Finite Element Approximation of the Linear Stochastic Wave Equation with Additive Noise

Kovács, Mihály; Larsson, Stig; Saedpanah, Fardin

doi:10.1137/090772241

Cited by 70 publications

(78 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We refer the reader to the introductions of [16] and [5] for relevant literature on the spatial, respectively, temporal, discretization of stochastic (linear) wave equations. Further, the recent publication [22] presents a full discretization of the wave equation with additive noise: a spectral Galerkin approximation is used in space, and an adapted stochastic trigonometric method, using linear functionals of the noise as in [12], is employed in time.…”

Section: Introductionmentioning

confidence: 99%

Full Discretization of Semilinear Stochastic Wave Equations Driven by Multiplicative Noise

Anton¹,

Cohen²,

Larsson³

et al. 2016

SIAM J. Numer. Anal.

Self Cite

View full text Add to dashboard Cite

Abstract.A fully discrete approximation of the semilinear stochastic wave equation driven by multiplicative noise is presented. A standard linear finite element approximation is used in space, and a stochastic trigonometric method is used for the temporal approximation. This explicit time integrator allows for mean-square error bounds independent of the space discretization and thus does not suffer from a step size restriction as in the often used Störmer-Verlet leapfrog scheme. Furthermore, it satisfies an almost trace formula (i.e., a linear drift of the expected value of the energy of the problem). Numerical experiments are presented and confirm the theoretical results.

show abstract

Section: Introductionmentioning

confidence: 99%

Full Discretization of Semilinear Stochastic Wave Equations Driven by Multiplicative Noise

Anton¹,

Cohen²,

Larsson³

et al. 2016

SIAM J. Numer. Anal.

Self Cite

View full text Add to dashboard Cite

show abstract

“…For partial differential equations (PDEs for short) with random coefficients, numerical realization of one MC "sample" entails the numerical solution of one deterministic PDE. Many of such "paths" are required for sufficient accuracy, causing suboptimal efficiency even if optimal algebraic solvers are used (see, e.g., [5,6,7,19,30]). Multi-level versions of MC were introduced, to the authors' knowledge, by M. Giles [20,21] for the numerical solution of Itô stochastic ordinary differential equations, following basic ideas in earlier work by S. Heinrich [27] on numerical quadrature.…”

Section: Introductionmentioning

confidence: 99%

Multilevel Monte Carlo Finite Element Methods for Stochastic Elliptic Variational Inequalities

Kornhuber

Schwab²,

Wolf³

2014

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

Abstract. Multi-Level Monte-Carlo Finite Element (MLMC-FE) methods for the solution of stochastic elliptic variational inequalities are introduced, analyzed, and numerically investigated. Under suitable assumptions on the random diffusion coefficient, the random forcing function, and the deterministic obstacle, we prove existence and uniqueness of solutions of "mean-square" and "pathwise" formulations. Suitable regularity results for deterministic, elliptic obstacle problems lead to uniform pathwise error bounds, providing optimal-order error estimates of the statistical error and upper bounds for the corresponding computational cost for classical Monte-Carlo and novel MLMC-FE methods. Utilizing suitable multigrid solvers for the occurring sample problems, in two space dimensions MLMC-FE methods then provide numerical approximations of the expectation of the random solution with the same order of efficiency as for a corresponding deterministic problem, up to logarithmic terms. Our theoretical findings are illustrated by numerical experiments.

show abstract

“…In the case of partial differential equations (PDEs) with random inputs, a MC method entails for each realization of the stochastic data the numerical solution of a deterministic PDE. For time dependent, parabolic problems driven by noise (see, e.g., [7,8,9,33,26]), numerous "realizations" of the PDE in space-time must be simulated.…”

Section: Introductionmentioning

confidence: 99%

Multilevel Monte Carlo Methods for Stochastic Elliptic Multiscale PDEs

Abdulle¹,

Barth²,

Schwab³

2013

Multiscale Model. Simul.

View full text Add to dashboard Cite

Abstract. In this paper Monte Carlo Finite Element (MC FE) approximations for elliptic homogenization problems with random coefficients which oscillate on n ∈ N a-priori known, separated length scales are considered. The convergence of multilevel MC FE (MLMC FE) discretizations is analyzed. In particular, it is considered that the multilevel FE discretization resolves the finest physical length scale, but the coarsest FE mesh does not, so that the socalled "resonance" case occurs at intermediate MLMC sampling levels. It is proved that switching to an Hierarchic Multiscale Finite Element method such as the Finite Element Heterogeneous Multiscale method (FE-HMM) to compute all MLMC FE samples on meshes which under-resolve the physical length scales implies once more optimal efficiency (in terms of accuracy versus computational work) for the numerical estimates of statistical moments with first and second order FE-HMMs. Specifically, the method proposed here allows to obtain estimates of the expectation of the random solution, with accuracy versus work that is identical to the solution of a single deterministic problem obtained by a FE-HMM, and which is, moreover, robust with respect to the physical length scales. Numerical experiments corroborate our analytical findings.

show abstract

Finite Element Approximation of the Linear Stochastic Wave Equation with Additive Noise

Cited by 70 publications

References 18 publications

Full Discretization of Semilinear Stochastic Wave Equations Driven by Multiplicative Noise

Full Discretization of Semilinear Stochastic Wave Equations Driven by Multiplicative Noise

Multilevel Monte Carlo Finite Element Methods for Stochastic Elliptic Variational Inequalities

Multilevel Monte Carlo Methods for Stochastic Elliptic Multiscale PDEs

Contact Info

Product

Resources

About