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On WaveletGalerkin Methods for Semilinear Parabolic Equations with Additive Noise
"... Abstract We consider the semilinear stochastic heat equation perturbed by additive noise. After timediscretization by Euler’s method the equation is split into a linear stochastic equation and a nonlinear random evolution equation. The linear stochastic equation is discretized in space by a nonad ..."
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Abstract We consider the semilinear stochastic heat equation perturbed by additive noise. After timediscretization by Euler’s method the equation is split into a linear stochastic equation and a nonlinear random evolution equation. The linear stochastic equation is discretized in space by a nonadaptive waveletGalerkin method. This equation is solved first and its solution is substituted into the nonlinear random evolution equation, which is solved by an adaptive wavelet method. We provide mean square estimates for the overall error. 1
Pathwise Hölder convergence of the implicit Euler scheme for semilinear SPDEs with multiplicative noise
, 2013
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DUALITY IN REFINED SOBOLEVMALLIAVIN SPACES AND WEAK APPROXIMATION OF SPDE
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CONSISTENCY AND STABILITY OF A MILSTEINGALERKIN FINITE ELEMENT SCHEME FOR SEMILINEAR SPDE
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Symplectic RungeKutta Semidiscretization for Stochastic Schrödinger Equation ∗
, 2014
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Simulation of SPDE’s for Excitable Media using Finite Elements
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On Weak Convergence, Malliavin Calculus and Kolmogorov Equa tions in Infinite Dimensions
"... This thesis is focused around weak convergence analysis of approximations of stochastic evolution equations in Hilbert space. This is a class of problems, which is sufficiently challenging to motivate new theoretical developments in stochastic analysis. The first paper of the thesis further develo ..."
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This thesis is focused around weak convergence analysis of approximations of stochastic evolution equations in Hilbert space. This is a class of problems, which is sufficiently challenging to motivate new theoretical developments in stochastic analysis. The first paper of the thesis further develops a known approach to weak convergence based on techniques from the Markov theory for the stochastic heat equation, such as the transition semigroup, Kolmogorov’s equation, and also integration by parts from the Malliavin calculus. The thesis then introduces a novel approach to weak convergence analysis, which relies on a duality argument in a Gelfand triple of refined SobolevMalliavin spaces. These spaces are introduced and a duality theory is developed for them. The family of refined SobolevMalliavin spaces contains the classical SobolevMalliavin spaces of Malliavin calculus as a special case. The novel approach is applied to the approximation in space and time of semilinear parabolic stochastic partial differential equations and to stochastic Volterra integrodifferential equations. The solutions to the latter type of equations are not Markov processes, and therefore classical proof