Adaptive wavelet methods for the stochastic Poisson equation (Q1759587)

This paper proposes numerical algorithms for the Poisson equation \(-\Delta U=X\) in a bounded Lipschitz domain \(D\subset \mathbb{R}^d\) with Dirichlet's boundary condition \(U=0\) on \(\partial D\) and \(X\) a random function with values in \(L^2(D)\). The field \(X\) is defined in terms of a stochastic wavelet expansion and its smoothness along Sobolev and Besov spaces is controlled by two parameters \(\alpha\) and \(\beta\), where \(\beta\) is a sparsity parameter. Efficient algorithms for the nonlinear approximation of the random functions \(X\) and \(U\) are constructed. Suitable adaptive wavelet algorithms achieve the nonlinear approximation of \(U\) at a computational cost that is proportional to the degrees of freedom. Numerical experiments are presented to complement the asymptotic error analysis.