Polynomial-Chaos Based Methods for Differential Algebraic Equations with Random Parameters

Abstract

Mathematical modelling of technical applications often yields systems of differential algebraic equations (DAEs), for example, in the simulation of electric circuits or mechanical multibody problems. Imperfections of a manufacturing procedure cause undesired variations in the produced devices. These variations can be taken as uncertainties of physical parameters in a DAE model. We replace the varying parameters by random variables to achieve an uncertainty quantification. The time-dependent solution of the DAEs becomes a random process, which is expanded into a series of the polynomial chaos. We can use either a stochastic Galerkin method or a stochastic collocation technique to determine the unknown coefficient functions. The Galerkin method yields a larger coupled system to be solved once, whereas the collocation approach requires to solve the original systems many times. We present numerical simulations of an illustrative example from electrical engineering.