Cubature Formulas for Multisymmetric Functions and Applications to Stochastic Partial Differential Equations

Abstract

The numerical solution of stochastic partial differential equations and numerical Bayesian estimation is computationally demanding. If the coefficients in a stochastic partial differential equation exhibit symmetries, they can be exploited to reduce the computational effort. To do so, we show that permutation-invariant functions can be approximated by permutation-invariant polynomials in the space of continuous functions as well as in the space of $p$-integrable functions defined on $[0,1]^s$ for $1\leq p < \infty$. We proceed to develop a numerical strategy to compute cubature formulas that exploit permutation-invariance properties related to multisymmetry groups in order to reduce computational work. We show that in a certain sense there is no curse of dimensionality if we restrict ourselves to multisymmetric functions, and we provide error bounds for formulas of this type. Finally, we present numerical results, comparing the proposed formulas to other integration techniques that are frequently applied to high-dimensional problems such as quasi-Monte Carlo rules and sparse grids.