Raphael Kruse

Optimal error estimates of Galerkin finite element methods for stochastic partial differential equations with multiplicative noise

We consider Galerkin finite element methods for semilinear sto- chastic partial differential equations (SPDEs) with multiplicative noise and Lipschitz continuous nonlinearities. We analyze the strong error of conver- gence for spatially semidiscrete approximations as well as a spatio-temporal discretization which is based on a linear implicit Euler-Maruyama method. In both cases we obtain optimal error estimates. The proofs are based on sharp integral versions of well-known error es- timates for the corresponding deterministic linear homogeneous equation to- gether with optimal regularity results for the mild solution of the SPDE. The results hold for different Galerkin methods such as the standard finite element method or spectral Galerkin approximations.

Strong and Weak Approximation of Semilinear Stochastic Evolution Equations

Introduction.- Stochastic Evolution Equations in Hilbert Spaces.- Optimal Strong Error Estimates for Galerkin Finite Element Methods.- A Short Review of the Malliavin Calculus in Hilbert Spaces.- A Malliavin Calculus Approach to Weak Convergence.- Numerical Experiments.- Some Useful Variations of Gronwalls Lemma.- Results on Semigroups and their Infinitesimal Generators.- A Generalized Version of Lebesgues Theorem.- References.- Index.

Duality in refined Sobolev–Malliavin spaces and weak approximation of SPDE

We introduce a new family of refined Sobolev–Malliavin spaces that capture the integrability in time of the Malliavin derivative. We consider duality in these spaces and derive a Burkholder type inequality in a dual norm. The theory we develop allows us to prove weak convergence with essentially optimal rate for numerical approximations in space and time of semilinear parabolic stochastic evolution equations driven by Gaussian additive noise. In particular, we combine a standard Galerkin finite element method with backward Euler timestepping. The method of proof does not rely on the use of the Kolmogorov equation or the Itō formula and is therefore non-Markovian in nature. Test functions satisfying polynomial growth and mild smoothness assumptions are allowed, meaning in particular that we prove convergence of arbitrary moments with essentially optimal rate.

Stochastic C-Stability and B-Consistency of Explicit and Implicit Euler-Type Schemes

This paper is concerned with the numerical approximation of stochastic ordinary differential equations, which satisfy a global monotonicity condition. This condition includes several equations with super-linearly growing drift and diffusion coefficient functions such as the stochastic Ginzburg–Landau equation and the 3/2-volatility model from mathematical finance. Our analysis of the mean-square error of convergence is based on a suitable generalization of the notions of C-stability and B-consistency known from deterministic numerical analysis for stiff ordinary differential equations. An important feature of our stability concept is that it does not rely on the availability of higher moment bounds of the numerical one-step scheme. While the convergence theorem is derived in a somewhat more abstract framework, this paper also contains two more concrete examples of stochastically C-stable numerical one-step schemes: the split-step backward Euler method from Higham et al. (SIAM J Numer Anal 40(3):1041–1063, 2002) and a newly proposed explicit variant of the Euler–Maruyama scheme, the so called projected Euler–Maruyama method. For both methods the optimal rate of strong convergence is proven theoretically and verified in a series of numerical experiments.

Characterization of bistability for stochastic multistep methods

We propose a numerical method for computing all eigenvalues (and the corresponding eigenvectors) of a nonlinear holomorphic eigenvalue problem that lie within a given contour in the complex plane. The method uses complex integrals of the resolvent operator, applied to at least k column vectors, where k is the number of eigenvalues inside the contour. The theorem of Keldysh is employed to show that the original nonlinear eigenvalue problem reduces to a linear eigenvalue problem of dimension k. No initial approximations of eigenvalues and eigenvectors are needed. The method is particularly suitable for moderately large eigenvalue problems where k is much smaller than the matrix dimension. We also give an extension of the method to the case where k is larger than the matrix dimension. The quadrature errors caused by the trapezoid sum are discussed for the case of analytic closed contours. Using well known techniques it is shown that the error decays exponentially with an exponent given by the product of the number of quadrature points and the minimal distance of the eigenvalues to the contour.

Mean-square convergence of the BDF2-Maruyama and backward Euler schemes for SDE satisfying a global monotonicity condition

In this paper the numerical approximation of stochastic differential equations satisfying a global monotonicity condition is studied. The strong rate of convergence with respect to the mean square norm is determined to be

A discrete stochastic Gronwall lemma

\frac{1}{2}

Stochastic Evolution Equations in Hilbert Spaces

12 for the two-step BDF-Maruyama scheme and for the backward Euler–Maruyama method. In particular, this is the first paper which proves a strong convergence rate for a multi-step method applied to equations with possibly superlinearly growing drift and diffusion coefficient functions. We also present numerical experiments for the