On Wavelet-Galerkin Methods for Semilinear Parabolic Equations with Additive Noise

@inproceedings{Kovcs2012OnWM,
  title={On Wavelet-Galerkin Methods for Semilinear Parabolic Equations with Additive Noise},
  author={Mih{\'a}ly Kov{\'a}cs and Stig Larsson and Karsten Urban},
  year={2012}
}
We consider the semilinear stochastic heat equation perturbed by additive noise. After time-discretization by Euler’s method the equation is split into a linear stochastic equation and a non-linear random evolution equation. The linear stochastic equation is discretized in space by a non-adaptive wavelet-Galerkin method. This equation is solved first and its solution is substituted into the nonlinear random evolution equation, which is solved by an adaptive wavelet method. We provide mean… 

On the Convergence Analysis of the Inexact Linearly Implicit Euler Scheme for a Class of Stochastic Partial Differential Equations

This paper is concerned with the adaptive numerical treatment of stochastic partial differential equations. Our method of choice is Rothe’s method. We use the implicit Euler scheme for the time

On the Convergence Analysis of the Inexact Linearly Implicit Euler Scheme for a Class of Stochastic Partial Differential Equations

This paper is concerned with the adaptive numerical treatment of stochastic partial differential equations. Our method of choice is Rothe’s method. We use the implicit Euler scheme for the time

On the Backward Euler Approximation of the Stochastic Allen-Cahn Equation

It is proved that the scheme converges uniformly in the strong Lp-sense but with no rate given, and it is shown that the method converges pathwise with a rate O(∆tγ) for any γ < 1 2 .

Adaptive Wavelet Schwarz Methods for the Navier-Stokes Equation

An adaptive additive Schwarz method based on discretization by means of a divergence-free wavelet frame is constructed and it is proved that the method is convergent and asymptotically optimal with respect to the degrees of freedom involved.

On the convergence analysis of the inexact linearly implicit Euler scheme for a class of SPDEs

This paper is concerned with the adaptive numerical treatment of stochastic partial differential equations. Our method of choice is Rothe's method. We use the implicit Euler scheme for the time

A Review of Wavelets Solution to Stochastic Heat Equation with Random Inputs

We consider a wavelet-based solution to the stochastic heat equation with random inputs. Computational methods based on the wavelet transform are analyzed for solving three types of stochastic heat

Numerical solution of nonlinear SPDEs using a multi-scale method

A new numerical method for solving a class of stochastic partial differential equations (SPDEs) based on B-splines wavelets based on implicit collocation with the multi-scale method is established.

Regularity theory for a new class of fractional parabolic stochastic evolution equations

. A new class of fractional-order stochastic evolution equations of the form ( ∂ t + A ) γ X ( t ) = ˙ W Q ( t ), t ∈ [0 ,T ], γ ∈ (0 , ∞ ), is introduced, where − A generates a C 0 -semigroup on a

Multilevel Monte Carlo Quadrature of Discontinuous Payoffs in the Generalized Heston Model Using Malliavin Integration by Parts

An integration by parts formula for the quadrature of discontinuous payoffs in a multidimensional Heston model is established and allows us to construct efficient multilevel Monte Carlo estimators.

Multilevel preconditioning for sparse optimization of functionals with nonconvex fidelity terms

It is shown that under certain boundedness and contraction conditions the resulting algorithm is linearly convergent to a global minimizer and that the iteration is monotone with respect to the Tikhonov functional.

References

SHOWING 1-10 OF 30 REFERENCES

Weak convergence of finite element approximations of linear stochastic evolution equations with additive noise

A unified approach is given for the analysis of the weak error of spatially semidiscrete finite element methods for linear stochastic partial differential equations driven by additive noise. An error

Strong convergence of the finite element method with truncated noise for semilinear parabolic stochastic equations with additive noise

It is shown that the orthogonal expansion of the finite-dimensional Wiener process, that appears in the discretized problem, can be truncated severely without losing the asymptotic order of the method, provided that the kernel of the covariance operator of theWiener process is smooth enough.

Adaptive wavelet methods for the stochastic Poisson equation

We study the Besov regularity as well as linear and nonlinear approximation of random functions on bounded Lipschitz domains in ℝd. The random functions are given either (i) explicitly in terms of a

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.

Spatial approximation of stochastic convolutions

Spatial Besov Regularity for Stochastic Partial Differential Equations on Lipschitz Domains

We use the scale of Besov spaces B^\alpha_{\tau,\tau}(O), \alpha>0, 1/\tau=\alpha/d+1/p, p fixed, to study the spatial regularity of the solutions of linear parabolic stochastic partial differential

On the convergence analysis of Rothe ’ s method

This paper is about the convergence analysis of the horizontal method of lines for deterministic and stochastic parabolic evolution equations. We use uniform discretizations in time and nonuniform

Adaptive Wavelet Schemes for Nonlinear Variational Problems

This work develops and analyzes wavelet based adaptive schemes for nonlinear variational problems and proves asymptotically optimal complexity for adaptive realizations of first order iterations and of Newton's method.

Wavelet Methods for Elliptic Partial Differential Equations

1. Introduction 2. Mulitscale Approximation and Multiresolution 3. Elliptic Boundary Value Problems 4. Multiresolution Galerkin Methods 5. Wavelets 6. Wavelet-Galerkin Methods 7. Adaptive Wavelet

Adaptive wavelet methods for elliptic operator equations: Convergence rates

The main result of the paper is the construction of an adaptive scheme which produces an approximation to u with error O(N -s ) in the energy norm, whenever such a rate is possible by N-term approximation.