The exit-time problem for a Markov jump process

Burch, Nathanial; D’Elia, Marta; Lehoucq, Richard B.

doi:10.1140/epjst/e2014-02331-7

Cited by 48 publications

(27 citation statements)

References 26 publications

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“…A nonlocal equation is characterized by integral operators acting on a lengthscale or "horizon", describing long-range forces and reducing the regularity requirements on the solutions. Engineering applications include surface or subsurface transport [3,4,21,30,31,47,48], fracture mechanics [29,34,52], turbulence [32,41], image processing [1,11,27,35] and stochastic processes [6,13,37,39,40].…”

Section: Introductionmentioning

confidence: 99%

Analysis of Anisotropic Nonlocal Diffusion Models: Well-posedness of Fractional Problems for Anomalous Transport

D’Elia¹,

Gulian²

2021

Preprint

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We analyze the well-posedness of an anisotropic, nonlocal diffusion equation. Establishing an equivalence between weighted and unweighted anisotropic nonlocal diffusion operators in the vein of unified nonlocal vector calculus, we apply our analysis to a class of fractional-order operators and present rigorous estimates for the solution of the corresponding anisotropic anomalous diffusion equation. Furthermore, we extend our analysis to the anisotropic diffusion-advection equation and prove well-posedness for fractional orders s ∈ [0.5, 1). We also present an application of the advection-diffusion equation to anomalous transport of solutes.

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Section: Introductionmentioning

confidence: 99%

Analysis of Anisotropic Nonlocal Diffusion Models: Well-posedness of Fractional Problems for Anomalous Transport

D’Elia¹,

Gulian²

2021

Preprint

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“…Nonlocal models that account for interaction occurring at a distance have been shown to provide improved simulation fidelity in the presence of long-range forces and anomalous behaviors [11]. For this reason, they have found interesting applications in a variety of research fields such as fracture mechanics [21,26,36], phase transitions [1,7,18], stochastic processes [5,9,28,32] and image processing [4,19,20,27].…”

Section: Introductionmentioning

confidence: 99%

On the convergence to local limit of nonlocal models with approximated interaction neighborhoods

Du¹,

Xie²,

Yin³

2021

Preprint

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Many nonlocal models have adopted a finite and radially symmetric nonlocal interaction domains. When solving them numerically, it is sometimes convenient to adopt polygonal approximations of such interaction domains. A crucial question is, to what extent such approximations affect the nonlocal operators and the corresponding nonlocal solutions. While recent works have analyzed this issue for nonlocal operators in the case of a fixed horizon parameter, the question remains open in the case of a small or vanishing horizon parameter, which happens often in many practical applications and has significant impact on the reliability and robustness of nonlocal modeling and simulations. In this work, we are interested in addressing this issue and establishing the convergence of new nonlocal solutions by polygonal approximations to the local limit of the original nonlocal solutions. Our finding reveals that the new nonlocal solution does not converge to the correct local limit when the number of sides of polygons is uniformly bounded. On the other hand, if the number of sides tends to infinity, the desired convergence can be shown. These results may be used to guide future computational studies of nonlocal problems.

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“…Volume constraints replace the boundary conditions associated with classical diffusion, and are needed to demonstrate that the problem is well posed. It also provides a link with a Markov jump process; see [7,8] for additional information, motivation and citations to the literature.…”

Section: Introductionmentioning

confidence: 99%

A meshless Galerkin method for non-local diffusion using localized kernel bases

et al. 2018

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We introduce a meshless method for solving both continuous and discrete variational formulations of a volume constrained, nonlocal diffusion problem. We use the discrete solution to approximate the continuous solution. Our method is nonconforming and uses a localized Lagrange basis that is constructed out of radial basis functions. By verifying that certain inf-sup conditions hold, we demonstrate that both the continuous and discrete problems are well-posed, and also present numerical and theoretical results for the convergence behavior of the method. The stiffness matrix is assembled by a special quadrature routine unique to the localized basis. Combining the quadrature method with the localized basis produces a well-conditioned, symmetric matrix. This then is used to find the discretized solution.

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The exit-time problem for a Markov jump process

Cited by 48 publications

References 26 publications

Analysis of Anisotropic Nonlocal Diffusion Models: Well-posedness of Fractional Problems for Anomalous Transport

Analysis of Anisotropic Nonlocal Diffusion Models: Well-posedness of Fractional Problems for Anomalous Transport

On the convergence to local limit of nonlocal models with approximated interaction neighborhoods

A meshless Galerkin method for non-local diffusion using localized kernel bases

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