature sensors inside the cube. Let us denote the resulting
dataset as D.
In this situation, we may use the heat equation in a different way. Since we know
that the inside of the cube is made up of the two materials A and B, the thermal
conductivity will be of the form
K(x) =
K
A
if A is the material at position x
K
B
otherwise
(4.165)
Thus, the material structure inside the cube will be known if we find a way to
compute K(x)(and,ofcourse,ifK
A
= K
B
). In principle, this can be done by solving
the following
Problem 7:
Determine K(x) according to Equation (4.165) such that an ‘‘optimal’’ approx-
imation of the data D is obtained if the heat equation (4.164) is solved using
K(x).
This type of problem is called an inverse problem since we are now looking for
parameters of the mathematical model (the thermal conductivity) based on a given
output of the system (the temperature measurements in the dataset D). Inverse
problems of this kind are abundant in all fields of science and technology. They
are used, for example, in geophysics, medical imaging, remote sensing, ocean
acoustic tomography, nondestructive testing, and astronomy [32]. Particularly, the
medical imaging applications have gained great public attention. These techniques
use e.g. X rays to produce a dataset (corresponding to D above), which is then
used to determine parameters of appropriate mathematical models (corresponding
to the heat equation and its parameter K(x)), and from this three-dimensional
reconstructions of internal structures of human bodies are obtained. The Radon
integral transform is one of the mathematical techniques that is used to solve this
reconstruction problem [236].
Inverse problems include parameter estimation problems such as the regression
problems that were treated in Chapter 2. Of course, it must be made precise in
316 4 Mechanistic Models II: PDEs
the above problem formulation what is meant by an ‘‘optimal’’ approximation
before the problem can be solved. In the regression problems treated in Chapter 2,
the minimization of the residual sum of squares was used to express an optimal
approximation of the data.
Note that it is often much more difficult to solve an inverse problem compared to
the direct solution of a mathematical model. Inverse problems often are ill posed,
that is, they may violate Hadamard’s well-posedness criteria that were discussed in
Section 4.3.2.1. Based on the above example, it is easy to understand why inverse
problems may suffer from ambiguities, since it obviously may be possible to explain
a particular dataset D based on two or more different distributions of the thermal
conductivity within the cube. Using a priori knowledge – such as Equation (4.165)
in the above example, which expresses the fact that the inside of the cube consists
of the materials A and B with thermal conductivities K
A
and K
B
, respectively – can
help to turn ill-posed inverse problems into well-posed problems [32, 236].
317
A
CAELinux and the Book Software
If you want to work with the examples and programs described in this book on your
own computer, you need OpenOffice/Calc, Maxima, R, Salome-Meca, Code-Saturne
and a zip file
MMS.zip which contains the programs and data files used in the
book. All this is open-source software that can be used by anybody free of charge.
Note, however, that the author makes no warranty, implied or expressed, that the
programs and procedures described in this book will work safely on your system,
and also no warranty is made for the recency, correctness, completeness or quality
of the software-related information provided in this book.
It is recommended to use Calc, Maxima, R, Salome-Meca and Code-Saturne based
on CAELinux,aLinux operating system which can be obtained as a Live-DVD
under www.caelinux.com. Based on the Live-DVD, CAELinux can be used without
the necessity to install anything on the hard drives of your computer, and without
affecting the original operating system installed on your computer. Once you have
obtained the CAELinux Live-DVD from www.caelinux.com, reboot your computer
using the Live-DVD. This will turn most machines (including most computers
running under Microsoft Windows) into CAELinux workstations.
Note In CAELinux, the main software instruments used in this book (Calc, R,
Maxima, Salome-Meca, Code-Saturne) can be used without further installations.
All contributed R packages used in the book – which are not a part of R’s
standard distribution – are also included.
If CAELinux should not work on your computer, you can of course install
Calc, R, Maxima, Salome-Meca and Code-Saturne separately on your computer,
using the installation procedures described at the appropriate internet sites (Calc:
http://www.openoffice.org, R: www.r-project.org, Maxima: maxima.sourceforge.net,
Salome-Meca: www.caelinux.com, Code-Saturne: www.code-saturne.org).
MMS.zip is available at the author’s homepage under www.fbg.fh-wiesbaden.de/
velten. It contains the programs and data files listed in Table A.1. The programs are
either R programs (extension ‘‘
.r’’) or Maxima programs (extension ‘‘.mac’’). See
the Appendices B and C for details on how you run the programs in R or Maxima.
In these Appendices it will be assumed that you have unzipped
MMS.zip on some
318 A CAELinux and the Book Software
Table A.1 Files in MMS.zip.
File Location Description
Euler.mac MechODE/MAC Euler method example (page 177).
Farm.mac Solution of the
wheat/barley problem
(page 31).
FeverDat.mac MechODE/MAC Body temperature example (page 120).
FeverExp.mac MechODE/MAC Body temperature example (page 121).
FeverODE.mac MechODE/MAC Body temperature example (page 180).
FeverSolve.mac MechODE/MAC Body temperature example (page 121).
Label.mac Principles Solution of tank labeling problem (page
29).
Mix.mac Principles Solution of a mixture problem (page 24).
Mix1.mac Principles Solution of a mixture problem (page 26).
ODEEx1.mac- MechODE/MAC ODE closed form solution examples
ODEEx16.mac (pages 159-173).
RoomDat.mac MechODE/MAC Alarm clock example (page 122).
RoomExp.mac MechODE/MAC Alarm clock example (page 126).
RoomODE.mac MechODE/MAC Alarm clock example (page 183).
RoomODED.mac MechODE/MAC Alarm clock example (page 141).
Stiff.mac MechODE/MAC Stiff ODE example (page 179).
Tin.mac Principles Solution of the tin problem (page 18).
VolPhase.mac MechODE/MAC Phase plot example (page 210).
Anova.r PhenMod/Stat Analysis of variance. (page 64).
Conway.r PhenMod Example cellular automaton. (page 311).
CRD.r PhenMod/DOE Completely randomized experimental
design (page 102).
FacBlock.r PhenMod/DOE Randomized factorial block design (page
107).
Fermentation.r MechODE/R Wine fermentation model (pages 203, 217).
HeatClos.r MechPDE Solution of heat equation using FD
(page 261).
LinRegEx1.r PhenMod/LinReg Linear regression using spring.csv
(page 67).
LinRegEx2.r PhenMod/LinReg Multiple regression using volz.csv
(page 76).
LinRegEx3.r PhenMod/LinReg Cross validation using volz.csv (page 79).
LinRegEx4.r PhenMod/LinReg Linear regression using gag.csv (page 73).
LinRegEx5.r PhenMod/LinReg Polynomial regression using gag.csv
(page 74).
LSD.r PhenMod/DOE Latin square design (page 105).
Mualem.r MechPDE Water retention curves (page 293).
NonRegEx1.r PhenMod/NonReg Nonlinear regression using klein.csv
(page
82).
NonRegEx2.r PhenMod/NonReg Nonlinear regression using stormer.csv
(page 84).
A CAELinux and the Book Software 319
Table A.1
(continued)
NNEx1.r PhenMod/NN Neural network using klein.csv (page 91).
NNEx2.r PhenMod/NN Neural network using rock.csv (page 98).
ODEEx1.r MechODE/R Body temperature ODE model (page 184).
ODEEx2.r MechODE/R Alarm clock ODE model (page 191).
ODEEx3.r MechODE/R Pharmacokinetic model (page 224).
ODEFitEx1.r MechODE/R Parameter estimation in the alarm clock
ODE model (page 195).
ODEFitEx2.r MechODE/R Parameter estimation in the alarm clock
ODE model (page 198).
ODEFitEx3.r MechODE/R Parameter estimation in the alarm clock
ODE model (page 199).
Plant1.r MechODE/R Exponential growth model (page 226).
Plant2.r MechODE/R Logistic growth model (page 226).
Plant3.r MechODE/R Asparagus growth model (page 228).
RCBD.r PhenMod/DOE Randomized complete block design
(page 103).
RNumbers.r PhenMod/Stat Probability density plots (page 55).
TTest.r PhenMod/Stat t-Test (page 62).
Volterra.r MechODE/R Predator-prey model (page 206).
VolterraND.r MechODE/R Predator-prey model (page 209).
asparagus.csv MechODE/R Asparagus growth data (page 228).
crop.csv Phenmod/Stat Crop yield data (page 61).
fermentation.csv MechODE/R wine fermentation data (page 218).
fever.csv MechODE/R Body temperature data (page 120).
fungicide.csv PhenMod/Stat Fungicide data (page 64).
gag.csv PhenMod/LinReg GAG urine concentration data (page 72).
klein.csv PhenMod/LinReg US investment data (pages 81,91).
rock.csv PhenMod/LinReg Rock permeability data (page 97).
room.csv MechODE/R Room temperature data (page 122).
spring.csv PhenMod/LinReg Spring elongation data (pages 32, 67).
stormer.csv PhenMod/LinReg Stormer viscometer data (page 84).
volz.csv PhenMod/LinReg Rose wilting data (page 75).
flowstep.med MechPDE/Backward Backward step problem (page 299).
result.med MechPDE/Backward Backward step problem (page 301).
CRD.ods PhenMod/DOE Completely randomized design example
(page 101).
external universal serial bus (USB) device, and that you are using the programs
and data files on that external USB device together with CAELinux as described
above (note that everything has been tested by the author using the CAELinux
2008 Live-DVD). Slight modifications of the procedures described there may be
necessary if you have installed Calc, R, Maxima, Salome-Meca and Code-Saturne
on the hard disk of your computer (based on CAELinux or a different operating
system).
Many of the programs can be used as software templates for the solution of
problems that you may have in one of the various problem classes treated in this
book. For example, to estimate parameters in an ordinary differential equation
320 A CAELinux and the Book Software
(ODE) system from data, you can take one of the R programs discussed in
Section 3.9 (e.g.
ODEFitEx1.r) as a template. To solve your problem, you will then
just have to replace the ODE and the parameters in the template with your own
ODE and parameters.
321
B
R (Programming Language and Software Environment)
This section gives you some information on R, focusing on the procedures used in
the book. Readers who need more information on R are referred to R’s help pages,
to the documentation available under www.r-project.org, and to a vast literature on
R (books such as [45, 237–241]).
Note It is assumed here that you have prepared your computer as described in
Appendix A.
B.1
Using R in a Konsole Window
B.1.1
Batch Mode
There are several R programs (which are also called R scripts) in the book software
which can be identified by their file extension ‘‘
.r’’ (see Appendix A). Suppose you
want to run e.g. the neural network program
NNEx2.r.InTableA.1youseethat
this program is in the
PhenMod/NN directory of MMS.zip. After you have unzipped
MMS.zip to the home directory of some external universal serial bus (USB) device
which appears under a name such as USBDEVICE on your computer,
NNEx2.r will
appear in the CAELinux directory
/media/USBDEVICE/MMS/PhenMod/NN (replace
‘‘USBDEVICE’’ by the appropriate name of your USB device). Open a Linux
Konsole window e.g. by clicking on the Konsole symbol on the desktop. Within
the Konsole window, navigate to
/media/USBDEVICE/MMS/PhenMod/NN using the
cd (‘‘change directory’’) command. Note that you can use the ls (‘‘list files’’)
command in the Konsole window if you want to see the files within a particular
directory. After arriving in
/media/USBDEVICE/MMS/PhenMod/NN,enterR to start
the R software. Wait a moment until you see the ‘‘>’’ command prompt of the R
software. To execute
NNEx2.r, enter the command: source("NNex2.r").Ifyou
prefer a GUI-based procedure to start your programs, you may use the
RKWard
program which you find in CAELinux if you select CAE Software/Math under the
322 B R (Programming Language and Software Environment)
PC button of the desktop.
B.1.2
Command Mode
As an alternative to running R programs in batch mode as described above, you
can also enter R commands directly into the Konsole window. For example, the
program
NNEx2.r begins with the command:
require(rgl)
After you have started the R software in the Konsole window, you may enter this
command directly after the R prompt in the Konsole window. If you go on and enter
the remaining commands of
NNEx2.r line by line, you will obtain the same result
that is obtained using the batch mode procedure described above. The advantage of
this ‘‘command mode’’ operation of R is that you can immediately see the results
of each of your commands and that you can then use this information to decide
how you go on. Beyond this, the command mode operation of R is useful if you
just want to perform a quick analysis that involves a few commands only. In such
cases, the command mode operation can be faster than the batch mode operation
where you have to invoke an editor to write the
.r program, save that program
somewhere, navigate to that directory within the Konsole window and so on.
B.2
R Commander
The R Commander is a GUI (graphical user interface) that facilitates the use
of R for basic statistical applications as described in Section 2.1. To start the R
Commander,openaKonsole window within CAELinux (in an arbitrary directory),
start the R software as described in Section B.1 and then enter the command
library(Rcmdr).
Several other GUIs for R are available, see www.r-project.org.
323
C
Maxima
This section gives you some information on Maxima, focusing on the proce-
dures used in the book. Readers who need more information on Maxima may
refer to books such as [18] which focus on Macsyma, a very similar commer-
cial version of Maxima, to the very comprehensive Maxima manual by W.
Schelter [106], toMaxima’s help pages and to the documentation available under
http://maxima.sourceforge.net/.
Note It is assumed here that you have prepared your computer as described in
Appendix A.
C.1
Using Maxima in a Konsole Window
C.1.1
Batch Mode
There are several Maxima programs in the book software which can be identified by
their file extension ‘‘
.mac’’ (see Appendix A). Suppose we want to run the program
Tin.mac which you find in the Principles directory of MMS.zip.Theprocedure
to run this program is very similar to the procedure to run R programs that is
described in Appendix B.1. As it is described there, open a Konsole window within
CAELinux and navigate into the appropriate directory on your universal serial bus
(USB) device. Enter
maxima in the Konsole window to start the Maxima software.
Then, enter the command
batch("Tin.mac") to run the program Tin.mac.If
you prefer a graphical user interface (GUI)-based procedure to start your programs,
you may use the wxMaxima program described below.
C.1.2
Command Mode
As an alternative to running Maxima programs in batch mode, you can also enter
Maxima commands directly into the Konsole window. For example, the program
324 CMaxima
Tin.mac begins with the command:
kill(all)$
After you have started the Maxima software in the Konsole window, you may
enter this command directly after the Maxima prompt in the Konsole window. If
you go on and enter the remaining commands of
Tin.mac line by line, you will
obtain the same result that is obtained using the batch mode procedure described
above. Again, the advantage of this ‘‘command mode’’ operation of Maxima is that
you can immediately see the results of each of your commands and that you can
then use this information to decide how you go on. Beyond this, the command
mode operation of Maxima is useful if you just want to perform a quick analysis
that involves a few commands only. In such cases, the command mode operation
can be faster than the batch mode operation where you have to invoke an editor to
write the
.mac program, save that program somewhere, navigate to that directory
within the Konsole window and so on.
C.2
wxMaxima
wxMaxima is a GUI that facilitates the use of Maxima in its batch mode as
well as in its command mode. It provides access to Maxima’s most important
commands through menus, a 2D formatted display of mathematical formulas,
the possibility to create documents made up of text and calculations and so on.
For example, to run the program
Tin.mac using wxMaxima,youwoulduseits
menu option
File/Batch File.Ifyoudothis,wxMaxima will also display the
command
batch("Tin.mac") that can be used in a Konsole window as described
in Section C.1 above, that is, using wxMaxima is also a way for beginners to
get acquainted with Maxima terminal commands. wxMaxima can be accessed in
CAELinux if you select
CAE Software/Math under the PC button of the desktop.