The Computational Mathematics research group at the Faculty of Mathematics of the University of Vienna is investigating methods for the reliable modeling of uncertainty in situations where a large (or even infinite) number of parameters is uncertain.

In this case, the data sets available are usually not sufficiently large to allow the estimation of a full high-dimensional probability model. Using simple-minded default distributions (such as Gaussians based on estimated means and covariance matrix) may lead to a severe underestimation of the risk (sometimes exponential in the dimension).

Thus different, more robust approaches are needed. We explore the use of clouds for objective (data-driven) uncertainty and of surprise functions for subjective uncertainty. Both can be integrated with traditional methods and work (in principle) in high dimensions. Box clouds can even be treated rigorously with interval techniques, and yield in certain cases reliable worst case bounds even for a very large number of uncertain parameters.

Current applications include geography (terrain modeling), medicine (radiation therapy planning), and engineering (truss structure analysis, robust space system design).

Extension of the techniques to applications involving partial differential equations are work in progress.

Clouds are a concept for uncertainty mediating between the concept of a fuzzy set and that of a probability distribution. A cloud is to a random variable more or less what an interval is to a number.

Clouds give - within the traditional probabilistic framework - a concept for imprecise probability with which quantitative conclusions can be derived from uncertain probabilistic information.

Problems involving box clouds (clouds whose alpha cuts are axiparallel boxes) can be rigorously treated by means of interval techniques, even in fairly high dimensions, in certain cases even in very large (truss structures) or infinite (partial differentail equations) dimension.

Problems involving ellipsoidal clouds, which are able to account for correlated uncertainity, can in principle be treated by ellipsoid arithmetics, although we currently have no practical experience with it in applications.

We currently treat correlated uncertainty by means of sparse polyhedral clouds, which seem to suffer least from the curse of dimensionality.

Surprise functions form an alternative approach to fuzzy modeling.
The surprise concept is related to the traditional membership function
by an antitone transformation.
Advantages of the surprise approach include:

1. It has a consistent semantic interpretation.

2. It allows the joint use of quantitative and qualitative knowledge,
using simple rules of logic.

3. It is a direct extension of (and allows combination with) the least
squares approach to reconciling conflicting approximate numerical data.

4. It is ideally suited to optimization under imprecise or conflicting
goals, specified by a combination of soft and hard interval constraints.

5. It gives a straightforward approach to constructing families of
functions consistent with fuzzy associative memories as used in fuzzy
control, with tuning parameters (reflecting linguistic ambiguity) that
can be adapted to available performance data.

Fuzzyland, a tale of wisdom

''Once upon a time, the country of Fuzzyland had a leadership that was
famous for ...''

**A. Neumaier**,
Towards optimization-based error bounds for uncertain PDEs,
Slides, 2008

Using tools from functional analysis and global optimization,
methods are presented for obtaining, given an approximate solution
of a partial differential equation, realistic error bounds for some
response functional of the solution.

The method is based on computable bounds for the inverse of linear
elliptic operators. Like in the dual weighted residual (DWR) method,
our error bounds for response functionals have the quadratic
approximation property (so that they are asymptotically optimal),
but in contrast to DWR, our bounds are rigorous and also capture
the higher order contributions to the error.

Using global optimization techniques, bounds can be found
that not only cover the errors in solving the differential equations
but also the errors caused by the uncertainty in the parameters.
This provides reliable tools for the assessment of uncertainty in the
solution of elliptic partial differential equations. Our bounds are
independent of the way the approximations are obtained, hence can be
used to independently verify the quality of an approximation computed
by an arbitrary solver.
The bounds not only account for discretization errors but also for other
numerical errors introduced through numerical integration and boundary
aproximations.

We also discuss how to represent model uncertainty in terms of
so-called clouds, which describe the rough shapes of typical samples
of various size, without fixing the details of the distribution.
Clouds use only information from 1- and 2-dimensional marginal
distributions, readily available in practice.

**A. Neumaier**,
Certified error bounds for uncertain elliptic equations,
J. Comput. Appl. Math. 218 (2008), 125--136.

Using tools from functional analysis and global optimization,
methods are presented for obtaining certificates for rigorous and
realistic error bounds on the solution of linear elliptic partial
differential equations in arbitrary domains, either in an energy norm,
or of key functionals of the solutions, given an approximate solution.

Uncertainty in the parameters specifying the partial differential
equations can be taken into account, either in a worst case setting,
or given limited probabilistic information in terms of clouds.

**A. Neumaier and A. Pownuk**,
Linear systems with large uncertainties, with applications to truss
structures,
Reliable Computing 13 (2007), 149-172.

Linear systems whose coefficients have large uncertainties
arise routinely in finite element calculations for
structures with uncertain geometry, material properties, or loads.
However, a true worst case analysis of the influence of such
uncertainties was previously possible only for very small systems
and uncertainties, or in special cases where the coefficients do
not exhibit dependence.

This paper presents a method for computing rigorous bounds on the
solution of such systems, with a computable overestimation factor that
is frequently quite small. The merits of the new approach are
demonstrated by computing realistic bounds for some large, uncertain
truss structures, some leading to linear systems with over
5000 variables and over 10000 interval parameters, with excellent
bounds for up to about 10% input uncertainty.

Also discussed are some counterexamples for the performance of
traditional approximate methods for worst case uncertainty analysis.

**A. Neumaier**,
Uncertainty modeling for robust verifiable design,
Slides, 2004.

pdf file (318K)

**A. Neumaier**,
Clouds, fuzzy sets and probability intervals,
Reliable Computing 10 (2004), 249-272.

This paper discusses the basic theoretical and numerical
properties of clouds, and relate them to histograms, cumulative
distribution functions, and likelihood ratios.

We show how to compute nonlinear transformations of clouds,
using global optimization and constraint satisfaction techniques.
We also show how to compute rigorous enclosures for the expectation
of arbitrary functions of random variables, and for probabilities
of arbitrary statements involving random variables,
even for problems involving more than a few variables.

Finally, we relate clouds to concepts from fuzzy set theory,
in particular to the consistent possibility and necessity measures
of Jamison and Lodwick.

**A. Neumaier**,
On the structure of clouds,
Manuscript (2003).

In this paper, some structural results about clouds are derived.
In particular, it is shown that every cloud contains some random
variable.

**A. Neumaier**,
Fuzzy modeling in terms of surprise,
Fuzzy Sets and Systems 135 (2003), 21-38.

This paper presents an approach to fuzzy modeling based on the
concept of surprise.

**A. Neumaier**,
The wrapping effect, ellipsoid arithmetic, stability and
confidence regions,
Computing Supplementum 9 (1993), 175-190.

The wrapping effect is one of the main reasons that the application
of interval arithmetic to the enclosure of dynamical systems is
difficult. In this paper the source of wrapping is analyzed
algebraically and geometrically.
A new method for reducing the wrapping effect is proposed,
based on an interval ellipsoid arithmetic.

Applications are given to the verification of stability regions for
nonlinear discrete dynamical systems and to the computation of rigorous
confidence regions for nonlinear functions of normally distributed
random vectors.

**M. Fuchs and A. Neumaier**,
Autonomous robust design optimization with potential clouds,
Int. J. Reliability Safety 3 (2009), 23-34.

The task of autonomous and robust design cannot be regarded as a single
task, but consists of two tasks that have to be accomplished
concurrently. First, the design should be found autonomously; this
indicates the existence of a method which is able to find the optimal
design choice automatically. Second, the design should be robust;
in other words: the design should be safeguarded against uncertain
perturbations. Traditional modeling of uncertainties faces several
problems. The lack of knowledge about distributions of uncertain
variables or about correlations between uncertain data, respectively,
typically leads to underestimation of error probabilities. Moreover,
in higher dimensions the numerical computation
of the error probabilities is very expensive, if not impossible, even
provided the knowledge of the multivariate probability distributions.
Based on the clouds formalism we have developed new methodologies to
gather all available uncertainty information from expert engineers,
process it to a reliable worst-case analysis and finally optimize the
design seeking the optimal robust design.

**M. Fuchs and A. Neumaier**,
Handling uncertainty in higher dimensions with potential clouds
towards robust design optimization,
pp. 376-382 in: Soft Methods for Handling Variability and Imprecision
(D. Dubois et al., eds.),
Advances in Soft Computing, Vol. 48, Springer 2008.

Robust design optimization methods applied to real life problems
face some major difficulties: how to deal with the estimation of
probability densities when data are sparse, how to cope with high
dimensional problems and how to use valuable information in the
form of unformalized expert knowledge.

We introduce the clouds formalism as means to process available
uncertainty information reliably, even if limited in amount and
possibly lacking a formal description. We provide a worst-case
analysis with confidence regions of relevant scenarios which
can be involved in an optimization problem formulation
for robust design.

**M. Fuchs and A. Neumaier**,
Potential based clouds in robust design optimization,
J. Stat. Theory Practice, Special Issue on Imprecision,
to appear (2008).

*pdf file (970K)*

*downloading/printing problems?*

Robust design optimization methods applied to real life problem
face some major difficulties: how to deal with the estimation of
probability densities when data are sparse, how to cope with high
dimensional problems and how to use valuable information
in the form of unformalized expert knowledge.

In this paper we introduce in detail the clouds formalism as means
to process available uncertainty information reliably, even if
limited in amount and possibly lacking a formal description,
to providing a worst-case analysis with confidence regions of
relevant scenarios which can be involved in an optimization problem
formulation for robust design.

**M. Fuchs and A. Neumaier**,
Uncertainty modeling with clouds in autonomous robust design
optimization,
pp. 1-22 in: Proc. 3rd Int. Workshop Reliable Engineering Computing,
Savannah, Georgia, USA, 2008.

The task of autonomous and robust design cannot be regarded as a
single task, but consists of two tasks that have to be accomplished
concurrently.

First, the design should be found autonomously;
this indicates the existence of a method which is able to find
the optimal design choice automatically.

Second, the design should be robust; in other words: the design
should be safeguarded against uncertain perturbations.
Traditional modeling of uncertainties faces several problems.
The lack of knowledge about distributions of uncertain variables
or about correlations between uncertain data, respectively,
typically leads to underestimation of error probabilities.
Moreover, in higher dimensions the numerical computation
of the error probabilities is very expensive, if not impossible,
even provided the knowledge of the multivariate probability
distributions.

Based on the clouds formalism we have developed new methodologies
to gather all available uncertainty information from expert
engineers, process it to a reliable worst-case analysis and
finally optimize the design seeking the optimal robust design.

The new methods are applied to problems for autonomous optimization
in robust spacecraft system design at the European Space Agency (ESA).

**M. Fuchs, D. Girimonte, D. Izzo, and A. Neumaier**,
Robust and automated space system design,
Chapter (pp. 251-272) in:
Robust intelligent systems (A. Schuster, ed.), Springer, 2008.

Over the last few years, much research has been dedicated to
the creation of decisions support systems for space system engineers
or even for completely automated design methods capturing
the reasoning of system experts. However, the problem of taking
into account the uncertainties of variables and models defining
an optimal and robust spacecraft design have not been tackled
effectively yet. This chapter proposes a novel, simple approach
based on the clouds formalism to elicit and process the uncertainty
information provided by expert designers and to incorporate this
information into the automated search for a robust, optimal design.

**M. Fuchs, A. Neumaier, and D. Girimonte**,
Uncertainty modeling in autonomous robust spacecraft system design
,
Proc. Appl. Math. Mech. 7 (2007), 2060041-2060042.

In the last few years, much research has been dedicated to the
development of decisions support systems for the space system
engineers or even of completely automated design methods capturing
the reasoning of the system experts. However, the
problem of taking into account the uncertainties of the variables
and models to determine an optimal and robust spacecraft
design has not been tackled effectively yet. Based on the
clouds formalism we propose a novel approach to process the
uncertainty information provided by expert designers and
incorporate it into the automated search for a robust optimal design.

**A. Neumaier, M. Fuchs, E. Dolejsi, T. Csendes,
J. Dombi, B. Bánhelyi, Z. Gera**,
Application of clouds for modeling uncertainties in robust space system
design,
Final Report,
ARIADNA Study 05/5201, European Space Agency (ESA), 2007.

*
Ariadna Completed Studies*

This report discusses a case study for modeling uncertainties with
clouds in a model for space system design, with the goal of determining
robust feasible designs for the model.

The problem structure involves all difficulties an optimization problem
can have: discrete variables, strong nonlinearities, discontinuities
due to branching decisions, multiple objectives, multiple local minima.

The formal modeling of uncertainty by means of clouds, and
the robust optimization of a resulting uncertain model was considered.
A heuristic approach using surrogate function modeling,
corner searches, and discrete line searches was used to solve the
robust optimization problem in case of interval uncertainties.
This approach works satisfactorily for the model problem for the case
of interval uncertainty.

Solving the model problem with uncertainty revealed significant
robustness advantages of the approach using uncertainty. The influence
on assumed knowledge about additional uncertainty information was
illustrated by for a few model choices.

**J. Santos, W.A. Lodwick and A. Neumaier**,
A New Approach to Incorporate Uncertainty in Terrain Modeling,
pp. 291-299 in:
Geographic Information Science: Second International Conference,
GIScience 2002, Proceedings (M. Egenhofer and D. Mark, eds.),
Lecture Notes in Computer Science Vol. 2478,
Springer, New York 2002.

A method for incorporating uncertainty in terrain modelling by
expressing elevations as fuzzy numbers is proposed. Given a finite set
of fuzzy elevations representative of the topographic surface in a
certain region, we develop methods to construct surfaces that
incorporate the uncertainty. The smoothness and continuity conditions
of the surface generating method are maintained. Using this approach,
we generalize some classic interpolators and compare them
qualitatively. Extensions to wider classes of interpolators follow
naturally from our approach. A numerical example is presented to
illustrate this idea.

**W.A. Lodwick, A. Neumaier and F. Newman**,
Optimization under uncertainity: methods and applications in radiation
therapy,
Proc. 10th IEEE Int. Conf. Fuzzy Systems,
December 2-5, 2001, Melbourne, Australia,
pp.1219-1222.

This research focuses on the methods and application of optimization
under uncertainty to radiation therapy planning, where it is natural
and useful to model the uncertainity of the problem directly.
In particular, we present methods of optimization under uncertainty in
radiation therapy of tumors and compare their results.
Two themes are developed in this study:
(1) the modeling of inherent uncertainty of the problems and
(2) the application of uncertainty optimization.

**
C. Elster and A. Neumaier,
Screening by conference designs**,
Biometrika 82 (1995), 589-602.

*dvi.gz file (28K)*,
*ps.gz file (92K)*,
*pdf file (215K)*,
*downloading/printing problems?*

Screening experiments are addressed to the identification
of the relevant variables within some process or experimental outcome
potentially depending on a large number of variables.
In this paper we introduce a new class of experimental
designs called edge designs. These designs are very useful for
screening experiments
since they allow a model-independent estimate of the set of relevant
variables, thus providing more robustness than traditional designs.

We give a bound on the determinant of the information matrix of certain
edge designs, and show that a large class of edge designs meeting
this bound can be constructed from conference matrices. We also
show that the resulting conference designs have an optimal space
exploration property which is important as a guard against unexpected
nonlinearities. We survey the existence of and constructions
for conference matrices, and give, for n<50 variables, explicit such
matrices when n is a prime, and references to explicit constructions
otherwise.

**A. Neumaier**,
Towards optimization-based error bounds for uncertain PDEs,
Slides, 2008

**A. Neumaier**,
Worst case analysis of mechanical structures by interval methods,
Slides, 2005.

pdf file (263K)

**A. Neumaier**,
Uncertainty modeling for robust verifiable design,
Slides, 2004.

pdf file (318K)

**A. Neumaier and J.-P. Merlet**,
Constraint satisfaction and global optimization in robotics,
Slides, 2002.

pdf file (449K)

CBDO: A Matlab toolbox for Cloud Based Design Optimization

Software for uncertainty modeling by clouds by Martin Fuchs

**M. Fuchs**,
Uncertainty modeling in higher dimensions: Towards robust design
optimization,
Ph.D. Thesis, Wien 2008.

Modern design problems impose multiple major tasks an engineer has to
accomplish.
* The design should account for the designated functionalities.
* It should be optimal with respect to a given design objective.
* Ultimately the design must be safeguarded against uncertain
perturbations which should not cause failure of the design.
These tasks are united in the problem of robust design optimization
giving rise to the development of computational methods for uncertainty
modeling and design optimization, simultaneously.

Methods for uncertainty modeling face some fundamental challenges:
The computational effort should not exceed certain limitations;
unjustified assumptions must be avoided as far as possible. However,
the most critical issues concern the handling of incomplete information
and of high dimensionality. While the low dimensional case is well
studied and several methods exist to handle incomplete information,
in higher dimensions there are only very few techniques. Imprecision
and lack of sufficient information cause severe difficulties - but the
situation is not hopeless.

In this dissertation, it is shown how to transfer the high-dimensional
to the one-dimensional case by means of the potential clouds formalism.
Using a potential function, this enables a worst-case analysis on
confidence regions of relevant scenarios. The confidence regions are
weaved into an optimization problem formulation for robust design as
safety constraints. Thus an interaction between optimization phase and
worst-case analysis is modeled which permits a posteriori adaptive
information updating. Finally, we apply our approach in two case
studies in 24 and 34 dimensions, respectively.

Interval Computations

Vladik Kreinovich's comprehensive interval archive

Intervals and Probability Distributions

Dan Berleant's comprehensive archive on interval probability

Reliable Engineering Computing

Rafi Muhanna's site at Georgia Tech Savannah,
with biannual conferences on reliable engineering.

Weldon Lodwick

who worked with me on applying surprise methods to
terrain modeling and radiation therapy planning

Interval Methods

FMathL - Formal mathematical language

Regularization

Mathematical Software

Computational Mathematics Links

Mathematics Links

Statistics Links

my home page (http://www.mat.univie.ac.at/~neum)

Arnold Neumaier (Arnold.Neumaier@univie.ac.at)