Table Of ContentALL MODELS ARE WRONG, SOME ARE USEFUL.
G.E. P. Box
I HAD COME TO AN ENTIRELY ERRONEOUS CON-
CLUSION WHICH SHOWS, MY DEAR WATSON,
HOW DANGEROUS IT ALWAYS IS TO REASON
FROM INSUFFICIENT DATA.
Sherlock Holmes 1
Preface
Modeling of real world phenomena is always accompanied by uncertainty.
Uncertainty in the selection of an adequate mathematical model, uncer-
tainty in the values of input data, uncertainty in the correctness of computer
codes, uncertainty in the error of numerical results; to list a few instances.
The accent of this book is on the uncertainty in input data and its
impact on the outputs of mathematical models. In pursuing this topic,
we use the worst scenario method, which searches for the most unfavorable
inputs among uncertain input data in the range of available information.
The word "unfavorable" indicates that a gauge is available to distinguish
between favorable and unfavorable data. Functionals evaluating outputs of
mathematical models are perfect examples of such gauges.
A simple modification in the mathematical formulation of the worst
scenario method leads to the best scenario method (optimal design) where
the most favorable inputs are sought. The difference between functional
values for the best and the worst scenario reflects the variety of model
outputs caused by the uncertain model inputs.
Critics may say that the worst scenario approach is too pessimistic be-
cause it does not take into account that the inputs might not be equally
1 The Adventure of the Speckled Band by Sir Arthur Conan Doyle.
vi PREFACE
distributed within the limits set by the amount of uncertainty. If this hap-
pens, then some input values are more frequent than others. Consequently,
it may happen that the worst scenario coincides with very rare input val-
ues. This rarity is not reflected by the method so that the importance of
worst scenario identification could be overrated. However, the worst sce-
nario method can be pronounced pessimistic only if information about a
data occurence is available, but not used.
Let us note that the rarity of data can be taken into account by cou-
pling the worst scenario method with other approaches to uncertainty. Take
fuzzy sets, for example. Among other things, this and other couplings are
illustrated in Chapter I, which we consider to be an easily accessible famil-
iarization with uncertainty in mathematical modeling.
From a mathematical point of view, the core of the worst scenario
method is presented in Chapter II. Then chapters devoted to particu-
lar models follow. Let us only mention Timoshenko beams, pseudoplates,
plates, and elastoplastic bodies in stability, thermal, and frictional contact
problems. Generally, nonlinear problems are treated. The content of the
book is outlined in more detail in the Introduction.
Prague and Austin, August 2004 Ivan Hlavs
Jan Chleboun
Ivo Babu~ka
List of Figures
1.1 Key points in modeling I .................................. 3
1.2 Key points in modeling II ................................. 4
2.1 Graph of r ]01 --, R ................................... 27
2.2 Membership function #I ................................... 28
2.3 Membership function #j of the fuzzy range of r ............ 29
2.4 Graph of 7r(y)-- Pl([y, y + 1]) ............................. 31
2.5 Graph of t3(y)= Bel([y, y + 1]) ............................ 31
2.6 Skyline graphs of histograms of input data ................. 33
2.7 Skyline graphs of histograms of output data ................ 34
2.8 Solution u(apq,.8) as a function of r and x .................. 35
2.9 Function x ................................................ 37
2.10 Interpolation need not match the original function ......... 41
2.11 Criterion-functional 1)I( is rather sensitive to uncertain r ... 42
2.12 Criterion-functional ~2 is rather insensitive to uncertain r.. 42
4.1 Domain ~, test subdomain G, and heat sources ............ 81
4.2 Thermal conductivities .................................... 82
4.3 Gradients V~ ............................................. 82
8.1 Function A ................................................ 143
8.2 Function 1A (k) plotted at discrete points ................... 144
12.1 The coordinate system ..................................... 189
12.2 Equilibrium paths for initial deflection in one halfwave ..... 193
xiii
xiv LIST OF FIGURES
12.3 Dependence of the lower buckling load dk on the initial de-
flection amplitude ......................................... 198
12.4 Dependence of the maximum mean reduced stress on the ini-
tial deflection amplitude for k = 1.4 ..... ~ ................. 198
12.5 Dependence of the maximum mean reduced stress on the ini-
tial deflection amplitude for k = 1.25 ....................... 199
12.6 Dependence of the maximum mean reduced stress on the ini-
tial deflection amplitude for k- 1.1 ........................ 199
12.7 Equilibrium paths for combined initial deflections .......... 202
12.8 Decisive part of the maximum mean reduced stress ......... 204
22.1 U-notched specimen ....................................... 352
22.2 Finite element mesh ....................................... 352
22.3 First cycle at the root of the notch ......................... 353
22.4 First cycle at the root of the notch ......................... 354
24.1 Boundaries ,wo12fO sq r..-.q~lco)w,p ix 0~"~ , ~--.-Q-uppipx, ppu~'~O qs ............... 387
24.2 Boundary Dopix and contour lines of u20 - u 0x ............ 388
*'~ WOl'~
List of Tables
2.1 Maximum range scenarios induced by a-cuts ............... 29
2.2 Material properties for plate and sheet aluminum .......... 48
12.1 Values of (I)*(0.2, t0) ....................................... 204
12.2 Values of (I)*(z0, 0.13) ..................................... 204
12.3 Values of (I)*(z0, 0) ........................................ 204
12.4 Values of 4)*(0.07, to) ...................................... 204
12.5 Values of (I)*(0.14, t0) ...................................... 205
22.1 Average stress (perfectly plastic model) .................... 355
22.2 Average stress (isotropic hardening model) ................. 355
22.3 Average stress (kinematic hardening model) ................ 356
22.4 Worst scenario for maximum average stress ................ 356
24.1 Estimates for pixel approximate domains .................. 387
24.2 Estimates for non-pixel approximate domains .............. 389
XV
Introduction
The immense progress in computational power and the promising perspec-
tive of its further evolution enable us to approach reality nearer and deeper
through modeling and solving problems in technical, natural, and social sci-
ences than one or two decades ago. More and more computational analysis
is used in engineering predictions and decisions (Oden, 2002).
Regardless of many achievements, modeling the real world is accompa-
nied with fairly general sorts of uncertainty. What mathematical model is
the best compromise between complexity, solvability, accuracy, safety, and
computational expenses? What is the relation between an exact solution
and its numerical approximation? How can we verify the trustworthiness of
the respective computer code?
These and other topics are briefly touched upon in the initial part of
Chapter I. However, the rest of the book concentrates on particular subjects
related to uncertain input data.
In contrast with the classical approach, where differential equations,
integral equations, or variational inequalities are equipped with uniquely
given input data (i.e., we have a complete knowledge of the input data as
coefficients, boundary or initial conditions, right-hand sides, etc.), we take
into account uncertainty in the data of the model.
A certain amount of uncertainty of this kind is more or less tacitly
present in many (if not all) technical and scientific problems. In fact, the
input data are usually obtained in two steps: first, experimental measure-
ments are made, then the corresponding inverse (identification) problem is
solved. Both these steps, however, are influenced by inaccuracy. Unavoid-
able "noise" in measurements is superimposed on errors of an approximate
solution of the inverse problem. A typical example can be the determination
of physical parameters in models of processes in the deep Earth core. Since
the range of our technical research methods is limited, we have to resort
to rather indirect identification approaches in this case. More direct means
XVII ..
xviii INTRODUCTION
can be used in the identification of properties of available materials.
Another source of uncertainty is the difference between the laboratory
environment, where tests of materials take place, and the harsh real world,
where materials are produced and used. When designing a structure, de-
signers are guided by handbooks of material coefficients. Do these tabular
materials exactly represent the real materials supplied by a manufacturer?
How do material properties change in time due to corrosion, for example?
The answers are often disappointingly vague and weak in information.
In this book, a number of particular examples of problems with uncertain
data will be explored.
Obviously, both the theory of problems with uncertain input data and
appropriate numerical methods are considerably more complex than those
of the classical approach with completely known inputs. Various methods
exist to model uncertainty in input data. Let us mention only two: the
stochastic (probabilistic) approach or the worst scenario method. Other
possibilities as well as their combinations with the worst scenario method
are discussed in Chapter I.
The theory of probability has proved extraordinarily useful in modeling
uncertainty; see (Ghanem and Spanos, 1991), (Holden et al., 1996), (Deb
et al., 2001), (Babu~ka et al., 2004b), or (Babu~ka et al., 2004c), for in-
stance. However, the information content of a probabilistic model is often
quite high and so it could be difficult to obtain relevant probabilistic data.
Moreover, the data are frequently generated by methods which have their
own additional uncertainties. In some cases, the resolution of an analysis
can be influenced by those parts of the probabilistic model that are most
difficult to establish precisely (Ben-Haim and Elishakoff, 1990). Also, the
interpretation of probabilistic results is not simple (Salmon, 1967, Chap-
ter 5).
In a sense, probabilistic models have deterministic features because the
probability of input data is considered completely known.
In any case, the required data are obtained by experimentation or expert
opinion, for instance, so that information about the probability of inputs
has to be related to problems defined and solved in the realm of statistics.
A powerful synthesis of many ideas surrounding uncertainty and the in-
terpretation of probability is presented in (Savage, 1972); see also (Cooke,
1991).
Whereas the stochastic approach requires information about the statis-
tical distribution of the data, the worst scenario method needs only bounds
for the input data to define a set of admissible data. As a consequence, the
xix
worst scenario method is applicable to a broad variety of problems, even
to those where the stochastic method has not yet been established. Also,
if the probability distribution in a stochastic model is uncertain, the entire
range of possible probabilistic outputs has to be determined, which is the
goal of the worst scenario method.
In the worst scenario approach, a criterion that evaluates a feature of
the solution to a state problem is defined in such a way that an increase in
the criterion value indicates a deterioration in the feature, i.e., the higher
the value, the "worse" the state. The goal is to maximize the criterion
value over a set of uncertain data entering the model. In other words, one
searches for the worst situation that can be determined by input data within
the scope of uncertain inputs.
Although this approach is related to the safe side rule used in all sorts
of engineering for centuries, the main idea of the worst scenario method
was probably first suggested in (Bulgakov, 1940), (Bulgakov, 1946), and
clearly formulated in (Ben-Haim and Elishakoff, 1990) as a convex modeling
of uncertainty. In convex modeling, the authors suppose that uncertain
data form a convex admissible set daU in ~l n.
Other terms for the worst scenario concept include the unknown-but-
dednuob uncertainty hcaorppa or the guaranteed performance .hcaorppa
Also known as anti-optimization (Elishakoff, 1990), the idea of the worst
scenario was incorporated into design and optimal design problems; see
(Lombardi and Haftka, 1998), (Qiu and Elishakoff, 2001), and references
therein.
Later, the concept of information-gap uncertainty was proposed and
analyzed (Ben-Haim, 1996), (Ben-Haim, 1999a), (Ben-Haim, 1999b), (Ben-
Haim, 2001a), (Ben-Haim, 2001b), (Ben-Haim, 2004), (Hemez and Ben-
Haim, 2004). Directed towards design evaluation and decision making, info-
gap models consider a continuum of nested sets Uad(a) controlled by a posi-
tive real parameter a, i.e., al <_ a2 implies Uad(al) C U~d(a2). An analogy
to the cost (criterion) function is a reward function depending not only on
the state solution, but also on the decision-maker's action. Then the robust-
ness function is defined as the greatest value of the uncertainty parameter
for which an acceptable performance (reward) is assured. Subsequently,
the trade-off between immunity-to-uncertainty and demanded reward is at
the center of attention. The worst scenario method represents a substantial
part of the information-gap theory.
Particular sorts of the worst scenario approaches have been intensively
investigated in linear algebra. Uncertain matrices are within the purview
of the theory of interval matrices; see (Rohn, 1994) or (Nedoma, 1998) and
xx INTRODUCTION
the references therein. As the structure of uncertainty in these uncertain
matrices not always matches the uncertainty in the matrices arising in design
problems, further research is required to fill the gaps.
Interval arithmetic is another example of an approach motivated by the
worst scenario approach. Concentrated on the inaccuracy of floating-point
arithmetic, interval arithmetic can deliver guaranteed bounds for a numer-
ical solution and it can help to control the accuracy of computation; see
for instance (Alefeld and Herzberger, 1983), (Adams and Kulisch, 1993),
(Hammer et al., 1995), (Kulisch, 1999).
Control theory also recognizes problems that lead to a sort of worst
scenario (Dullerud and Paganini, 2000).
By comparison with the stochastic approach with completely known
information, the worst scenario method is pessimistic because it does not
consider information used in the stochastic model. Nevertheless, the method
helps a designer to stay on the safe side. It emphasizes the worst, i.e., the
most dangerous data, even if the probability of their occurrence may be
low. It is possible, however, to couple the worst scenario approach with
probability-, likelihood-, or possibility-based methods; see Chapter I.
If reliable probabilistic information is unavailable and, consequently, lit-
tle is known about the input data distribution, then a stochastic approach
should not be proposed. In such circumstances, a non-stochastic approach
should be used; see relevant sections of (Ben-Haim and Elishakoff, 1990)
and (Elishakoff et al., 2001). Even if probabilistic information is available to
substantiate a stochastic analysis, one may prefer a simpler non-stochastic
worst scenario method. This preference occurs if the distribution of un-
certain data is "close to uniform" (with large deviation) because then the
worst scenario approach yields results comparable with those of a stochastic
approach; see (Elishakoff et al., 1994a), (Elishakoff et al., 2001, Section 5.2),
or (Elishakoff and Zingales, 2003). For a survey of probabilistic techniques
with an emphasis on environmental engineering, we refer to (Cullen and
Frey, 1999), where a large list of references is given.
Various aspects of uncertainty modeling are treated in (Natke and Ben-
Haim, 1997), (Haldar et al., 1997), and (Elishakoff, 1999), for instance.
Let us illustrate the leading idea of the worst scenario method on an ex-
ample of a quasilinear elliptic boundary value problem in a bounded domain
~CI~ d
div(a(u) grad u) = f (0.1)
with u - 0 on the boundary 012. Let the scalar function a(.) be uncertain.
We will assume that a(.) belongs to a given set daU of admissible functions.
JXX
The above problem may represent a model of a steady heat flow if a(-)
denotes a temperature dependent heat conductivity coefficient and u is the
temperature.
Let a unique solution u(a) of problem (0.1) exist in a function space V
for any data a E Uad. The existence can be proved under relatively mild
assumptions (Hlavs et al., 1994). Let the quantity that we are interested
in be the mean temperature over an a priori chosen small subdomain G C
(or G C 0f~). We wish to find the maximum value of the quantity of interest
under the assumption a E Uad. To this end, we identify the quantity of
interest with a suitable criterion, i.e., a criterion-based functional (used
here as a criterion-functional) ~(v) : V ~ R is defined as the mean value of
u(a) over G. Then we solve the problem
a ~ = argmax ~b(u(a)). (0.2)
aaUEa
Problem (0.2) can have more than one solution. Each solution, i.e.,
each conductivity coefficient solving (0.2) consequently implies the max-
imum mean temperature in G. Either a ~ or ~(u(a~ or even the pair
(a ~ ~(u(a~ can be called the worst scenario.
In practice, instead of a ~ itself, the value of ~(u(a~ is more important.
We get ~(u(a~ through a ~
Often, we are also interested in a minimization version of problem (0.2)
and in the difference between the maximum and minimum value of the
quantity of interest.
The worst scenario method is only one of the approaches to uncertain
data. Moreover, uncertainty in input data is only one of the numerous
facets of uncertainty in modeling. Although this latter, more general sub-
ject deserves much attention, a detailed treatment lies outside the scope of
this book. Nevertheless, Chapter I is intended as an introduction to uncer-
tainty in modeling. It touches various aspects of uncertainty, and briefly
presents other approaches to uncertain data and their coupling with the
worst scenario method. Certain topics, namely verification and validation,
are further elucidated in the Appendix at the end of the book.
A general formulation of the worst scenario method is proposed in Chap-
ter II, where both a general abstract scheme and an analysis of the method
are presented. Also, conditions sufficient for the existence of a worst sce-
nario are given. Then an approximate worst scenario problem is formulated.
To this end, respective discretizations of V and daU are necessary. A conver-
gence analysis with respect to discretization parameters is also presented.
This general framework is further applied to particular families of problems.
