Bernhard Schmelzer

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

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.

On solutions of stochastic differential equations with parameters modeled by random sets

We consider ordinary stochastic differential equations whose coefficients depend on parameters. After giving conditions under which the solution processes continuously depend on the parameters random compact sets are used to model the parameter uncertainty. This leads to continuous set-valued stochastic processes whose properties are investigated. Furthermore, we define analogues of first entrance times for set-valued processes called first entrance and inclusion times. The theoretical concept is applied to a simple example from mechanics.

Efficiency of tuned mass dampers with uncertain parameters on the performance of structures under stochastic excitation

Abstract Tuned mass dampers serve the purpose of damping vibrations of structures such as earthquake-induced vibrations. In their design, two types of uncertainty are relevant: the stochastic excitation (e.g. earthquake record) and the inherent uncertainty of internal parameters of the devices themselves. This paper presents a new framework that admits the combination of stochastic processes and interval-type parameter uncertainty, modelled by random sets. The approach is applied to show how the efficiency of tuned mass dampers can be realistically assessed in the presence of uncertainty.

Joint distributions of random sets and their relation to copulas

Random sets are set-valued random variables. They have been applied in various fields like stochastic geometry, statistics, economics, engineering or computer science, and are often used for modeling uncertainty. In an earlier paper the author has defined joint capacity and joint containment functionals which are multivariate set functions describing the joint distribution of random sets. This paper is concerned with the question how copulas can be used to describe or model the dependence of random sets. It is demonstrated that a joint containment functional can be related to its margins by a family of copulas. Furthermore, the paper provides a first insight how copulas can be used to define joint containment functionals.

Sklar's theorem for minitive belief functions

In a former paper, the author has investigated how copulas can be used to express the dependence relation between two random sets. It has been proven that a joint belief function is related to its marginal belief functions by a family of copulas and that, in general, a single copula is not sufficient. In this paper the results are investigated under the assumption that the involved belief functions are minitive which corresponds to the important case where the associated random sets are consonant. It is proven that under this additional assumption a single copula is sufficient to express the dependence relation. In other words, this means that Sklars theorem remains valid if joint and marginal distribution functions are replaced by joint and marginal minitive belief functions. The dependence relation between random sets can be expressed by a family of copulas.For consonant random sets one copula is sufficient.The approach only makes use of the belief functions, no reference to specific random set representations is needed.The relation to other approaches is discussed.

Characterizing joint distributions of random sets by multivariate capacities

By the Choquet theorem, distributions of random closed sets can be characterized by a certain class of set functions called capacity functionals. In this paper a generalization to the multivariate case is presented, that is, it is proved that the joint distribution of finitely many random sets can be characterized by a multivariate set function being completely alternating in each component, or alternatively, by a capacity functional defined on complements of cylindrical sets. For the special case of finite spaces a multivariate version of the Moebius inversion formula is derived. Furthermore, we use this result to formulate an existence theorem for set-valued stochastic processes.

Seismic Performance of Tuned Mass Dampers with Uncertain Parameters

This chapter addresses the seismic performance of Tuned Mass Dampers (TMDs). In the design of a TMD, two types of uncertainty are relevant: the stochastic excitation modeling the earthquake, and the inherent uncertainty of internal parameters of the damping device and the subsoil. Modeling the excitation by a continuous-time stochastic process the structure-damper system can be described by a linear system of stochastic differential equations. The response is a stochastic process depending on the uncertain parameters of the damping device and the subsoil. These uncertainties are modeled by random sets, i.e., interval-valued random variables. A framework is presented here that admits the combination of these two types of uncertainty leading to a set-valued stochastic process, which is interpreted as containing the true system response. The approach is applied to show how the efficiency of TMDs can be realistically assessed in the presence of uncertainty. The main focus of this paper is on non-stationary models for the excitation based on colored noise multiplied by a prescribed intensity function.

Set-Valued Stochastic Processes and Sets of Probability Measures Induced by Stochastic Differential Equations with Random Set Parameters

We consider stochastic differential equations depending on parameters whose uncertainty is modeled by random compact sets. Several approaches are discussed how to construct set-valued processes from the solutions. The induced lower and upper probabilities are compared to a set of probability measures constructed from the distributions of the solutions and the selections of the random set.

Sensitivity analysis through random and fuzzy sets

Sensitivity analysis has become a major tool in the assessment of the reliability of engineering structures. Given an input-output system, the question is which input variables have the most decisive influence on the output. Random and/or fuzzy sets offer a framework for modelling the data variability. Propagating random set data or fuzzy set data through a deterministic system provides a valuable impression of the output variability. The sensitivity can be assessed by pinching individual variables, changing their correlations or by varying their degree of interactivity. An important ingredient in the quantification of the changes derives from generalized information theory, namely, measures of nonspecificity, in particular, Hartley-like measures. The purpose of this contribution is to present an investigation of various methods of modelling correlations and interactivity, quantifying the results by Hartley-like measures and exhibiting a number of concrete applications in engineering.