| 1. |
- Anton, Rikard, et al.
(författare)
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A fully discrete approximation of the one-dimensional stochastic heat equation
- 2020
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Ingår i: IMA Journal of Numerical Analysis. - : Oxford University Press. - 0272-4979 .- 1464-3642. ; 40:1, s. 247-284
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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.
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| 2. |
- Anton, R., et al.
(författare)
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Full Discretization of Semilinear Stochastic Wave Equations Driven by Multiplicative Noise
- 2016
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Ingår i: Siam Journal on Numerical Analysis. - : Society for Industrial & Applied Mathematics (SIAM). - 0036-1429 .- 1095-7170. ; 54:2, s. 1093-1119
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Tidskriftsartikel (refereegranskat)abstract
- A fully discrete approximation of the semilinear stochastic wave equation driven by multiplicative noise is presented. A standard linear finite element approximation is used in space, and a stochastic trigonometric method is used for the temporal approximation. This explicit time integrator allows for mean-square error bounds independent of the space discretization and thus does not suffer from a step size restriction as in the often used Stormer-Verlet leapfrog scheme. Furthermore, it satisfies an almost trace formula (i.e., a linear drift of the expected value of the energy of the problem). Numerical experiments are presented and confirm the theoretical results.
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| 3. |
- Araujo-Cabarcas, Juan Carlos, 1981-
(författare)
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Reliable hp finite element computations of scattering resonances in nano optics
- 2019
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Eigenfrequencies are commonly studied in wave propagation problems, as they are important in the analysis of closed cavities such as a microwave oven. For open systems, energy leaks into infinity and therefore scattering resonances are used instead of eigenfrequencies. An interesting application where resonances take an important place is in whispering gallery mode resonators.The objective of the thesis is the reliable and accurate approximation of scattering resonances using high order finite element methods. The discussion focuses on the electromagnetic scattering resonances in metal-dielectric nano-structures using a Drude-Lorentz model for the description of the material properties. A scattering resonance pair satisfies a reduced wave equationand an outgoing wave condition. In this thesis, the outgoing wave condition is replaced by a Dirichlet-to-Neumann map, or a Perfectly Matched Layer. For electromagnetic waves and for acoustic waves, the reduced wave equation is discretized with finite elements. As a result, the scattering resonance problem is transformed into a nonlinear eigenvalue problem.In addition to the correct approximation of the true resonances, a large number of numerical solutions that are unrelated to the physical problem are also computed in the solution process. A new method based on a volume integral equation is developed to remove these false solutions.The main results of the thesis are a novel method for removing false solutions of the physical problem, efficient solutions of non-linear eigenvalue problems, and a new a-priori based refinement strategy for high order finite element methods. The overall material in the thesis translates into a reliable and accurate method to compute scattering resonances in physics and engineering.
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| 4. |
- Berg, Andre, et al.
(författare)
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Approximated exponential integrators for the stochastic Manakov equation
- 2021
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Tidskriftsartikel (övrigt vetenskapligt/konstnärligt)abstract
- This article presents and analyses an exponential integrator for the stochastic Manakov equation, a system arising in the study of pulse propagation in randomly birefringent optical fibers. We first prove that the strong order of the numerical approximation is 1/2 if the nonlinear term in the system is globally Lipschitz-continuous. Then, we use this fact to prove that the 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 exponential integrator as well as a modified version of it.
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| 5. |
- Berg, André, 1990-
(författare)
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Numerical analysis and simulation of stochastic partial differential equations with white noise dispersion
- 2023
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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.
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| 6. |
- Bréhier, Charles-Edouard, et al.
(författare)
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Splitting integrators for linear Vlasov equations with stochastic perturbations
- 2024
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Ingår i: Journal of Computational Dynamics. - 2158-2505.
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Tidskriftsartikel (refereegranskat)abstract
- We consider a class of linear Vlasov partial differential equations driven by Wiener noise. Different types of stochastic perturbations are treated: additive noise, multiplicative Itô and Stratonovich noise, and transport noise. We propose to employ splitting integrators for the temporal discretization of these stochastic partial differential equations. These integrators are designed in order to preserve qualitative properties of the exact solutions depending on the stochastic perturbation, such as preservation of norms or positivity of the solutions. We provide numerical experiments in order to illustrate the properties of the proposed integrators and investigate mean-square rates of convergence.
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| 7. |
- Bréhier, Charles-Edouard, et al.
(författare)
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Splitting integrators for stochastic Lie-Poisson systems
- 2021
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Tidskriftsartikel (övrigt vetenskapligt/konstnärligt)abstract
- We study stochastic Poisson integrators for a class of stochastic Poisson systems driven by Stratonovich noise. Such geometric integrators preserve Casimir functions and the Poisson map property. For this purpose, we propose explicit stochastic Poisson integrators based on a splitting strategy, and analyse their qualitative and quantitative properties: preservation of Casimir functions, existence of almost sure or moment bounds, asymptotic preserving property, and strong and weak rates of convergence. The construction of the schemes and the theoretical results are illustrated through extensive numerical experiments for three examples of stochastic Lie--Poisson systems, namely: stochastically perturbed Maxwell--Bloch, rigid body and sine--Euler equations.
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| 8. |
- Chen, C. C., et al.
(författare)
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Drift-preserving numerical integrators for stochastic Hamiltonian systems
- 2020
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Ingår i: Advances in Computational Mathematics. - : Springer Science and Business Media LLC. - 1019-7168 .- 1572-9044. ; 46:2
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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.
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| 9. |
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| 10. |
- Cohen, David, 1977, et al.
(författare)
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Exponential integrators for stochastic Maxwell's equations driven by Itô noise
- 2020
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Ingår i: Journal of Computational Physics. - : Elsevier BV. - 1090-2716 .- 0021-9991. ; 410
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Tidskriftsartikel (refereegranskat)abstract
- This article presents explicit exponential integrators for stochastic Maxwell's equations driven by both multiplicative and additive noises. By utilizing the regularity estimate of the mild solution, we first prove that the strong order of the numerical approximation is [Formula presented] for general multiplicative noise. Combining a proper decomposition with the stochastic Fubini's theorem, the strong order of the proposed scheme is shown to be 1 for additive noise. Moreover, for linear stochastic Maxwell's equation with additive noise, the proposed time integrator is shown to preserve exactly the symplectic structure, the evolution of the energy as well as the evolution of the divergence in the sense of expectation. Several numerical experiments are presented in order to verify our theoretical findings.
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