Approximating Stochastic Partial Differential Equations with Finite Elements: Computation and Analysis
Doctoral thesis, 2019
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
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
Mean-square stability analysis of approximations of stochastic differential equations in infinite dimensions
BIT Numerical Mathematics,;Vol. 57(2017)p. 963-990
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
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