Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients

Abstract

Strong convergence rates for (temporal, spatial, and noise) numerical approximations of semilinear stochastic evolution equations (SEEs) with smooth and regular nonlinearities are well understood in the scientific literature. Weak convergence rates for numerical approximations of such SEEs have been investigated for about two decades and are far away from being well understood: roughly speaking, no essentially sharp weak convergence rates are known for parabolic SEEs with nonlinear diffusion coefficient functions; see Remark 2.3 in [A. Debussche, Weak approximation of stochastic partial differential equations: the nonlinear case, Math. Comp. 80 (2011), no. 273, 89-117] for details. In this article we solve the weak convergence problem emerged from Debussche's article in the case of spectral Galerkin approximations and establish essentially sharp weak convergence rates for spatial spectral Galerkin approximations of semilinear SEEs with nonlinear diffusion coefficient functions. Our solution to the weak convergence problem does not use Malliavin calculus. Rather, key ingredients in our solution to the weak convergence problem emerged from Debussche's article are the use of appropriately modified versions of the spatial Galerkin approximation processes and applications of a mild Itô type formula for solutions and numerical approximations of semilinear SEEs. This article solves the weak convergence problem emerged from Debussche's article merely in the case of spatial spectral Galerkin approximations instead of other more complicated numerical approximations. Our method of proof extends, however, to a number of other kinds of spatial and temporal numerical approximations for semilinear SEEs.

Influence of the regularity of the test functions for weak convergence in numerical discretization of SPDEs

“…Indeed, as emphasized in [2], [6], [14], Sobolev-type regularity properties for the spatial derivatives of the solution of this infinite dimensional PDE are required to treat the most irregular terms in the error expansion. Similar arguments appear in [11], [15] and related articles. The regularity estimates have singularities at the initial time, even when the test function (seen as the initial condition of the Kolmogorov equation) is regular.…”

“…The results in Theorem 2 in the regime r ∈ [0, 1 4 ) are not new, they are straightforward applications of the strong convergence estimates in (11). The case r ∈ ( 1 4 , 1 2 ) is treated in Section 4.2.…”

“…In the case of multiplicative noise, the results require that φ is at least of class C 3 , however the order of convergence remains equal to 1 2 for such test functions. The case r ∈ ( 1 2 , 1) is obtained using the lower bounds from [11].…”

“…For semilinear equations, see [2], [6], [14], [30], [32], [33], for an approach related to the Kolmogorov equation. See [11], [18], [20], where a mild Itô formula is used. Finally, for semilinear equations with additive noise, see [1] and [7] for different approaches.…”

Malliavin regularity and weak approximation of semilinear SPDEs with Lévy noise

“…In the context of the numerical approximation of the solution processes of such equations, the quantity of interest is typically the expected value of some functional of the solution and one is thus interested in the weak convergence rate of the considered numerical scheme. While the weak convergence analysis for numerical approximations of SPDE with Gaussian noise is meanwhile relatively far developed, see, e.g., [1,2,3,7,8,9,10,11,12,14,15,16,17,18,23,30], available results for non-Gaussian Lévy noise have been restricted to linear equations so far [4,5,21,25]. In this article, we analyze for the first time the weak convergence rate of numerical approximations for a class of semi-linear SPDE with non-Gaussian Lévy noise.…”