Polynomial Chaos Methods

Abstract

In this chapter we review methods for formulating partial differential equations based on the random field representations outlined in Chap. 2. These include the stochastic Galerkin method, which is the predominant choice in this book, as well as other methods that frequently occur in the literature. We also briefly discuss methods that are not polynomial chaos methods themselves but are viable alternatives. Intrusive MethodsIn the context of gPC, problem formulations result in a new set of equations that are distinctly different from the original set of equations and thus require the design of new numerical solvers. These solvers are referred to as intrusive methodsas opposed to non-intrusive stochastic methods that exclusively rely on existing deterministic codes. Stochastic Galerkin MethodsThe stochastic Galerkin method was introduced by Ghanem and Spanos in order to solve linear stochastic equations [11]. It relies on a weak problem formulation where the set of solution basis functions (trial functions) is the same as the space of stochastic test functions. Consider a general scalar conservation law defined on a spatial domain˝x with boundary x subject to initial and boundary conditions, given by @u.x; t; / @t C @f .u.

A Geometric Approach to Dynamical Model Order Reduction

“…Many methods, such as polynomial chaos expansions [82], Fourier decomposition [79], or Proper Orthogonal Decomposition [30] rely on the choice of a predefined, timeindependent orthonormal basis either for the modes, (u i ), or the coefficients, (ζ i ), and obtain equations for the respective unknown coefficients or modes by Galerkin projection [58]. However, the use of modes and coefficients that are simultaneously dynamic has been shown to be efficient [44,45].…”