# The effect of finite element discretization on the stationary distribution of SPDEs

@article{Voss2012TheEO, title={The effect of finite element discretization on the stationary distribution of SPDEs}, author={Jochen Voss}, journal={Communications in Mathematical Sciences}, year={2012}, volume={10}, pages={1143-1159} }

This article studies the effect of discretisation error on the stationary distribution of stochastic partial differential equations (SPDEs). We restrict the analysis to the effect of space discretisation, performed by finite element schemes. The main result is that under appropriate assumptions the stationary distribution of the finite element discretisation converges in total variation norm to the stationary distribution of the full SPDE.

## Figures from this paper

## 4 Citations

### Accurate stationary densities with partitioned numerical methods for stochastic partial differential equations

- Computer Science
- 2014

The reverse leapfrog method and stochastic Runge–Kutta Leapfrog methods are introduced, their performance applied to linear SPDEs are analysed and numerical experiments are performed to examine their accuracy applied to a type of nonlinear SPDE.

### An Introduction to Computational Stochastic PDEs

- Computer Science
- 2014

This book offers graduate students and researchers powerful tools for understanding uncertainty quantification for risk analysis and theory is developed in tandem with state-of-the art computational methods through worked examples, exercises, theorems and proofs.

### A Computational Framework for Infinite-Dimensional Bayesian Inverse Problems Part I: The Linearized Case, with Application to Global Seismic Inversion

- MathematicsSIAM J. Sci. Comput.
- 2013

A computational framework for estimating the uncertainty in the numerical solution of linearized infinite-dimensional statistical inverse problems that incorporates algorithms for manipulating the prior, constructing a low rank approximation of the data-informed component of the posterior covariance operator, and exploring the posterior that together ensure scalability of the entire framework to very high parameter dimensions.

### Stochastic Dynamics of $$\phi ^4$$ Kinks: Numerics and Analysis

- PhysicsNonlinear Systems and Complexity
- 2019

The stationary density of the overdamped \(\phi ^4\) SPDE corresponds to a mean number of kinks and antikinks that is maintained by a balance between nucleation of new kink-antikink pairs and…

## References

SHOWING 1-10 OF 24 REFERENCES

### On implicit and explicit discretization schemes for parabolic SPDEs in any dimension

- Computer Science
- 2005

### Finite Element Approximation of Stochastic Partial Differential Equations driven by Poisson Random Measures of Jump Type

- MathematicsSIAM J. Numer. Anal.
- 2007

Stochastic partial differential equations driven by Poisson random measures of jump type and their numerical approximation are solved and the accuracy of space and time approximation is investigated.

### Higher Order Pathwise Numerical Approximations of SPDEs with Additive Noise

- Mathematics, Computer ScienceSIAM J. Numer. Anal.
- 2011

A new numerical scheme for the time and space discretization of semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive noise is proposed, and this scheme is shown to converge for such SPDEs faster than standard numerical schemes.

### Rate of Convergence of Space Time Approximations for Stochastic Evolution Equations

- Mathematics
- 2007

Stochastic evolution equations in Banach spaces with unbounded nonlinear drift and diffusion operators driven by a finite dimensional Brownian motion are considered. Under some regularity condition…

### Finite Element Methods for Parabolic Stochastic PDE’s

- Mathematics
- 2005

Abstract
We study the rate of convergence of some explicit and implicit numerical schemes for the solution of a parabolic stochastic partial differential equation driven by white noise. These include…

### Numerical solution of partial differential equations by the finite element method

- Mathematics
- 1988

Professor Johnson presents an easily accessible introduction to one of the most important methods used to solve partial differential equations. The bulk of the text focuses on linear problems,…

### ANALYSIS OF SPDES ARISING IN PATH SAMPLING PART II: THE NONLINEAR CASE

- Mathematics
- 2007

In many applications, it is important to be able to sample paths of SDEs conditional on observations of various kinds. This paper studies SPDEs which solve such sampling problems. The SPDE may be…

### Analysis of SPDEs arising in path sampling. Part I: The Gaussian case

- MathematicsCommunications in Mathematical Sciences
- 2005

In many applications it is important to be able to sample paths of SDEs conditional on observations of various kinds. This paper studies SPDEs which solve such sampling problems. The SPDE may be…