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| 2. |
- Lang, Annika, 1980, et al.
(författare)
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Mean-square stability analysis of approximations of stochastic differential equations in infinite dimensions
- 2017
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Ingår i: ArXiv.
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- The (asymptotic) behaviour of the second moment of solutions to stochastic differential equations is treated in mean-square stability analysis. The purpose of this article is to discuss this property for approximations of infinite-dimensional stochastic differential equations and give necessary and sufficient conditions that ensure mean-square stability of the considered finite-dimensional approximations. Stability properties of typical discretization schemes such as combinations of spectral Galerkin, finite element, Euler-Maruyama, Milstein, Crank-Nicolson, and forward and backward Euler methods are characterized. Furthermore, results on their relationship to stability properties of the analytical solutions are provided. Simulations of the stochastic heat equation confirm the theory.
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| 3. |
- Lang, Annika, 1980, et al.
(författare)
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Mean-square stability analysis of approximations of stochastic differential equations in infinite dimensions
- 2017
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Ingår i: BIT Numerical Mathematics. - : Springer Science and Business Media LLC. - 0006-3835 .- 1572-9125. ; 57:4, s. 963-990
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Tidskriftsartikel (refereegranskat)abstract
- © 2017, The Author(s). The (asymptotic) behaviour of the second moment of solutions to stochastic differential equations is treated in mean-square stability analysis. This property is discussed for approximations of infinite-dimensional stochastic differential equations and necessary and sufficient conditions ensuring mean-square stability are given. They are applied to typical discretization schemes such as combinations of spectral Galerkin, finite element, Euler–Maruyama, Milstein, Crank–Nicolson, and forward and backward Euler methods. Furthermore, results on the relation to stability properties of corresponding analytical solutions are provided. Simulations of the stochastic heat equation illustrate the theory.
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| 6. |
- Andersson, Adam, 1979, et al.
(författare)
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Finite element approximation of Lyapunov equations for the computation of quadratic functionals of SPDEs
- 2019
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Tidskriftsartikel (övrigt vetenskapligt/konstnärligt)abstract
- The computation of quadratic functionals of the solution to a linear stochastic partial differential equation with multiplicative noise is considered. An operator valued Lyapunov equation, whose solution admits a deterministic representation of the functional, is used for this purpose and error estimates are shown in suitable operator norms for a fully discrete approximation of this equation. Weak error rates are also derived for a fully discrete approximation of the stochastic partial differential equation, using the results obtained from the approximation of the Lyapunov equation. In the setting of finite element approximations, a computational complexity comparison reveals that approximating the Lyapunov equation allows for cheaper computation of quadratic functionals compared to applying Monte Carlo or covariance-based methods directly to the discretized stochastic partial differential equation. Numerical simulations illustrates the theoretical results.
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| 7. |
- Kovacs, Mihaly, 1977, et al.
(författare)
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Hilbert-Schmidt regularity of symmetric integral operators on bounded domains with applications to SPDE approximations
- 2023
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Ingår i: Stochastic Analysis and Applications. - : Informa UK Limited. - 0736-2994 .- 1532-9356. ; 41:3, s. 564-90
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Tidskriftsartikel (refereegranskat)abstract
- Regularity estimates for an integral operator with a symmetric continuous kernel on a convex bounded domain are derived. The covariance of a mean-square continuous random field on the domain is an example of such an operator. The estimates are of the form of Hilbert-Schmidt norms of the integral operator and its square root, composed with fractional powers of an elliptic operator equipped with homogeneous boundary conditions of either Dirichlet or Neumann type. These types of estimates, which couple the regularity of the driving noise with the properties of the differential operator, have important implications for stochastic partial differential equations on bounded domains as well as their numerical approximations. The main tools used to derive the estimates are properties of reproducing kernel Hilbert spaces of functions on bounded domains along with Hilbert-Schmidt embeddings of Sobolev spaces. Both non-homogeneous and homogeneous kernels are considered. In the latter case, results in a general Schatten class norm are also provided. Important examples of homogeneous kernels covered by the results of the paper include the class of Matern kernels.
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| 8. |
- Kovacs, Mihaly, 1977, et al.
(författare)
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Weak convergence of fully discrete finite element approximations of semilinear hyperbolic SPDE with additive noise
- 2020
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Ingår i: Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique Et Analyse Numerique. - : EDP Sciences. - 0764-583X .- 2822-7840 .- 1290-3841. ; 54:6, s. 2199-2227
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Tidskriftsartikel (refereegranskat)abstract
- The numerical approximation of the mild solution to a semilinear stochastic wave equation driven by additive noise is considered. A standard finite element method is employed for the spatial approximation and a a rational approximation of the exponential function for the temporal approximation. First, strong convergence of this approximation in both positive and negative order norms is proven. With the help of Malliavin calculus techniques this result is then used to deduce weak convergence rates for the class of twice continuously differentiable test functions with polynomially bounded derivatives. Under appropriate assumptions on the parameters of the equation, the weak rate is found to be essentially twice the strong rate. This extends earlier work by one of the authors to the semilinear setting. Numerical simulations illustrate the theoretical results.
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| 9. |
- Lang, Annika, 1980, et al.
(författare)
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Monte Carlo versus multilevel Monte Carlo in weak error simulations of SPDE approximations
- 2015
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- The simulation of weak error rates for the stochastic heat equation driven by multiplicative noise is presented. It is shown why conventional Monte Carlo approximations fail for these computationally expensive problems with small errors, and two different estimators for the weak error are presented that perform better in theory and in practice. One is another Monte Carlo estimator while the other one includes a multilevel Monte Carlo approximation in the computation of error plots.
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| 10. |
- Lang, Annika, 1980, et al.
(författare)
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Monte Carlo versus multilevel Monte Carlo in weak error simulations of SPDE approximations
- 2018
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Ingår i: Mathematics and Computers in Simulation. - : Elsevier BV. - 0378-4754. ; 143, s. 99-113
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Tidskriftsartikel (refereegranskat)abstract
- The simulation of the expectation of a stochastic quantity E[Y] by Monte Carlo methods is known to be computationally expensive especially if the stochastic quantity or its approximation Y-n is expensive to simulate, e.g., the solution of a stochastic partial differential equation. If the convergence of Y-n to Y in terms of the error |E[Y - Y-n]| is to be simulated, this will typically be done by a Monte Carlo method, i.e., |E[Y] - E-N [Y-n]| is computed. In this article upper and lower bounds for the additional error caused by this are determined and compared to those of |E-N [Y - Y-n]|, which are found to be smaller. Furthermore, the corresponding results for multilevel Monte Carlo estimators, for which the additional sampling error converges with the same rate as |E[Y - Y-n]|, are presented. Simulations of a stochastic heat equation driven by multiplicative Wiener noise and a geometric Brownian motion are performed which confirm the theoretical results and show the consequences of the presented theory for weak error simulations. (C) 2017 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.
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