Adaptive wavelet methods for a class of stochastic partial differential equations
Loading...
Files
Date
relationships.isAuthorOf
Publisher
Philipps-Universität Marburg
item.page.supervisor-of-thesis
Abstract
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.
Review
Metadata
Contributors
Supervisor:
Dates
Created: 2016Issued: 2016-02-04
Faculty
Fachbereich Mathematik und Informatik
Publisher
Philipps-Universität Marburg
Language
eng
Data types
DoctoralThesis
Keywords
WaveletsAdaptive MethodsBesov-RegularitätAdaptive MethodenBesov-RegularityWavelets
DFG-subjects
EvolutionsgleichungStochastikNumerische Mathematik
DDC-Numbers
510
show more
Kinzel, Stefan: Adaptive wavelet methods for a class of stochastic partial differential equations. : Philipps-Universität Marburg 2016-02-04. DOI: https://doi.org/10.17192/z2016.0058.
License
This item has been published with the following license: In Copyright