Leonhard, Claudine (2016) Derivative-Free Numerical Schemes for Stochastic Partial Differential Equations. (PhD/ Doctoral thesis), University of Lübeck, Lübeck , 126 pp.

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Abstract

Analytical solutions to stochastic partial differential equations are in most cases not explicitly
computable. Therefore, the goal of this work is to derive numerical schemes to solve these
equations. Particularly, we focus on schemes with higher orders of convergence that are free
of derivatives. These numerical methods involve, in general, less computational effort with the
same high order of convergence compared to schemes that include the derivative of the diffusion
operator.
The convergence of the numerical schemes is proved analytically and the computational costs of
these schemes are analyzed. We derive the effective order of convergence for various schemes by
minimizing the mean-square error given some fixed computational cost. In general, this number
is higher for the derivative-free methods.
In the approximation of stochastic partial differential equations that are not commutative, it-
erated stochastic integrals have to be simulated. We present and analyze numerical schemes to
complete this task. These schemes are incorporated into a derivative-free numerical method to
approximate the mild solution of such equations. Moreover, the effective order of convergence of
these algorithms is derived and compared to the order of established approximation methods.
The theoretical results are illustrated and confirmed with numerical simulations for both types
of equation.

Document Type:	Thesis (PhD/ Doctoral thesis)
Research affiliation:	Kiel University Kiel University > Kiel Marine Science OceanRep > The Future Ocean - Cluster of Excellence
Open Access Journal?:	Yes
Projects:	Future Ocean
Date Deposited:	10 Oct 2017 14:33
Last Modified:	23 Sep 2019 23:58
URI:	https://oceanrep.geomar.de/id/eprint/39848

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