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Adaptive wavelet methods for a class of stochastic partial differential equations
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 line...
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| Format: | Doctoral Thesis |
| Language: | English |
| Published: |
Philipps-Universität Marburg
2016
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| Subjects: | |
| Online Access: | PDF Full Text |
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| 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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| DOI: | 10.17192/z2016.0058 |