A new ‘walk on spheres’ type method for fractional diffusion equation in high dimensions based on the Feynman–Kac formulas

Su, Bihao; Xu, Chenglong; Sheng, Changtao

doi:10.1016/j.aml.2023.108597

Applied Mathematics Letters

2023

DOI: 10.1016/j.aml.2023.108597

|View full text |Cite

A new ‘walk on spheres’ type method for fractional diffusion equation in high dimensions based on the Feynman–Kac formulas

Bihao Su

Chenglong Xu

Changtao Sheng

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2024

Publication Types

Select...

Article1

Relationship

Self Cite0

Independent1

Authors

Journals

Cited by 1 publication

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

An implicit-explicit Monte Carlo method for semi-linear PDEs driven by 𝛼-stable Lévy process and its error estimates

Sheng,

Su,

2024

Math. Comp.

View full text Add to dashboard Cite

In this paper, we propose an implicit-explicit Monte Carlo method (IMEX-MCM) for solving semi-linear parabolic partial differential equations (PDEs) driven by α \alpha -stable Lévy process on bounded domains, also known as semi-linear PDEs involving integral fractional Laplacian operator (i.e., α ∈ ( 0 , 2 ) \alpha \in (0,2) ) and classical Laplacian operator (i.e., α = 2 \alpha =2 ). The main idea behind our approach is to divide the time interval into N N sub-intervals and derive the Feynman-Kac formula within each time sub-interval. Then, we develop a fully discretized parallel scheme using the implicit-explicit technique for temporal discretization, followed by applying a novel walk-on-sphere method to obtain the numerical solution. Notably, the proposed method takes advantage of the Poisson kernel associated with the stochastic processes to accommodate the full discretization of the classical and fractional semi-linear PDEs in a single framework. Moreover, we rigorously analyze the error bound for the proposed scheme that is totally explicit with respect to the number of simulation paths M M and the time step size Δ t \Delta t . As a by-product, we prove the maximum principle of the proposed schemes for fractional Allen Cahn equation with double well potential and provide an error estimate by using the established maximum principle. Extensive numerical experiments in two and three-dimensional spaces are presented to verify our theoretical results and to show the robustness and accuracy of our method.

show abstract

An implicit-explicit Monte Carlo method for semi-linear PDEs driven by 𝛼-stable Lévy process and its error estimates

Sheng,

Su,

2024

Math. Comp.

View full text Add to dashboard Cite

show abstract

scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.

Contact Info

customersupport@researchsolutions.com

10624 S. Eastern Ave., Ste. A-614

Henderson, NV 89052, USA

This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply.

Blog Terms and Conditions API Terms Privacy Policy Contact Cookie Preferences Do Not Sell or Share My Personal Information

Made with 💙 for researchers

Part of the Research Solutions Family.

A new ‘walk on spheres’ type method for fractional diffusion equation in high dimensions based on the Feynman–Kac formulas

Cited by 1 publication

References 10 publications

An implicit-explicit Monte Carlo method for semi-linear PDEs driven by 𝛼-stable Lévy process and its error estimates

An implicit-explicit Monte Carlo method for semi-linear PDEs driven by 𝛼-stable Lévy process and its error estimates

Contact Info

Product

Resources

About