Utilize este identificador para referenciar este registo:
http://hdl.handle.net/10071/26202
Autoria: | Rodriguez-Rozas, A. Acebron, J.A. Spigler, R. |
Editor: | Luisa Beghin Francesco Mainardi Roberto Garrappa |
Data: | 2021 |
Título próprio: | The PDD method for solving linear, nonlinear, and fractional PDEs problems |
Título e volume do livro: | Nonlocal and fractional operators |
Paginação: | 239 - 273 |
Título e número da coleção: | SEMA SIMAI Springer Series |
Referência bibliográfica: | 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 |
Palavras-chave: | 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 |
Resumo: | 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. |
Arbitragem científica: | yes |
Acesso: | Acesso Aberto |
Aparece nas coleções: | IT-CLI - Capítulos de livros internacionais |
Ficheiros deste registo:
Ficheiro | Tamanho | Formato | |
---|---|---|---|
bookPart_84725.pdf | 609,87 kB | Adobe PDF | Ver/Abrir |
Todos os registos no repositório estão protegidos por leis de copyright, com todos os direitos reservados.