| 1. |
- Berg, André, et al.
(författare)
-
APPROXIMATED EXPONENTIAL INTEGRATORS FOR THE STOCHASTIC MANAKOV EQUATION
- 2023
-
Ingår i: Journal of Computational Dynamics. - : American Institute of Mathematical Sciences (AIMS). - 2158-2491 .- 2158-2505. ; 10:2, s. 323-344
-
Tidskriftsartikel (refereegranskat)abstract
- . This article presents and analyzes an approximated exponential integrator for the (inhomogeneous) stochastic Manakov system. This system of SPDE occurs in the study of pulse propagation in randomly birefringent optical fibers. For a globally Lipschitz-continuous nonlinearity, we prove that the strong order of the time integrator is 1/2. This is then used to prove that the approximated exponential integrator has convergence order 1/2 in probability and almost sure order 1/2-, in the case of the cubic nonlinear coupling which is relevant in optical fibers. Finally, we present several numerical experiments in order to support our theoretical findings and to illustrate the efficiency of the approximated exponential integrator as well as a modified version of it.
|
|
| 2. |
- Berg, André, et al.
(författare)
-
Lie-Trotter Splitting for the Nonlinear Stochastic Manakov System
- 2021
-
Ingår i: Journal of Scientific Computing. - : Springer Science and Business Media LLC. - 0885-7474 .- 1573-7691. ; 88:1
-
Tidskriftsartikel (refereegranskat)abstract
- This article analyses the convergence of the Lie-Trotter splitting scheme for the stochastic Manakov equation, a system arising in the study of pulse propagation in randomly birefringent optical fibers. First, we prove that the strong order of the numerical approximation is 1/2 if the nonlinear term in the system is globally Lipschitz. Then, we show that the splitting scheme has convergence order 1/2 in probability and almost sure order 1/2- in the case of a cubic nonlinearity. We provide several numerical experiments illustrating the aforementioned results and the efficiency of the Lie-Trotter splitting scheme. Finally, we numerically investigate the possible blowup of solutions for some power-law nonlinearities.
|
|
| 3. |
- Brehier, C. E., et al.
(författare)
-
Analysis of a splitting scheme for a class of nonlinear stochastic Schrodinger equations
- 2023
-
Ingår i: Applied Numerical Mathematics. - : Elsevier BV. - 0168-9274. ; 186, s. 57-83
-
Tidskriftsartikel (refereegranskat)abstract
- We analyze the qualitative properties and the order of convergence of a splitting scheme for a class of nonlinear stochastic Schrodinger equations driven by additive noise. The class of nonlinearities of interest includes nonlocal interaction cubic nonlinearities. We show that the numerical solution is symplectic and preserves the expected mass for all times (trace formula). On top of that, for the convergence analysis, some exponential moment bounds for the exact and numerical solutions are proved. This enables us to provide strong orders of convergence as well as orders of convergence in probability and almost surely. Finally, extensive numerical experiments illustrate the performance of the proposed numerical scheme. (c) 2023 The Author(s). Published by Elsevier B.V. on behalf of IMACS. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).
|
|
| 4. |
- Cohen, David, et al.
(författare)
-
Numerical approximation and simulation of the stochastic wave equation on the sphere
- 2022
-
Ingår i: Calcolo. - : Springer Science and Business Media LLC. - 0008-0624 .- 1126-5434. ; 59:3
-
Tidskriftsartikel (refereegranskat)abstract
- Solutions to the stochastic wave equation on the unit sphere are approximated by spectral methods. Strong, weak, and almost sure convergence rates for the proposed numerical schemes are provided and shown to depend only on the smoothness of the driving noise and the initial conditions. Numerical experiments confirm the theoretical rates. The developed numerical method is extended to stochastic wave equations on higher-dimensional spheres and to the free stochastic Schrodinger equation on the unit sphere.
|
|
| 5. |
- Berg, André, 1990-
(författare)
-
Numerical analysis and simulation of stochastic partial differential equations with white noise dispersion
- 2023
-
Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This doctoral thesis provides a comprehensive numerical analysis and exploration of several stochastic partial differential equations (SPDEs). More specifically, this thesis investigates time integrators for SPDEs with white noise dispersion. The thesis begins by examining the stochastic nonlinear Schrödinger equation with white noise dispersion (SNLSE), see Paper 1. The investigation probes the performance of different numerical integrators for this equation, focusing on their convergences, L2-norm preservation, and computational efficiency. Further, this thesis thoroughly investigates a conjecture on the critical exponent of the SNLSE, related to a phenomenon known as blowup, through numerical means. The thesis then introduces and studies exponential integrators for the stochastic Manakov equation (SME) by presenting two new time integrators - the explicit and symmetric exponential integrators - and analyzing their convergence properties, see Paper 2. Notably, this study highlights the flexibility and efficiency of these integrators compared to traditional schemes. The narrative then turns to the Lie-Trotter splitting integrator for the SME, see Paper 3, comparing its performance to existing time integrators. Theoretical proofs for convergence in various senses, alongside extensive numerical experiments, shed light on the efficacy of the proposed numerical scheme. The thesis also deep dives into the critical exponents of the SME, proposing a conjecture regarding blowup conditions for this SPDE.Lastly, the focus shifts to the stochastic generalized Benjamin-Bona-Mahony equation, see Paper 4. The study introduces and numerically assesses four novel exponential integrators for this equation. A primary finding here is the superior performance of the symmetric exponential integrator. This thesis also offers a succinct and novel method to depict the order of convergence in probability.
|
|
| 6. |
- Brehier, C. E., et al.
(författare)
-
SPLITTING SCHEMES FOR FITZHUGH-NAGUMO STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS
- 2023
-
Ingår i: Discrete and Continuous Dynamical Systems-Series B. - 1531-3492. ; 29:1, s. 214-44
-
Tidskriftsartikel (refereegranskat)abstract
- . We design and study splitting integrators for the temporal discretization of the stochastic FitzHugh-Nagumo system. This system is a model for signal propagation in nerve cells where the voltage variable is the solution of a one-dimensional parabolic PDE with a cubic nonlinearity driven by additive space-time white noise. We first show that the numerical solutions have finite moments. We then prove that the splitting schemes have, at least, the strong rate of convergence 1/4. Finally, numerical experiments illustrating the performance of the splitting schemes are provided.
|
|
| 7. |
- Cohen, David, et al.
(författare)
-
Drift-preserving numerical integrators for stochastic Poisson systems
- 2021
-
Ingår i: International Journal of Computer Mathematics. - : Informa UK Limited. - 0020-7160 .- 1029-0265. ; 99:1
-
Tidskriftsartikel (refereegranskat)abstract
- We perform a numerical analysis of a class of randomly perturbed Hamiltonian systems and Poisson systems. For the considered additive noise perturbation of such systems, we show the long-time behaviour of the energy and quadratic Casimirs for the exact solution. We then propose and analyse a drift-preserving splitting scheme for such problems with the following properties: exact drift preservation of energy and quadratic Casimirs, mean-square order of convergence 1, weak order of convergence 2. These properties are illustrated with numerical experiments.
|
|
| 8. |
- Anton, Rikard, et al.
(författare)
-
A fully discrete approximation of the one-dimensional stochastic heat equation
- 2020
-
Ingår i: IMA Journal of Numerical Analysis. - : Oxford University Press. - 0272-4979 .- 1464-3642. ; 40:1, s. 247-284
-
Tidskriftsartikel (refereegranskat)abstract
- A fully discrete approximation of the one-dimensional stochastic heat equation driven by multiplicative space–time white noise is presented. The standard finite difference approximation is used in space and a stochastic exponential method is used for the temporal approximation. Observe that the proposed exponential scheme does not suffer from any kind of CFL-type step size restriction. When the drift term and the diffusion coefficient are assumed to be globally Lipschitz this explicit time integrator allows for error bounds in Lq(Ω), for all q ≥ 2, improving some existing results in the literature. On top of this we also prove almost sure convergence of the numerical scheme. In the case of nonglobally Lipschitz coefficients, under a strong assumption about pathwise uniqueness of the exact solution, convergence in probability of the numerical solution to the exact solution is proved. Numerical experiments are presented to illustrate the theoretical results.
|
|
| 9. |
- Chen, C. C., et al.
(författare)
-
Drift-preserving numerical integrators for stochastic Hamiltonian systems
- 2020
-
Ingår i: Advances in Computational Mathematics. - : Springer Science and Business Media LLC. - 1019-7168 .- 1572-9044. ; 46:2
-
Tidskriftsartikel (refereegranskat)abstract
- The paper deals with numerical discretizations of separable nonlinear Hamiltonian systems with additive noise. For such problems, the expected value of the total energy, along the exact solution, drifts linearly with time. We present and analyze a time integrator having the same property for all times. Furthermore, strong and weak convergence of the numerical scheme along with efficient multilevel Monte Carlo estimators are studied. Finally, extensive numerical experiments illustrate the performance of the proposed numerical scheme.
|
|
| 10. |
- Cohen, David, et al.
(författare)
-
Energy-preserving integrators for stochastic Poisson systems
- 2014
-
Ingår i: Communications in Mathematical Sciences. - : International Press of Boston. - 1539-6746 .- 1945-0796. ; 12:8, s. 1523-1539
-
Tidskriftsartikel (refereegranskat)abstract
- A new class of energy-preserving numerical schemes for stochastic Hamiltonian systems with non-canonical structure matrix (in the Stratonovich sense) is proposed. These numerical integrators are of mean-square order one and also preserve quadratic Casimir functions. In the deterministic setting, our schemes reduce to methods proposed in [E. Hairer, JNAIAM. J. Numer. Anal. Ind. Appl. Math., 5(1-2), 73–84, 2011] and [D. Cohen, and E. Hairer, BIT, 51(1), 91–101, 2011].
|
|
