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Cohen, Davidorcid.org/0000-0001-6490-1957

Open this publication in new window or tab >>Drift-preserving numerical integrators for stochastic Poisson systems### Cohen, David

### Vilmart, Gilles

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_0_j_idt208_some",{id:"formSmash:j_idt204:0:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_0_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_0_j_idt208_otherAuthors",{id:"formSmash:j_idt204:0:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_0_j_idt208_otherAuthors",multiple:true}); 2022 (English)In: International Journal of Computer Mathematics, ISSN 0020-7160, E-ISSN 1029-0265, Vol. 99, no 1, p. 4-20Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Taylor & Francis, 2022
##### Keywords

Casimir, energy, numerical schemes, Stochastic differential equations, stochastic Hamiltonian systems, stochastic Poisson systems, strong convergence, trace formula, weak convergence
##### National Category

Computational Mathematics Probability Theory and Statistics
##### Identifiers

urn:nbn:se:umu:diva-183794 (URN)10.1080/00207160.2021.1922679 (DOI)000652767900001 ()2-s2.0-85106279840 (Scopus ID)
#####

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##### Funder

Swedish Research Council, 2018-04443Swedish National Infrastructure for Computing (SNIC)
Available from: 2021-06-01 Created: 2021-06-01 Last updated: 2022-07-08Bibliographically approved

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Department of Mathematical Sciences, Chalmers University of Technology, University of Gothenburg, Gothenburg, Sweden.

Section de mathématiques, Université de Genève, Geneva, Switzerland.

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.

Open this publication in new window or tab >>A fully discrete approximation of the one-dimensional stochastic heat equation### Anton, Rikard

### Cohen, David

### Quer-Sardanyons, Lluis

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_1_j_idt208_some",{id:"formSmash:j_idt204:1:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_1_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_1_j_idt208_otherAuthors",{id:"formSmash:j_idt204:1:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_1_j_idt208_otherAuthors",multiple:true}); 2020 (English)In: IMA Journal of Numerical Analysis, ISSN 0272-4979, E-ISSN 1464-3642, Vol. 40, no 1, p. 247-284Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Oxford University Press, 2020
##### Keywords

stochastic heat equation, multiplicative noise, finite difference scheme, stochastic exponential integrator, Lq(Ω)-convergence
##### National Category

Computational Mathematics
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:umu:diva-146948 (URN)10.1093/imanum/dry060 (DOI)000544720400008 ()2-s2.0-85084810614 (Scopus ID)
#####

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##### Funder

Swedish Research Council
##### Note

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

Department of Mathematics, Universitat Autònoma de Barcelona, Catalonia, Spain.

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 L^{q}(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.

Originally included in thesis in manuscript form.

Available from: 2018-04-24 Created: 2018-04-24 Last updated: 2023-03-23Bibliographically approvedOpen this publication in new window or tab >>Drift-preserving numerical integrators for stochastic Hamiltonian systems### Chen, Chuchu

### Cohen, David

### D'Ambrosio, Raffaele

### Lang, Annika

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_2_j_idt208_some",{id:"formSmash:j_idt204:2:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_2_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_2_j_idt208_otherAuthors",{id:"formSmash:j_idt204:2:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_2_j_idt208_otherAuthors",multiple:true}); 2020 (English)In: Advances in Computational Mathematics, ISSN 1019-7168, E-ISSN 1572-9044, Vol. 46, no 2, article id 27Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Springer-Verlag New York, 2020
##### Keywords

Stochastic differential equations, Stochastic Hamiltonian systems, Energy, Trace formula, Numerical schemes, Strong convergence, Weak convergence, Multilevel Monte Carlo
##### National Category

Computational Mathematics
##### Identifiers

urn:nbn:se:umu:diva-167824 (URN)10.1007/s10444-020-09771-5 (DOI)000521127000001 ()2-s2.0-85081949646 (Scopus ID)
#####

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##### Funder

Swedish Research Council, 20134562Swedish Research Council, 201804443The Swedish Foundation for International Cooperation in Research and Higher Education (STINT), CH20166729Swedish National Infrastructure for Computing (SNIC)
Available from: 2020-02-05 Created: 2020-02-05 Last updated: 2023-03-24Bibliographically approved

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

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.

Open this publication in new window or tab >>Exponential integrators for stochastic Maxwell's equations driven by Itô noise### Cohen, David

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.### Cui, Jianbo

### Hong, Jialin

### Sun, Liying

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_3_j_idt208_some",{id:"formSmash:j_idt204:3:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_3_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_3_j_idt208_otherAuthors",{id:"formSmash:j_idt204:3:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_3_j_idt208_otherAuthors",multiple:true}); 2020 (English)In: Journal of Computational Physics, ISSN 0021-9991, E-ISSN 1090-2716, Vol. 410, article id 109382Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Elsevier, 2020
##### Keywords

Stochastic Maxwell's equation, Exponential integrator, Strong convergence, Trace formula, Average energy, Average divergence
##### National Category

Computational Mathematics
##### Identifiers

urn:nbn:se:umu:diva-171813 (URN)10.1016/j.jcp.2020.109382 (DOI)000534234000032 ()2-s2.0-85082680185 (Scopus ID)
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Available from: 2020-06-12 Created: 2020-06-12 Last updated: 2020-06-12Bibliographically approved

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 ½ 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.

Open this publication in new window or tab >>High order numerical integrators for single integrand Stratonovich SDEs### Cohen, David

### Debrabant, Kristian

### Rößler, Andreas

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_4_j_idt208_some",{id:"formSmash:j_idt204:4:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_4_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_4_j_idt208_otherAuthors",{id:"formSmash:j_idt204:4:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_4_j_idt208_otherAuthors",multiple:true}); 2020 (English)In: Applied Numerical Mathematics, ISSN 0168-9274, E-ISSN 1873-5460, Vol. 158, p. 264-270Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Elsevier, 2020
##### Keywords

Stratonovich stochastic differential equation, Single integrand SDEs, Geometric numerical integration, B-series methods, Strong error, Weak error, High order
##### National Category

Computational Mathematics
##### Identifiers

urn:nbn:se:umu:diva-174411 (URN)10.1016/j.apnum.2020.08.002 (DOI)000571481300014 ()2-s2.0-85089486357 (Scopus ID)
#####

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##### Funder

Swedish Research Council, 2018-04443
Available from: 2020-08-24 Created: 2020-08-24 Last updated: 2023-03-24Bibliographically approved

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, Gothenburg, Sweden.

Department of Mathematics and Computer Science, University of Southern Denmark, Denmark.

Institut für Mathematik, Universität zu Lübeck, Germany.

We show that applying any deterministic B-series method of order pdwith a random step size to single integrand SDEs gives a numerical method converging in the mean-square and weak sense with order [_{Pd}/2]. As an application, we derive high order energy-preserving methods for stochastic Poisson systems as well as further geometric numerical schemes for this wide class of Stratonovich SDEs.

Open this publication in new window or tab >>Exponential integrators for stochastic Schrödinger equations driven by Itô noise### Anton, Rikard

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.### Cohen, David

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_5_j_idt208_some",{id:"formSmash:j_idt204:5:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_5_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_5_j_idt208_otherAuthors",{id:"formSmash:j_idt204:5:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_5_j_idt208_otherAuthors",multiple:true}); 2018 (English)In: Journal of Computational Mathematics, ISSN 0254-9409, E-ISSN 1991-7139, Vol. 36, no 2, p. 276-309Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Global Science Press, 2018
##### Keywords

Stochastic partial differential equations, Stochastic Schr¨odinger equations, Numerical methods, Geometric numerical integration, Stochastic exponential integrators, Strong convergence, Trace formulas
##### National Category

Computational Mathematics
##### Identifiers

urn:nbn:se:umu:diva-137247 (URN)10.4208/jcm.1701-m2016-0525 (DOI)000455995300007 ()2-s2.0-85072292610 (Scopus ID)
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##### Funder

Swedish Research Council, 2013-4562
Available from: 2017-06-28 Created: 2017-06-28 Last updated: 2023-09-05Bibliographically approved

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Department of Mathematics, University of Innsbruck, A–6020 Innsbruck, Austria.

We study an explicit exponential scheme for the time discretisation of stochastic Schr¨odinger Equations Driven by additive or Multiplicative Itô Noise. The numerical scheme is shown to converge with strong order 1 if the noise is additive and with strong order 1/2 for multiplicative noise. In addition, if the noise is additive, we show that the exact solutions of the linear stochastic Schr¨odinger equations satisfy trace formulas for the expected mass, energy, and momentum (i. e., linear drifts in these quantities). Furthermore, we inspect the behaviour of the numerical solutions with respect to these trace formulas. Several numerical simulations are presented and confirm our theoretical results.

Open this publication in new window or tab >>Numerical discretisations of stochastic wave equations### Cohen, David

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_6_j_idt208_some",{id:"formSmash:j_idt204:6:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_6_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_6_j_idt208_otherAuthors",{id:"formSmash:j_idt204:6:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_6_j_idt208_otherAuthors",multiple:true}); 2018 (English)In: International conference of numerical analysis and applied mathematics (ICNAAM 2017), American Institute of Physics (AIP), 2018, Vol. 1978, p. 1-5, article id 1Conference paper, Oral presentation with published abstract (Refereed)
##### Abstract [en]

##### Place, publisher, year, edition, pages

American Institute of Physics (AIP), 2018
##### Series

AIP Conference Proceedings, ISSN 0094-243X, E-ISSN 1551-7616 ; 1078
##### Keywords

Stochastic partial differential equations, Stochastic wave equations, Numerical methods, Convergence, Long-time behaviour
##### National Category

Computational Mathematics
##### Identifiers

urn:nbn:se:umu:diva-150238 (URN)10.1063/1.5043646 (DOI)000445105400001 ()2-s2.0-85049977367 (Scopus ID)978-0-7354-1690-1 (ISBN)
##### Conference

International conference of numerical analysis and applied mathematics (ICNAAM 2017), Thessaloniki, Greece, September 25-30, 2017
#####

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##### Funder

Swedish Research Council, 2013-4562
##### Note

This extended abstract starts with a brief introduction to stochastic partial differential equations with a particular focus on stochastic wave equations. Various numerical experiments for this stochastic partial differential equation are presented. Finally, we point out results from the literature on the numerical analysis of stochastic wave equations.

Extended abstract

Available from: 2018-07-22 Created: 2018-07-22 Last updated: 2023-09-05Bibliographically approvedOpen this publication in new window or tab >>Exponential integrators for nonlinear Schrödinger equations with white noise dispersion### Cohen, David

### Dujardin, Guillaume

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_7_j_idt208_some",{id:"formSmash:j_idt204:7:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_7_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_7_j_idt208_otherAuthors",{id:"formSmash:j_idt204:7:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_7_j_idt208_otherAuthors",multiple:true}); 2017 (English)In: Stochastics and Partial Differential Equations: Analysis and Computations, ISSN 2194-0401, Vol. 5, no 4, p. 592-613Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

New York: Springer, 2017
##### Keywords

Stochastic partial differential equations, Nonlinear Schrödinger equation, White noise dispersion, Numerical methods, Geometric numerical integration, Exponential integrators, Mean-square convergence
##### National Category

Computational Mathematics
##### Identifiers

urn:nbn:se:umu:diva-137244 (URN)10.1007/s40072-017-0098-1 (DOI)000415876800005 ()2-s2.0-85055971953 (Scopus ID)
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##### Funder

Swedish Research Council, 2013-4562
Available from: 2017-06-28 Created: 2017-06-28 Last updated: 2023-03-23Bibliographically approved

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Department of Mathematics, University of Innsbruck, Innsbruck, Austria.

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 L^{2}-norm of the solution.

Open this publication in new window or tab >>Exponential integrators for stochastic Schrödinger equations driven by Itô noise### Cohen, David

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_8_j_idt208_some",{id:"formSmash:j_idt204:8:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_8_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_8_j_idt208_otherAuthors",{id:"formSmash:j_idt204:8:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_8_j_idt208_otherAuthors",multiple:true}); 2017 (English)In: Oberwolfach mini-Workshop: Stochastic Differential Equations:Regularity and Numerical Analysis in Finite and InfiniteDimensions: Workshop ID: 1706b, Oberwolfach, 2017, p. 11-12Conference paper, Oral presentation with published abstract (Other academic)
##### Place, publisher, year, edition, pages

Oberwolfach: , 2017
##### Series

Oberwolfach report ; 9/2017
##### National Category

Computational Mathematics
##### Identifiers

urn:nbn:se:umu:diva-138342 (URN)
##### Conference

Mini-Workshop: Stochastic Differential Equations: Regularity and Numerical Analysis in Finite and Infinite Dimensions, Oberwolfach Research Institute for Mathematics, Oberwolfach, 5 Feb - 11 Feb 2017
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##### Funder

Swedish Research Council, 2013-4562
Available from: 2017-08-21 Created: 2017-08-21 Last updated: 2018-06-09Bibliographically approved

Open this publication in new window or tab >>Geometric numerical integrators for Hunter–Saxton-like equations### Miyatake, Yuto

### Cohen, David

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.### Furihata, Daisuke

### Matsuo, Takayasu

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_9_j_idt208_some",{id:"formSmash:j_idt204:9:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_9_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_9_j_idt208_otherAuthors",{id:"formSmash:j_idt204:9:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_9_j_idt208_otherAuthors",multiple:true}); 2017 (English)In: Japan journal of industrial and applied mathematics, ISSN 0916-7005, E-ISSN 1868-937X, Vol. 34, no 2, p. 441-472Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

Hunter-Saxton equation; Modified Hunter-Saxton equation; Two-component Hunter-Saxton equation; Multi-symplectic formulation; Numerical discretisation; Geometric numerical integration; Discrete variational derivative method; Multi-symplectic schemes; Euler box scheme
##### National Category

Computational Mathematics
##### Identifiers

urn:nbn:se:umu:diva-137245 (URN)10.1007/s13160-017-0252-1 (DOI)000406362100007 ()2-s2.0-85020744830 (Scopus ID)
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##### Funder

The Royal Swedish Academy of Sciences, FY2012
Available from: 2017-06-28 Created: 2017-06-28 Last updated: 2023-03-24Bibliographically approved

We present novel geometric numerical integrators for Hunter-Saxton-like equations by means of new multi-symplectic formulations and known Hamiltonian structures of the problems. We consider the Hunter-Saxton equation, the modified Hunter-Saxton equation, and the two-component Hunter-Saxton equation. Multi-symplectic discretisations based on these new formulations of the problems are exemplified by means of the explicit Euler box scheme, and Hamiltonian-preserving discretisations are exemplified by means of the discrete variational derivative method. We explain and justify the correct treatment of boundary conditions in a unified manner. This is necessary for a proper numerical implementation of these equations and was never explicitly clarified in the literature before, to the best of our knowledge. Finally, numerical experiments demonstrate the favourable behaviour of the proposed numerical integrators.