Dokument

Summary:
An abstract interpretation of Rothe’s method for the discretization of evolution equations is derived. The error propagation is analyzed and condition on the tolerances are proven, which ensure convergence in the case of inexact operator evaluations. Substantiating the abstract analysis, the linearly implicit Euler scheme on a uniform time discretization is applied to a class of semi-linear parabolic stochastic partial differential equations. Using the existence of asymptotically optimal adaptive solver for the elliptic subproblems, sufficient conditions for convergence with corresponding convergence orders also in the case of inexact operator evaluations are shown. Upper complexity bounds are proven in the deterministic case. The stochastic Poisson equation with random right hand sides is used as model equation for the elliptic subproblems. The random right hand sides are introduced based on wavelet decompositions and a stochastic model that, as is shown, provides an explicit regularity control of their realizations and induces sparsity of the wavelet coefficients. For this class of equations, upper error bounds for best N-term wavelet approximation on different bounded domains are proven. They show that the use of nonlinear (adaptive) methods over uniform linear methods is justified whenever sparsity is present, which in particularly holds true on Lipschitz domains of two or three dimensions. By providing sparse variants of general Gaussian random functions, the class of random functions derived from the stochastic model is interesting on its own. The regularity of the random functions is analyzed in certain smoothness spaces, as well as linear and nonlinear approximation results are proven, which clarify their applicability for numerical experiments.

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