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Adaptive wavelet methods for the stochastic Poisson equation. (English) Zbl 1260.65003

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.

MSC:

65C30 Numerical solutions to stochastic differential and integral equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
65T60 Numerical methods for wavelets
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