| 1. |
- Andersson, Adam, 1979, et al.
(författare)
-
Duality in refined Sobolev–Malliavin spaces and weak approximation of SPDE
- 2016
-
Ingår i: Stochastic Partial Differential Equations: Analysis and Computations. - : Springer Science and Business Media LLC. - 2194-0401 .- 2194-041X. ; 4:1, s. 113-149
-
Tidskriftsartikel (refereegranskat)abstract
- 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.
|
|
| 2. |
- Bachouch, Achref, et al.
(författare)
-
Euler time discretization of backward doubly SDEs and application to semilinear SPDEs
- 2016
-
Ingår i: Stochastics and Partial Differential Equations: Analysis and Computations. - : Springer Science and Business Media LLC. - 2194-0401 .- 2194-041X. ; 4, s. 592-634
-
Tidskriftsartikel (refereegranskat)abstract
- This paper investigates a numerical probabilistic method for the solution of some semilinear stochastic partial differential equations (SPDEs in short). The numerical scheme is based on discrete time approximation for solutions of systems of decoupled forward-backward doubly stochastic differential equations. Under standard assumptions on the parameters, the convergence and the rate of convergence of the numerical scheme is proven. The proof is based on a generalization of the result on the path regularity of the backward equation.
|
|
| 3. |
- Cohen, David, et al.
(författare)
-
Exponential integrators for nonlinear Schrödinger equations with white noise dispersion
- 2017
-
Ingår i: Stochastics and Partial Differential Equations: Analysis and Computations. - New York : Springer. - 2194-0401 .- 2194-041X. ; 5:4, s. 592-613
-
Tidskriftsartikel (refereegranskat)abstract
- This article deals with the numerical integration in time of the nonlinear Schrödinger equation with power law nonlinearity and random dispersion. We introduce a new explicit exponential integrator for this purpose that integrates the noisy part of the equation exactly. We prove that this scheme is of mean-square order 1 and we draw consequences of this fact. We compare our exponential integrator with several other numerical methods from the literature. We finally propose a second exponential integrator, which is implicit and symmetric and, in contrast to the first one, preserves the L2-norm of the solution.
|
|
| 4. |
- Dareiotis, Konstantinos
(författare)
-
Symmetrization of exterior parabolic problems and probabilistic interpretation
- 2017
-
Ingår i: STOCHASTICS AND PARTIAL DIFFERENTIAL EQUATIONS-ANALYSIS AND COMPUTATIONS. - : SPRINGER. - 2194-0401 .- 2194-041X. ; 5:1, s. 38-52
-
Tidskriftsartikel (refereegranskat)abstract
- We prove a comparison theorem for the spatial mass of the solutions of two exterior parabolic problems, one of them having symmetrized geometry, using approximation of the Schwarz symmetrization by polarizations, as it was introduced in Brock and Solynin (TransAmMath Soc 352(4): 1759-1796, 2000). This comparison provides an alternative proof, based on PDEs, of the isoperimetric inequality for the Wiener sausage, which was proved in Peres and Sousi (Geom Funct Anal 22(4): 10001014, 2012).
|
|
| 5. |
- Fahim, K., et al.
(författare)
-
Some approximation results for mild solutions of stochastic fractional order evolution equations driven by Gaussian noise
- 2023
-
Ingår i: Stochastics and Partial Differential Equations: Analysis and Computations. - : Springer Science and Business Media LLC. - 2194-0401 .- 2194-041X. ; 11:3, s. 1044-1088
-
Tidskriftsartikel (refereegranskat)abstract
- We investigate the quality of space approximation of a class of stochastic integral equations of convolution type with Gaussian noise. Such equations arise, for example, when considering mild solutions of stochastic fractional order partial differential equations but also when considering mild solutions of classical stochastic partial differential equations. The key requirement for the equations is a smoothing property of the deterministic evolution operator which is typical in parabolic type problems. We show that if one has access to nonsmooth data estimates for the deterministic error operator together with its derivative of a space discretization procedure, then one obtains error estimates in pathwise Hölder norms with rates that can be read off the deterministic error rates. We illustrate the main result by considering a class of stochastic fractional order partial differential equations and space approximations performed by spectral Galerkin methods and finite elements. We also improve an existing result on the stochastic heat equation.
|
|
| 6. |
- Herrmann, L., et al.
(författare)
-
Numerical analysis of lognormal diffusions on the sphere
- 2018
-
Ingår i: Stochastics and Partial Differential Equations: Analysis and Computations. - : Springer Science and Business Media LLC. - 2194-0401 .- 2194-041X. ; 6:1, s. 1-44
-
Tidskriftsartikel (refereegranskat)abstract
- Numerical solutions of stationary diffusion equations on the unit sphere with isotropic lognormal diffusion coefficients are considered. Holder regularity in L-P sense for isotropic Gaussian random fields is obtained and related to the regularity of the driving lognormal coefficients. This yields regularity in L-P sense of the solution to the diffusion problem in Sobolev spaces. Convergence rate estimates of multilevel Monte Carlo Finite and Spectral Element discretizations of these problems are then deduced. Specifically, a convergence analysis is provided with convergence rate estimates in terms of the number of Monte Carlo samples of the solution to the considered diffusion equation and in terms of the total number of degrees of freedom of the spatial discretization, and with bounds for the total work required by the algorithm in the case of Finite Element discretizations. The obtained convergence rates are solely in terms of the decay of the angular power spectrum of the (logarithm) of the diffusion coefficient. Numerical examples confirm the presented theory.
|
|
| 7. |
- Lang, Annika, 1980, et al.
(författare)
-
Covariance structure of parabolic stochastic partial differential equations
- 2013
-
Ingår i: Stochastic Partial Differential Equations: Analysis and Computations. - : Springer Science and Business Media LLC. - 2194-0401 .- 2194-041X. ; 1:2, s. 351-364
-
Tidskriftsartikel (refereegranskat)abstract
- In this paper parabolic random partial differential equations and parabolic stochastic partial differential equations driven by a Wiener process are considered. A deterministic, tensorized evolution equation for the second moment and the covariance of the solutions of the parabolic stochastic partial differential equations is derived. Well-posedness of a space–time weak variational formulation of this tensorized equation is established.
|
|
