| 1. |
- Kovacs, Mihaly, 1977, et al.
(författare)
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Nonlinear semigroups for nonlocal conservation laws
- 2023
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Ingår i: Partial Differential Equations and Applications. - 2662-2963 .- 2662-2971. ; 4:4
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Tidskriftsartikel (refereegranskat)abstract
- We investigate a class of nonlocal conservation laws in several space dimensions, where the continuum average of weighted nonlocal interactions are considered over a finite horizon. We establish well-posedness for a broad class of flux functions and initial data via semigroup theory in Banach spaces and, in particular, via the celebrated Crandall–Liggett Theorem. We also show that the unique mild solution satisfies a Kružkov-type nonlocal entropy inequality. Similarly to the local case, we demonstrate an efficient way of proving various desirable qualitative properties of the unique solution.
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| 2. |
- Andersson, Adam, 1979, et al.
(författare)
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Weak error analysis for semilinear stochastic Volterra equations with additive noise
- 2014
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- We prove a weak error estimate for the approximation in space and time of a semilinear stochastic Volterra integro-differential equation driven by additive space-time Gaussian noise. We treat this equation in an abstract framework, in which parabolic stochastic partial differential equations are also included as a special case. The approximation in space is performed by a standard finite element method and in time by an implicit Euler method combined with a convolution quadrature. The weak rate of convergence is proved to be twice the strong rate, as expected. Our weak convergence result concerns not only the solution at a fixed time but also integrals of the entire path with respect to any finite Borel measure. The proof does not rely on a Kolmogorov equation. Instead it is based on a duality argument from Malliavin calculus.
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| 3. |
- Andersson, Adam, 1979, et al.
(författare)
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Weak error analysis for semilinear stochastic Volterra equations with additive noise
- 2016
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Ingår i: Journal of Mathematical Analysis and Applications. - : Elsevier BV. - 0022-247X .- 1096-0813. ; 437:2, s. 1283-1304
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Tidskriftsartikel (refereegranskat)abstract
- We prove a weak error estimate for the approximation in space and time of a semilinear stochastic Volterra integro-differential equation driven by additive space-time Gaussian noise. We treat this equation in an abstract framework, in which parabolic stochastic partial differential equations are also included as a special case. The approximation in space is performed by a standard finite element method and in time by an implicit Euler method combined with a convolution quadrature. The weak rate of convergence is proved to be twice the strong rate, as expected. Our convergence result concerns not only functionals of the solution at a fixed time but also more complicated functionals of the entire path and includes convergence of covariances and higher order statistics. The proof does not rely on a Kolmogorov equation. Instead it is based on a duality argument from Malliavin calculus.
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| 4. |
- Baeumer, B., et al.
(författare)
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Boundary conditions for fractional diffusion
- 2018
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Ingår i: Journal of Computational and Applied Mathematics. - : Elsevier BV. - 0377-0427. ; 336, s. 408-424
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Tidskriftsartikel (refereegranskat)abstract
- This paper derives physically meaningful boundary conditions for fractional diffusion equations, using a mass balance approach. Numerical solutions are presented, and theoretical properties are reviewed, including well-posedness and steady state solutions. Absorbing and reflecting boundary conditions are considered, and illustrated through several examples. Reflecting boundary conditions involve fractional derivatives. The Caputo fractional derivative is shown to be unsuitable for modeling fractional diffusion, since the resulting boundary value problem is not positivitypreserving.
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| 5. |
- Baeumer, Boris, et al.
(författare)
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Fractional partial differential equations with boundary conditions
- 2018
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Ingår i: Journal of Differential Equations. - : Elsevier BV. - 0022-0396 .- 1090-2732. ; 264, s. 1377-1410
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Tidskriftsartikel (refereegranskat)abstract
- We identify the stochastic processes associated with one-sided fractional partial differential equations on a bounded domain with various boundary conditions. This is essential for modelling using spatial fractional derivatives. We show well-posedness of the associated Cauchy problems in C 0 (Ω) and L 1 (Ω). In order to do so we develop a new method of embedding finite state Markov processes into Feller processes on bounded domains and then show convergence of the respective Feller processes. This also gives a numerical approximation of the solution. The proof of well-posedness closes a gap in many numerical algorithm articles approximating solutions to fractional differential equations that use the Lax–Richtmyer Equivalence Theorem to prove convergence without checking well-posedness.
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| 6. |
- Baeumer, Boris, et al.
(författare)
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Reprint of: Boundary conditions for fractional diffusion
- 2018
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Ingår i: Journal of Computational and Applied Mathematics. - : Elsevier BV. - 0377-0427. ; 339, s. 414-430
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Tidskriftsartikel (refereegranskat)abstract
- This paper derives physically meaningful boundary conditions for fractional diffusion equations, using a mass balance approach. Numerical solutions are presented, and theoretical properties are reviewed, including well-posedness and steady state solutions. Absorbing and reflecting boundary conditions are considered, and illustrated through several examples. Reflecting boundary conditions involve fractional derivatives. The Caputo fractional derivative is shown to be unsuitable for modeling fractional diffusion, since the resulting boundary value problem is not positivity preserving.
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| 7. |
- Bolin, David, 1983, et al.
(författare)
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Numerical solution of fractional elliptic stochastic PDEs with spatial white noise
- 2020
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Ingår i: Ima Journal of Numerical Analysis. - : Oxford University Press (OUP). - 0272-4979 .- 1464-3642. ; 40:2, s. 1051-1073
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Tidskriftsartikel (refereegranskat)abstract
- The numerical approximation of solutions to stochastic partial differential equations with additive spatial white noise on bounded domains in R-d is considered. The differential operator is given by the fractional power L-beta, beta is an element of (0, 1) of an integer-order elliptic differential operator L and is therefore nonlocal. Its inverse L-beta is represented by a Bochner integral from the Dunford-Taylor functional calculus. By applying a quadrature formula to this integral representation the inverse fractional-order operator L-beta is approximated by a weighted sum of nonfractional resolvents (I + exp(2yl)L)(-1) at certain quadrature nodes t(j) > 0. The resolvents are then discretized in space by a standard finite element method. This approach is combined with an approximation of the white noise, which is based only on the mass matrix of the finite element discretization. In this way an efficient numerical algorithm for computing samples of the approximate solution is obtained. For the resulting approximation the strong mean-square error is analyzed and an explicit rate of convergence is derived. Numerical experiments for L = kappa(2) - Delta, kappa > 0 with homogeneous Dirichlet boundary conditions on the unit cube (0, 1)(d) in d = 1, 2, 3 spatial dimensions for varying beta is an element of (0, 1) attest to the theoretical results.
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| 8. |
- Bolin, David, et al.
(författare)
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REGULARITY AND NUMERICAL APPROXIMATION OF FRACTIONAL ELLIPTIC DIFFERENTIAL EQUATIONS ON COMPACT METRIC GRAPHS
- 2024
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Ingår i: Mathematics of Computation. - 1088-6842 .- 0025-5718. ; 93:349, s. 2439-2472
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Tidskriftsartikel (refereegranskat)abstract
- The fractional differential equation L(beta)u = f posed on a compact metric graph is considered, where beta > 0 and L = kappa(2) - del(a del ) is a second order elliptic operator equipped with certain vertex conditions and sufficiently smooth and positive coefficients kappa, a. We demonstrate the existence of a unique solution for a general class of vertex conditions and derive the regularity of the solution in the specific case of Kirchhoff vertex conditions. These results are extended to the stochastic setting when f is replaced by Gaussian white noise. For the deterministic and stochastic settings under generalized Kirchhoff vertex conditions, we propose a numerical solution based on a finite element approximation combined with a rational approximation of the fractional power L-beta. For the resulting approximation, the strong error is analyzed in the deterministic case, and the strong mean squared error as well as the L-2( Gamma x Gamma )error of the covariance function of the solution are analyzed in the stochastic setting. Explicit rates of convergences are derived for all cases. Numerical experiments for L = kappa(2) - del, kappa > 0 are performed to illustrate the results.
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| 9. |
- Bolin, David, 1983, et al.
(författare)
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Weak convergence of Galerkin approximations for fractional elliptic stochastic PDEs with spatial white noise
- 2018
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Ingår i: BIT (Copenhagen). - : Springer Science and Business Media LLC. - 0006-3835 .- 1572-9125. ; 58:4, s. 881-906
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Tidskriftsartikel (refereegranskat)abstract
- The numerical approximation of the solution to a stochastic partial differential equation with additive spatial white noise on a bounded domain is considered. The differential operator is assumed to be a fractional power of an integer order elliptic differential operator. The solution is approximated by means of a finite element discretization in space and a quadrature approximation of an integral representation of the fractional inverse from the Dunford–Taylor calculus. For the resulting approximation, a concise analysis of the weak error is performed. Specifically, for the class of twice continuously Fréchet differentiable functionals with second derivatives of polynomial growth, an explicit rate of weak convergence is derived, and it is shown that the component of the convergence rate stemming from the stochasticity is doubled compared to the corresponding strong rate. Numerical experiments for different functionals validate the theoretical results.
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| 10. |
- 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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