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Author(s): Rodriguez-Rozas, A.
Acebron, J.A.
Spigler, R.
Editor: Luisa Beghin
Francesco Mainardi
Roberto Garrappa
Date: 2021
Title: The PDD method for solving linear, nonlinear, and fractional PDEs problems
Book title/volume: Nonlocal and fractional operators
Pages: 239 - 273
Collection title and number: SEMA SIMAI Springer Series
Reference: Rodriguez-Rozas, A., Acebron, J.A., & Spigler, R. (2021). The PDD method for solving linear, nonlinear, and fractional PDEs problems. Em Luisa Beghin, Francesco Mainardi, Roberto Garrappa (Eds.). Nonlocal and fractional operators (pp.239-273). Springer. 10.1007/978-3-030-69236-0_13
ISBN: 978-3-030-69235-3
DOI (Digital Object Identifier): 10.1007/978-3-030-69236-0_13
Keywords: Probabilistic domain decomposition
Domain decomposition methods
Partial differential equations
Método de Monte Carlo
Quasi-Monte Carlo
Elliptic operators
Transport equations
Vlasov-Poisson system
Nonlocal and fractional operators
Abstract: We review the Probabilistic Domain Decomposition (PDD) method for the numerical solution of linear and nonlinear Partial Differential Equation (PDE) problems. This Domain Decomposition (DD) method is based on a suitable probabilistic representation of the solution given in the form of an expectation which, in turns, involves the solution of a Stochastic Differential Equation (SDE). While the structure of the SDE depends only upon the corresponding PDE, the expectation also depends upon the boundary data of the problem. The method consists of three stages: (i) only few values of the sought solution are solved by Monte Carlo or Quasi-Monte Carlo at some interfaces; (ii) a continuous approximation of the solution over these interfaces is obtained via interpolation; and (iii) prescribing the previous (partial) solutions as additional Dirichlet boundary conditions, a fully decoupled set of sub-problems is finally solved in parallel. For linear parabolic problems, this is based on the celebrated Feynman-Kac formula, while for semilinear parabolic equations requires a suitable generalization based on branching diffusion processes. In case of semilinear transport equations and the Vlasov-Poisson system, a generalization of the probabilistic representation was also obtained in terms of the Method of Characteristics (characteristic curves). Finally, we present the latest progress towards the extension of the PDD method for nonlocal fractional operators. The algorithm notably improves the scalability of classical algorithms and is suited to massively parallel implementation, enjoying arbitrary scalability and fault tolerance properties. Numerical examples conducted in 1D and 2D, including some for the KPP equation and Plasma Physics, are given.
Peerreviewed: yes
Access type: Open Access
Appears in Collections:IT-CLI - Capítulos de livros internacionais

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