| 1. |
- Bågmark, Kasper, 1995, et al.
(författare)
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An energy-based deep splitting method for the nonlinear filtering problem
- 2023
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Ingår i: Partial Differential Equations and Applications. - : Springer Science and Business Media LLC. - 2662-2963 .- 2662-2971. ; 4:2
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Tidskriftsartikel (refereegranskat)abstract
- The purpose of this paper is to explore the use of deep learning for the solution of the nonlinear filtering problem. This is achieved by solving the Zakai equation by a deep splitting method, previously developed for approximate solution of (stochastic) partial differential equations. This is combined with an energy-based model for the approximation of functions by a deep neural network. This results in a computationally fast filter that takes observations as input and that does not require re-training when new observations are received. The method is tested on four examples, two linear in one and twenty dimensions and two nonlinear in one dimension. The method shows promising performance when benchmarked against the Kalman filter and the bootstrap particle filter.
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| 2. |
- Eisenmann, Monika, et al.
(författare)
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Error estimates of the backward Euler-Maruyama method for multi-valued stochastic differential equations
- 2022
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Ingår i: Bit Numerical Mathematics. - : Springer Science and Business Media LLC. - 0006-3835 .- 1572-9125. ; 62:3, s. 803-48
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we derive error estimates of the backward Euler-Maruyama method applied to multi-valued stochastic differential equations. An important example of such an equation is a stochastic gradient flow whose associated potential is not continuously differentiable but assumed to be convex. We show that the backward Euler-Maruyama method is well-defined and convergent of order at least 1/4 with respect to the root-mean-square norm. Our error analysis relies on techniques for deterministic problems developed in Nochetto et al. (Commun Pure Appl Math 53(5):525-589, 2000). We verify that our setting applies to an overdamped Langevin equation with a discontinuous gradient and to a spatially semi-discrete approximation of the stochastic p-Laplace equation.
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| 3. |
- Eisenmann, Monika, et al.
(författare)
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On a randomized backward Euler method for nonlinear evolution equations with time-irregular coefficients
- 2019
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Ingår i: Foundations of Computational Mathematics. - : Springer Science and Business Media LLC. - 1615-3375 .- 1615-3383. ; 19:6, s. 1387-1430
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we introduce a randomized version of the backward Euler method that is applicable to stiff ordinary differential equations and nonlinear evolution equations with time-irregular coefficients. In the finite-dimensional case, we consider Carathéodory-type functions satisfying a one-sided Lipschitz condition. After investigating the well-posedness and the stability properties of the randomized scheme, we prove the convergence to the exact solution with a rate of 0.5 in the root-mean-square norm assuming only that the coefficient function is square integrable with respect to the temporal parameter. These results are then extended to the approximation of infinite-dimensional evolution equations under monotonicity and Lipschitz conditions. Here, we consider a combination of the randomized backward Euler scheme with a Galerkin finite element method. We obtain error estimates that correspond to the regularity of the exact solution. The practicability of the randomized scheme is also illustrated through several numerical experiments.
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| 4. |
- Kovacs, Mihaly, 1977, et al.
(författare)
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On the discretisation in time of the stochastic Allen-Cahn equation
- 2018
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Ingår i: Mathematische Nachrichten. - : Wiley. - 0025-584X .- 1522-2616. ; 291:5-6, s. 966-995
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Tidskriftsartikel (refereegranskat)abstract
- We consider the stochastic Allen-Cahn equation perturbed by smooth additive Gaussian noise in a bounded spatial domain with smooth boundary in dimension d3, and study the semidiscretisation in time of the equation by an Euler type split-step method with step size k>0. We show that the method converges strongly with a rate O(k(1/2)). By means of a perturbation argument, we also establish the strong convergence of the standard backward Euler scheme with the same rate.
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| 5. |
- Cohen, David, et al.
(författare)
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A trigonometric method for the linear stochastic wave equation
- 2013
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Ingår i: SIAM Journal on Numerical Analysis. - 0036-1429 .- 1095-7170. ; 51:1, s. 204-222
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Tidskriftsartikel (refereegranskat)abstract
- A fully discrete approximation of the linear stochastic wave equation driven by additive noise is presented. A standard finite element method is used for the spatial discretization and a stochastic trigonometric scheme for the temporal approximation. This explicit time integrator allows for error bounds independent of the space discretization and thus does not have a step-size restriction as in the often used Störmer--Verlet-leap-frog scheme. Moreover, it enjoys a trace formula as does the exact solution of our problem. These favorable properties are demonstrated with numerical experiments. Read More: http://epubs.siam.org/doi/abs/10.1137/12087030X
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| 6. |
- Kovacs, Mihaly, 1977, et al.
(författare)
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Mittag-leffler euler integrator for a stochastic fractional order equation with additive noise
- 2020
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Ingår i: SIAM Journal on Numerical Analysis. - 0036-1429 .- 1095-7170. ; 58:1, s. 66-85
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Tidskriftsartikel (refereegranskat)abstract
- Motivated by fractional derivative models in viscoelasticity, a class of semilinear stochastic Volterra integro-differential equations, and their deterministic counterparts, are considered. A generalized exponential Euler method, named here the Mittag-Leffler Euler integrator, is used for the temporal discretization, while the spatial discretization is performed by the spectral Galerkin method. The temporal rate of strong convergence is found to be (almost) twice compared to when the backward Euler method is used together with a convolution quadrature for time discretization. Numerical experiments that validate the theory are presented. © 2020 Society for Industrial and Applied Mathematics.
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| 7. |
- Larsson, Stig, 1952, et al.
(författare)
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A weak space-time formulation for the linear stochastic heat equation
- 2017
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Ingår i: International Journal of Applied and Computational Mathematics. - : Springer Science and Business Media LLC. - 2349-5103 .- 2199-5796. ; 3:2, s. 787-806
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Tidskriftsartikel (refereegranskat)abstract
- We apply the well-known Banach–Nečas–Babuška inf–sup theory in a stochastic setting to introduce a weak space-time formulation of the linear stochastic heat equation with additive noise. We give sufficient conditions on the data and on the covariance operator associated to the driving Wiener process, in order to have existence and uniqueness of the solution. We show the relation of the obtained solution to the mild solution and to the variational solution of the same problem. The spatial regularity of the solution is also discussed. Finally, an extension to the case of linear multiplicative noise is presented.
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| 8. |
- Furihata, Daisuke, et al.
(författare)
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Strong convergence of a fully discrete finite element approximation of the stochastic cahn–hilliard equation
- 2018
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Ingår i: SIAM Journal on Numerical Analysis. - 0036-1429 .- 1095-7170. ; 56, s. 708-731
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Tidskriftsartikel (refereegranskat)abstract
- © 2018 Society for Industrial and Applied Mathematics. We consider the stochastic Cahn–Hilliard equation driven by additive Gaussian noise in a convex domain with polygonal boundary in dimension d ≤ 3. We discretize the equation using a standard finite element method in space and a fully implicit backward Euler method in time. By proving optimal error estimates on subsets of the probability space with arbitrarily large probability and uniform-in-time moment bounds we show that the numerical solution converges strongly to the solution as the discretization parameters tend to zero.
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| 9. |
- Kovacs, Mihaly, 1977, et al.
(författare)
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Introduction to stochastic partial differential equations
- 2008
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Ingår i: Publications of the ICMCS. ; 4, s. 159-232
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Tidskriftsartikel (övrigt vetenskapligt/konstnärligt)abstract
- We introduce the Hilbert space-valued Wiener process and the corresponding stochastic integral of It\^o type. This is then used together with semigroup theory to obtain existence and uniqueness of weak solutions of linear and semilinear stochastic evolution problems in Hilbert space. Finally, this abstract theory is applied to the linear heat and wave equations driven by additive noise.
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| 10. |
- Mesforush, Ali, 1971, et al.
(författare)
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A posteriori error analysis for the Cahn-Hilliard equation
- 2022
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Ingår i: Journal of Mathematical Modeling. ; 10:4, s. 437-452
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Tidskriftsartikel (refereegranskat)abstract
- The Cahn-Hilliard equation is discretized by a Galerkin finite element method based on continuous piecewise linear functions in space and discontinuous piecewise constant functions in time. A posteriori error estimates are proved by using the methodology of dual weighted residuals.
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