Michael Oberguggenberger

Propagation of uncertainty through multivariate functions in the framework of sets of probability measures

Abstract This paper proposes a solution to Challenge Problem 1 posed by Oberkampf et al. [Reliab. Engng Syst. Safety (this issue)] and focuses on questions of dependence or independence of variables. The specification of uncertainties is consistently reduced to sets of probability measures. For whatever interpretation, the results essentially depend on the choice of combination rule in defining the joint distributions. The approach adopted here is shown to be consistent with interval analysis, random sets and fuzzy sets, and admits a clear description of the influence of the concept of independence used. A MATLAB-tool has been developed that allows to compute the lower and upper probability distributions for all choices of framework and combination rule.

Generalized solutions to the Korteweg-de Vries and the regularized long-wave equations

In this article generalized solutions to two model equations describing nonlinear dispersive waves are studied. The solutions are found in certain algebras of new generalized functions containing spaces of distributions. On the one hand, this allows the handling of initial data with strong singularities. On the other hand, suitable scaling allows one to introduce an infinitesimally small coefficient; thereby the authors produce generalized solutions in the sense of Colombeau to the inviscid Burgers equation.

Hyperbolic systems with discontinuous coefficients: generalized solutions and a transmission problem in acoustics

where V= (V,, . . . . V,), G = (G,, . . . . G,), A and F are (n x n) matrices, A is real-valued and diagonal, in the case of discontinuous coeffkients. More precisely, we wish to assume solely that A is a bounded measurable function and F is the distributional derivative of such a function. The example of Hurd and Sattinger [3, p. 1661 shows that, in general, such a system will not have solutions in the sense of distributions, even if it is in the form of a conservation law. Observe that, a priori, 8, V will not be a function, and so multiplication of distributions is involved in (0.1). Our approach will be to solve the problem in the Colombeau algebra Y(@), a differential algebra which contains the distributions and, in addition, has the algebra of infinitely differentiable functions as a sub-algebra. In Section 1 we shall show that if A, F, G are generalized functions belonging to 9(lR2) and A belongs to 9(R), then system (0.1) has a unique solution VE a( R*), provided the generalized functions A and F satisfy some assumptions corresponding to A E L”(lR*), FE W-L*m (lR*). Distributional solutions are recovered from the generalized solutions via a partial projection 452 0022-247X/89

Microlocal hypoellipticity of linear partial differential operators with generalized functions as coefficients

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Generalized solutions to Burgers' equation

We investigate microlocal properties of partial differential operators with generalized functions as coefficients. The main result is an extension of a corresponding (microlocalized) distribution theoretic result on operators with smooth hypoelliptic symbols. Methodological novelties and technical refinements appear embedded into classical strategies of proof in order to cope with most delicate interferences by non-smooth lower order terms. We include simplified conditions which are applicable in special cases of interest.

Classical and imprecise probability methods for sensitivity analysis in engineering: A case study

where v is a positive constant or generalized constant. The solutions will belong to an algebra 4,, of generalized functions, to be defined below, which contains the space of bounded distributions 9im. In particular, the initial data may be arbitrary bounded distributions. Before we go on describing our results, let us explain the setting we use. Our aim is to look for solutions in large differential algebras of generalized functions, so that all differentiations and nonlinear operations involved can be performed unrestrictedly. Following Colombeau [4, 51 and also Rosinger [lS], we construct these algebras by putting an algebraic structure on certain spaces made up by nets of smooth functions on open or closed subsets of W. In order not to complicate matters, we shall take all of Iw” as the underlying domain for the purpose of the introduction.

Fuzzy models in geotechnical engineering and construction management

This article addresses questions of sensitivity of output values in engineering models with respect to variations in the input parameters. Such an analysis is an important ingredient in the assessment of the safety and reliability of structures. A major challenge in engineering applications lies in the fact that high computational costs have to be faced. Methods have to be developed that admit assertions about the sensitivity of the output with as few computations as possible. This article serves to explore various techniques from precise and imprecise probability theory that may contribute to achieving this goal. It is a case study using an aerospace engineering example and compares sensitivity analysis methods based on random sets, fuzzy sets, interval spreads simulated with the aid of the Cauchy distribution, and sensitivity indices calculated by direct Monte Carlo simulation. Computational cost, accuracy, interpretability, ability to incorporate correlated input and applicability to large scale problems will be discussed.

Generalized solutions to semilinear hyperbolic systems

This article is devoted to a variety of applications of fuzzy models in civil engineering, presenting current work of a group of researchers at the University of Innsbruck. With fuzzy methods and possibility theory as an encompassing framework, the following areas are addressed: uncertainties in geotechnical engineering, fuzzy finite element computation of a foundation raft, fuzzy dynamical systems, processing uncertainty in project scheduling and cost planning.

Group analysis of differential equations and generalized functions

In this article semilinear hyperbolic first order systems in two variables are considered, whose nonlinearity satisfies a global Lipschitz condition. It is shown that these systems admit unique global solutions in the Colombeau algebraG(ℝ2). In particular, this provides unique generalized solutions for arbitrary distributions as initial data. The solution inG(ℝ2) is shown to be consistent with the locally integrable or the distributional solutions, when they exist.

Generalized solutions to partial differential equations of evolution type

We present an extension of the methods of classical Lie group analysis of differential equations to equations involving generalized functions (in particular: distributions). A suitable framework for such a generalization is provided by Colombeaus theory of algebras of generalized functions. We show that under some mild conditions on the differential equations, symmetries of classical solutions remain symmetries for generalized solutions. Moreover, we introduce a generalization of the infinitesimal methods of group analysis that allows us to compute symmetries of linear and nonlinear differential equations containing generalized function terms. Thereby, the group generators and group actions may be given by generalized functions themselves.