Weak error analysis for the stochastic Allen-Cahn equation

@article{Breit2022WeakEA,
  title={Weak error analysis for the stochastic Allen-Cahn equation},
  author={Dominic Breit and Andreas Prohl},
  journal={ArXiv},
  year={2022},
  volume={abs/2210.02051}
}
. We prove strong rate resp. weak rate O ( τ ) for a structure preserving temporal discretization (with τ the step size) of the stochastic Allen-Cahn equation with additive resp. multiplicative colored noise in d = 1 , 2 , 3 dimensions. Direct variational arguments exploit the one-sided Lipschitz property of the cubic nonlinearity in the first setting to settle first order strong rate. It is the same property which allows for uniform bounds for the derivatives of the solution of the related… 

References

SHOWING 1-10 OF 22 REFERENCES

Strong convergence rates of semidiscrete splitting approximations for the stochastic Allen–Cahn equation

This article analyses an explicit temporal splitting numerical scheme for the stochastic Allen–Cahn equation driven by additive noise in a bounded spatial domain with smooth boundary in dimension

Analysis of some splitting schemes for the stochastic Allen-Cahn equation

This work introduces and analyzes an explicit time discretization scheme for the one-dimensional stochastic Allen-Cahn, driven by space-time white noise, and proves boundedness of moments of the numerical solution.

Optimal Error Estimates of Galerkin Finite Element Methods for Stochastic Allen–Cahn Equation with Additive Noise

By introducing two auxiliary approximation processes, an appropriate decomposition of the considered error terms is proposed and a novel approach of error analysis is introduced, to successfully recover the convergence rates of the numerical schemes.

Optimal Strong Rates of Convergence for a Space-Time Discretization of the Stochastic Allen–Cahn Equation with Multiplicative Noise

The fact that 𝒜⁢(x) satisfies a weak monotonicity property is used to deduce uniform bounds in strong norms for solutions of the temporal, as well as of the spatio-temporal discretization of the problem.

Weak approximation of stochastic partial differential equations: the nonlinear case

It is proved that as it is often the case, the weak order of convergence is twice the strong order and Malliavin calculus is a key ingredient in this proof.

Numerical analysis of the Allen-Cahn equation and approximation for mean curvature flows

Optimal order and quasi-optimal order error bounds are shown for the semi-discrete and fully discrete schemes under different constraints on the mesh size h and the time step size k and different regularity assumptions on the initial datum function u0.

On the discretization in time of parabolic stochastic partial differential equations

  • J. Printems
  • Mathematics, Computer Science
    Monte Carlo Methods Appl.
  • 2001
In an abstract framework, results on the order of convergence of a semi-discretization in time by an implicit Euler scheme of a stochastic parabolic equation are generalized.

Strong Solutions for Stochastic Partial Differential Equations of Gradient Type

Second Order Pde's in Finite and Infinite Dimension: A Probabilistic Approach

Kolmogorov equations in Rd with unbounded coefficients.- Asymptotic behaviour of solutions.- Analyticity of the semigroup in a degenerate case.- Smooth dependence on data for the SPDE: the Lipschitz