| 61. |
- 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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| 62. |
- Hausenblas, E., et al.
(författare)
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GLOBAL SOLUTIONS TO STOCHASTIC VOLTERRA EQUATIONS DRIVEN BY LEVY NOISE
- 2018
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Ingår i: Fractional Calculus and Applied Analysis. - : Springer Science and Business Media LLC. - 1311-0454 .- 1314-2224. ; 21:5, s. 1170-1202
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we investigate the existence and uniqueness of semilinear stochastic Volterra equations driven by multiplicative Levy noise of pure jump type. In particular, we consider the equation {du(t) = (A integral(t)(0) b(t - s)u(s) ds) dt vertical bar F(t, u(t)) dt + integral(Z) G(t, u(t), z) (eta) over tilde (dz, dt) + integral(ZL) G(L)(t, u(t), z)eta(L)(dz, dt), t is an element of (0, T], where Z and Z(L) are Banach spaces, (eta) over tilde is a time-homogeneous compensated Poisson random measure on Z with intensity measure. (capturing the small jumps), and eta(L) is a time-homogeneous Poisson random measure on ZL independent to (eta) over tilde. with finite intensity measure nu(L) (capturing the large jumps). Here, A is a selfadjoint operator on a Hilbert space H, b is a scalar memory function and F, G and G(L) are nonlinear mappings. We provide conditions on b, F G and GL under which a unique global solution exists. We also present an example from the theory of linear viscoelasticity where our result is applicable. The specific kernel b(t) = c(rho)t(rho-2), 1 < rho < 2, corresponds to a fractional-in-time stochastic equation and the nonlinear maps F and G can include fractional powers of A.
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| 63. |
- Kovacs, Mihaly, 1977, et al.
(författare)
-
Hilbert-Schmidt regularity of symmetric integral operators on bounded domains with applications to SPDE approximations
- 2023
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Ingår i: Stochastic Analysis and Applications. - : Informa UK Limited. - 0736-2994 .- 1532-9356. ; 41:3, s. 564-90
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Tidskriftsartikel (refereegranskat)abstract
- Regularity estimates for an integral operator with a symmetric continuous kernel on a convex bounded domain are derived. The covariance of a mean-square continuous random field on the domain is an example of such an operator. The estimates are of the form of Hilbert-Schmidt norms of the integral operator and its square root, composed with fractional powers of an elliptic operator equipped with homogeneous boundary conditions of either Dirichlet or Neumann type. These types of estimates, which couple the regularity of the driving noise with the properties of the differential operator, have important implications for stochastic partial differential equations on bounded domains as well as their numerical approximations. The main tools used to derive the estimates are properties of reproducing kernel Hilbert spaces of functions on bounded domains along with Hilbert-Schmidt embeddings of Sobolev spaces. Both non-homogeneous and homogeneous kernels are considered. In the latter case, results in a general Schatten class norm are also provided. Important examples of homogeneous kernels covered by the results of the paper include the class of Matern kernels.
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| 64. |
- Kovacs, Mihaly, 1977, et al.
(författare)
-
On the discretisation in time of the stochastic Allen-Cahn equation
- 2018
-
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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| 65. |
- Kovacs, Mihaly, 1977, et al.
(författare)
-
On the stochastic Allen-Cahn equation on networks with multiplicative noise
- 2021
-
Ingår i: Electronic Journal of Qualitative Theory of Differential Equations. - : University of Szeged. - 1417-3875. ; :7, s. 1-24
-
Tidskriftsartikel (refereegranskat)abstract
- We consider a system of stochastic Allen-Cahn equations on a finite network represented by a finite graph. On each edge in the graph a multiplicative Gaussian noise driven stochastic Allen-Cahn equation is given with possibly different potential barrier heights supplemented by a continuity condition and a Kirchhoff-type law in the vertices. Using the semigroup approach for stochastic evolution equations in Banach spaces we obtain existence and uniqueness of solutions with sample paths in the space of continuous functions on the graph. We also prove more precise space-time regularity of the solution.
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| 66. |
- Kovacs, Mihaly, 1977, et al.
(författare)
-
Stochastic reaction-diffusion equations on networks
- 2021
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Ingår i: Journal of Evolution Equations. - : Springer Science and Business Media LLC. - 1424-3199 .- 1424-3202. ; 21, s. 4213-4260
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Tidskriftsartikel (refereegranskat)abstract
- We consider stochastic reaction-diffusion equations on a finite network represented by a finite graph. On each edge in the graph, a multiplicative cylindrical Gaussian noise-driven reaction-diffusion equation is given supplemented by a dynamic Kirchhoff-type law perturbed by multiplicative scalar Gaussian noise in the vertices. The reaction term on each edge is assumed to be an odd degree polynomial, not necessarily of the same degree on each edge, with possibly stochastic coefficients and negative leading term. We utilize the semigroup approach for stochastic evolution equations in Banach spaces to obtain existence and uniqueness of solutions with sample paths in the space of continuous functions on the graph. In order to do so, we generalize existing results on abstract stochastic reaction-diffusion equations in Banach spaces.
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| 67. |
- Kovacs, Mihaly, 1977, et al.
(författare)
-
Weak convergence of fully discrete finite element approximations of semilinear hyperbolic SPDE with additive noise
- 2020
-
Ingår i: Esaim-Mathematical Modelling and Numerical Analysis-Modelisation Mathematique Et Analyse Numerique. - : EDP Sciences. - 0764-583X .- 2822-7840 .- 1290-3841. ; 54:6, s. 2199-2227
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Tidskriftsartikel (refereegranskat)abstract
- The numerical approximation of the mild solution to a semilinear stochastic wave equation driven by additive noise is considered. A standard finite element method is employed for the spatial approximation and a a rational approximation of the exponential function for the temporal approximation. First, strong convergence of this approximation in both positive and negative order norms is proven. With the help of Malliavin calculus techniques this result is then used to deduce weak convergence rates for the class of twice continuously differentiable test functions with polynomially bounded derivatives. Under appropriate assumptions on the parameters of the equation, the weak rate is found to be essentially twice the strong rate. This extends earlier work by one of the authors to the semilinear setting. Numerical simulations illustrate the theoretical results.
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| 68. |
- Kow, P. Z., et al.
(författare)
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Quadrature Domains for the Helmholtz Equation with Applications to Non-scattering Phenomena
- 2023
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Ingår i: Potential Analysis. - : Springer Science and Business Media LLC. - 0926-2601 .- 1572-929X. ; 60, s. 387-424
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we introduce quadrature domains for the Helmholtz equation. We show existence results for such domains and implement the so-called partial balayage procedure. We also give an application to inverse scattering problems, and show that there are non-scattering domains for the Helmholtz equation at any positive frequency that have inward cusps.
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| 69. |
- Lang, Annika, 1980, et al.
(författare)
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Galerkin-Chebyshev approximation of Gaussian random fields on compact Riemannian manifolds
- 2023
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Ingår i: Bit Numerical Mathematics. - 0006-3835 .- 1572-9125. ; 63:4
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Tidskriftsartikel (refereegranskat)abstract
- A new numerical approximation method for a class of Gaussian random fields on compact connected oriented Riemannian manifolds is introduced. This class of random fields is characterized by the Laplace-Beltrami operator on the manifold. A Galerkin approximation is combined with a polynomial approximation using Chebyshev series. This so-called Galerkin-Chebyshev approximation scheme yields efficient and generic sampling algorithms for Gaussian random fields on manifolds. Strong and weak orders of convergence for the Galerkin approximation and strong convergence orders for the Galerkin-Chebyshev approximation are shown and confirmed through numerical experiments.
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| 70. |
- Lindo, Alexey, 1987, et al.
(författare)
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General linear-fractional branching processes with discrete time.
- 2018
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Ingår i: Stochastics. - : Informa UK Limited. - 1744-2508 .- 1744-2516. ; 90:3, s. 364-378
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Tidskriftsartikel (refereegranskat)abstract
- We study a linear-fractional Bienayme-Galton-Watson process with a general type space. The corresponding tree contour process is described by an alternating random walk with the downward jumps having a geometric distribution. This leads to the linear-fractional distribution formula for an arbitrary observation time, which allows us to establish transparent limit theorems for the subcritical, critical and supercritical cases. Our results extend recent findings for the linear-fractional branching processes with countably many types.
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