Approximating Stochastic Partial Differential Equations with Finite Elements: Computation and Analysis
Doctoral thesis, 2019

Stochastic partial differential equations (SPDE) must be approximated in space and time to allow for the simulation of their solutions. In this thesis fully discrete approximations of such equations are considered, with an emphasis on finite element methods combined with rational semigroup approximations.

A quantity of interest for SPDE simulations often takes the form of an expected value of a functional applied to the solution. This is the major theme of this thesis, which divides into two minor themes. The first is how to analyze the error resulting from the fully discrete approximation of an SPDE with respect to a given functional, which is referred to as the weak error of the approximation. The second is how to efficiently compute the quantity of interest as well as the weak error itself. The Monte Carlo (MC) and multilevel Monte Carlo (MLMC) methods are common approaches for this.

The thesis consists of five papers. In the first paper the additional error caused by MC and MLMC methods in simulations of the weak error is analyzed. Upper and lower bounds are derived for the different methods and simulations illustrate the results. The second paper sets up a framework for the analysis of the asymptotic mean square stability, the stability as measured in a quadratic functional, of a general stochastic recursion scheme, which is applied to several discretizations of an SPDE. In the third paper, a novel technique for efficiently generating samples of SPDE approximations is introduced, based on the computation of discrete covariance operators. The computational complexities of the resulting MC and MLMC methods are analyzed. The fourth paper considers the analysis of the weak error for the approximation of the semilinear stochastic wave equation. In the fifth paper, a Lyapunov equation is derived, which allows for the deterministic approximation of the expected value of a quadratic functional applied to the solution of an SPDE. The paper also includes an error analysis of an approximation of this equation and an analysis of the weak error, with respect to the quadratic functional, of an approximation of the considered SPDE.

Lévy process

Lyapunov equation

white noise

finite element method

multilevel Monte Carlo

Monte Carlo

multiplicative noise

asymptotic mean square stability

stochastic heat equation

covariance operator

weak convergence

generalized Wiener process

numerical approximation

stochastic wave equation

Stochastic partial differential equations

Pascal, Chalmers tvärgata 3
Opponent: Prof. Gabriel J. Lord, Department of Mathematics, Radboud University, Netherlands

Author

Andreas Petersson

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

Monte Carlo versus multilevel Monte Carlo in weak error simulations of SPDE approximations

Mathematics and Computers in Simulation,;Vol. 143(2018)p. 99-113

Journal article

Petersson, A. Rapid covariance-based sampling of linear SPDE approximations in the multilevel Monte Carlo method

Kovács, M., Lang, A., Petersson, A. Weak convergence of fully discrete finite element approximations of semilinear hyperbolic SPDE with additive noise

Andersson, A., Lang, A., Petersson, A., Schroer, L. Finite element approximation of {L}yapunov equations for the computation of quadratic functionals of SPDE

Matematiska ekvationer används av forskare för att förstå och förutsäga världen omkring oss. Ett vanligt exempel är partiella differentialekvationer, som bland annat används för att beskriva utveckling i tid och rum. De kan beskriva väder, aktiemarknaden, vågrörelser, värmespridning, med mera. Dessa fenomen kan involvera en stor del osäkerhet, eller slump. Alla variabler som ingår ekvationen kanske inte kan mätas, eller så finns det osäkerhet kring hur sambanden dem emellan ser ut. Om storleken eller strukturen på osäkerheten är någorlunda känd så kan den dock modelleras matematiskt vilket leder till slumpmässiga, eller stokastiska, partiella differentialekvationer. Dessa kan ge bättre förståelse av de fenomen som modelleras men kan vanligen inte lösas exakt utan måste approximeras med komplexa datoralgoritmer.

I denna avhandling studeras matematiska frågor som uppstår när stokastiska partiella differentialekvationer approximeras med den så kallade finita elementmetoden. Hur litet kommer approximationsfelet att bli och hur mäts det på bästa sätt? Är algoritmerna stabila, dvs. kan vi vara säkra på att små förändringar i indata inte resulterar i stora förändringar i utdata? Hur kan snabbare algoritmer designas om vi bara är intresserade av vissa egenskaper hos lösningen? Med verktyg från matematiska fält såsom sannolikhetsteori och funktionalanalys svarar artiklarna som utgör avhandlingen på delar av dessa frågor så att rätt algoritm kan väljas innan den översätts till datorkod.

Approximation and simulation of Lévy-driven SPDE

Swedish Research Council (VR) (2014-3995), 2015-01-01 -- 2018-12-31.

Subject Categories

Computational Mathematics

Probability Theory and Statistics

Mathematical Analysis

ISBN

978-91-7905-206-5

Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie: 4673

Publisher

Chalmers

Pascal, Chalmers tvärgata 3

Opponent: Prof. Gabriel J. Lord, Department of Mathematics, Radboud University, Netherlands

More information

Latest update

11/1/2019