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**Harvard**

Lindgren, F. (2012) *On weak and strong convergence of numerical approximations of stochastic partial differential equations*. Göteborg : Chalmers University of Technology (Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie, nr: 3468).

** BibTeX **

@book{

Lindgren2012,

author={Lindgren, Fredrik},

title={On weak and strong convergence of numerical approximations of stochastic partial differential equations},

isbn={978-91-7385-787-1},

abstract={This thesis is concerned with numerical approximation of linear stochastic partial
differential equations driven by additive noise. In the first part, we develop a
framework for the analysis of weak convergence and within this framework we
analyze the stochastic heat equation, the stochastic wave equation, and the linearized
stochastic Cahn-Hilliard, or the linearized Cahn-Hilliard-Cook equation.
The general rule of thumb, that the rate of weak convergence is twice the rate of
strong convergence, is confirmed.
In the second part, we investigate various ways to approximate the driving
noise and analyze its effect on the rate of strong convergence. First, we consider
the use of frames to represent the noise. We show that if the frame is chosen in a
way that is well suited for the covariance operator, then the number of elements
of the frame needed to represent the noise without effecting the overall convergence
rate of the numerical method may be quite low. Second, we investigate the
use of finite element approximations of the eigenpairs of the covariance operator.
It turns out that if the kernel of the operator is smooth, then the number of basis
functions needed may be substantially reduced.
Our analysis is done in a framework based on operator semigroups. It is performed
in a way that reduces our results to results about approximation of the
respective (deterministic) semigroup.},

publisher={Institutionen för matematiska vetenskaper, matematik, Chalmers tekniska högskola,},

place={Göteborg},

year={2012},

series={Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie, no: 3468},

keywords={Additive noise, Cahn-Hilliard-Cook equation, Error estimate, Finite element, Hyperbolic equation, Parabolic equation, Rational approximation, Stochastic partial differential equation, Strong convergence, Truncation, Wiener process, Weak convergence},

note={51},

}

** RefWorks **

RT Dissertation/Thesis

SR Electronic

ID 166946

A1 Lindgren, Fredrik

T1 On weak and strong convergence of numerical approximations of stochastic partial differential equations

YR 2012

SN 978-91-7385-787-1

AB This thesis is concerned with numerical approximation of linear stochastic partial
differential equations driven by additive noise. In the first part, we develop a
framework for the analysis of weak convergence and within this framework we
analyze the stochastic heat equation, the stochastic wave equation, and the linearized
stochastic Cahn-Hilliard, or the linearized Cahn-Hilliard-Cook equation.
The general rule of thumb, that the rate of weak convergence is twice the rate of
strong convergence, is confirmed.
In the second part, we investigate various ways to approximate the driving
noise and analyze its effect on the rate of strong convergence. First, we consider
the use of frames to represent the noise. We show that if the frame is chosen in a
way that is well suited for the covariance operator, then the number of elements
of the frame needed to represent the noise without effecting the overall convergence
rate of the numerical method may be quite low. Second, we investigate the
use of finite element approximations of the eigenpairs of the covariance operator.
It turns out that if the kernel of the operator is smooth, then the number of basis
functions needed may be substantially reduced.
Our analysis is done in a framework based on operator semigroups. It is performed
in a way that reduces our results to results about approximation of the
respective (deterministic) semigroup.

PB Institutionen för matematiska vetenskaper, matematik, Chalmers tekniska högskola,

T3 Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie, no: 3468

LA eng

LK http://publications.lib.chalmers.se/records/fulltext/166946.pdf

OL 30