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Exponential integrators for stochastic partial differential equations
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
2018 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Stochastic partial differential equations (SPDEs) have during the past decades become an important tool for modeling systems which are influenced by randomness. Because of the complex nature of SPDEs, knowledge of efficient numerical methods with good convergence and geometric properties is of considerable importance. Due to this, numerical analysis of SPDEs has become an important and active research field.

The thesis consists of four papers, all dealing with time integration of different SPDEs using exponential integrators. We analyse exponential integrators for the stochastic wave equation, the stochastic heat equation, and the stochastic Schrödinger equation. Our primary focus is to study strong order of convergence of temporal approximations. However, occasionally, we also analyse space approximations such as finite element and finite difference approximations. In addition to this, for some SPDEs, we consider conservation properties of numerical discretizations.

As seen in this thesis, exponential integrators for SPDEs have many benefits over more traditional integrators such as Euler-Maruyama schemes or the Crank-Nicolson-Maruyama scheme. They are explicit and therefore very easy to implement and use in practice. Also, they are excellent at handling stiff problems, which naturally arise from spatial discretizations of SPDEs. While many explicit integrators suffer step size restrictions due to stability issues, exponential integrators do not in general.

In Paper 1 we consider a full discretization of the stochastic wave equation driven by multiplicative noise. We use a finite element method for the spatial discretization, and for the temporal discretization we use a stochastic trigonometric method. In the first part of the paper, we prove mean-square convergence of the full approximation. In the second part, we study the behavior of the total energy, or Hamiltonian, of the wave equation. It is well known that for deterministic (Hamiltonian) wave equations, the total energy remains constant in time. We prove that for stochastic wave equations with additive noise, the expected energy of the exact solution grows linearly with time. We also prove that the numerical approximation produces a small error in this linear drift.

In the second paper, we study an exponential integrator applied to the time discretization of the stochastic Schrödinger equation with a multiplicative potential. We prove strong convergence order 1 and 1/2 for additive and multiplicative noise, respectively. The deterministic linear Schrödinger equation has several conserved quantities, including the energy, the mass, and the momentum. We first show that for Schrödinger equations driven by additive noise, the expected values of these quantities grow linearly with time. The exponential integrator is shown to preserve these linear drifts for all time in the case of a stochastic Schrödinger equation without potential. For the equation with a multiplicative potential, we obtain a small error in these linear drifts.

The third paper is devoted to studying a full approximation of the one-dimensional stochastic heat equation. For the spatial discretization we use a finite difference method and an exponential integrator is used for the temporal approximation. We prove mean-square convergence and almost sure convergence of the approximation when the coefficients of the problem are assumed to be Lipschitz continuous. For non-Lipschitz coefficients, we prove convergence in probability.

In Paper 4 we revisit the stochastic Schrödinger equation. We consider this SPDE with a power-law nonlinearity. This nonlinearity is not globally Lipschitz continuous and the exact solution is not assumed to remain bounded for all times. These difficulties are handled by considering a truncated version of the equation and by working with stopping times and random time intervals. We prove almost sure convergence and convergence in probability for the exponential integrator as well as convergence orders of ½ − 𝜀, for all 𝜀 > 0, and 1/2, respectively.

Place, publisher, year, edition, pages
Umeå: Umeå universitet , 2018. , p. 24
Series
Research report in mathematics, ISSN 1653-0810 ; 61
Keywords [en]
Stochastic partial differential equations, numerical methods, stochastic exponential integrator, strong convergence, trace formulas
National Category
Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-146949ISBN: 978-91-7601-880-4 (print)OAI: oai:DiVA.org:umu-146949DiVA, id: diva2:1200403
Public defence
2018-05-18, MA121, MIT-huset, Umeå, 10:00 (English)
Opponent
Supervisors
Available from: 2018-04-27 Created: 2018-04-24 Last updated: 2018-06-09Bibliographically approved
List of papers
1. Full discretization of semilinear stochastic wave equations driven by multiplicative noise
Open this publication in new window or tab >>Full discretization of semilinear stochastic wave equations driven by multiplicative noise
2016 (English)In: SIAM Journal on Numerical Analysis, ISSN 0036-1429, E-ISSN 1095-7170, Vol. 54, no 2, p. 1093-1119Article in journal (Refereed) Published
Abstract [en]

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.

Keywords
semilinear stochastic wave equation, multiplicative noise, strong convergence, trace formula, stochastic trigonometric methods, geometric numerical integration
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:umu:diva-121620 (URN)10.1137/15M101049X (DOI)000375488100024 ()
Available from: 2016-06-20 Created: 2016-06-03 Last updated: 2018-06-07Bibliographically approved
2. Exponential integrators for stochastic Schrödinger equations driven by Itô noise
Open this publication in new window or tab >>Exponential integrators for stochastic Schrödinger equations driven by Itô noise
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]

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.

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)
Funder
Swedish Research Council, 2013-4562
Available from: 2017-06-28 Created: 2017-06-28 Last updated: 2018-06-09Bibliographically approved
3. A fully discrete approximation of the one-dimensional stochastic heat equation
Open this publication in new window or tab >>A fully discrete approximation of the one-dimensional stochastic heat equation
(English)Manuscript (preprint) (Other academic)
National Category
Mathematics
Identifiers
urn:nbn:se:umu:diva-146948 (URN)
Available from: 2018-04-24 Created: 2018-04-24 Last updated: 2018-06-09
4. Convergence of an exponential method for the stochastic Schrödinger equation with power-law nonlinearity
Open this publication in new window or tab >>Convergence of an exponential method for the stochastic Schrödinger equation with power-law nonlinearity
(English)Manuscript (preprint) (Other academic)
Abstract [en]

A temporal approximation of the stochastic Schrödinger equation with a power-law nonlinearity and driven by multiplicative Ito noise is considered. Observe that the nonlinearity is not globally Lipschitz continuous and, in general, the exact solution cannot be assumed to remain bounded for all times. The first of these issues is handled by considering a truncated version of the equation. The second issue is handled by working with stopping times and random time intervals, on which the solution is almost surely bounded. For the time integration we use a stochastic exponential method. This exponential method has the advantage of being explicit and it does not suffer any CFL-type step size restrictions in general. In this work we prove almost sure convergence and convergence in probability, accompanied by convergence orders of 1/2- and 1/2, respectively. We also find that the regularity assumptions required on the noise and initial value are less restrictive for the convergence of the exponential scheme, compared to the Crank-Nicolson scheme. In addition, we provide numerical experiments that illustrate our theoretical results.

National Category
Mathematics
Identifiers
urn:nbn:se:umu:diva-146946 (URN)
Available from: 2018-04-24 Created: 2018-04-24 Last updated: 2018-06-09

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