A new parallel solver suited for arbitrary semilinear parabolic partial differential equations based on generalized random trees

A probabilistic representation for initial value semilinear parabolic problems based on generalized random trees has been derived. Two different strategies have been proposed, both requiring generating suitable random trees combined with a Pade approximant for approximating accurately a given divergent series. Such series are obtained by summing the partial contribution to the solution coming from trees with arbitrary number of branches. The new representation greatly expands the class of problems amenable to be solved probabilistically, and was used successfully to develop a generalized probabilistic domain decomposition method. Such a method has been shown to be suited for massively parallel computers, enjoying full scalability and fault tolerance. Finally, a few numerical examples are given to illustrate the remarkable performance of the algorithm, comparing the results with those obtained with a classical method.

[1]  Ronald L. Graham,et al.  Concrete Mathematics, a Foundation for Computer Science , 1991, The Mathematical Gazette.

[2]  M. Freidlin.  Functional Integration And Partial Differential Equations , 1985 .

[3]  Barry Smith,et al.  Domain Decomposition Methods for Partial Differential Equations , 1997 .

[4]  Juan A. Acebrón,et al.  Efficient Parallel Solution of Nonlinear Parabolic Partial Differential Equations by a Probabilistic Domain Decomposition , 2010, J. Sci. Comput..

[5]  H. McKean.  Application of brownian motion to the equation of kolmogorov-petrovskii-piskunov , 1975 .

[6]  Jorge M. Ramírez,et al.  Multiplicative cascades applied to PDEs (two numerical examples) , 2006, J. Comput. Phys..

[7]  Paul C. Du Chateau,et al.  Applied Partial Differential Equations , 1990, The Mathematical Gazette.

[8]  Juan A. Acebrón,et al.  Domain decomposition solution of nonlinear two-dimensional parabolic problems by random trees , 2009, J. Comput. Phys..

[9]  H. M. Antia.  Numerical Methods for Scientists and Engineers , 2002 .

[10]  Vivek Sarkar,et al.  Software challenges in extreme scale systems , 2009 .

[11]  T. Chan,et al.  Domain decomposition algorithms , 1994, Acta Numerica.

[12]  Piero Lanucara,et al.  Domain Decomposition Solution of Elliptic Boundary-Value Problems via Monte Carlo and Quasi-Monte Carlo Methods , 2005, SIAM J. Sci. Comput..

[13]  P. Kloeden,et al.  Numerical Solution of Stochastic Differential Equations , 1992 .

[14]  Jean-Christophe Aval,et al.  Multivariate Fuss-Catalan numbers , 2007, Discret. Math..

[15]  O. Widlund.  Domain Decomposition Algorithms , 1993 .

[16]  M. V. Tretyakov,et al.  Stochastic Numerics for Mathematical Physics , 2004, Scientific Computation.

[17]  Ioannis Karatzas,et al.  Brownian Motion and Stochastic Calculus , 1987 .

[18]  R. Vilela Mendes,et al.  Poisson-Vlasov: stochastic representation and numerical codes , 2008 .

[19]  Piero Lanucara,et al.  Probabilistically induced domain decomposition methods for elliptic boundary-value problems , 2005 .

[20]  Luc Wuytack.  On the conditioning of the Padé approximation problem , 1981 .

[21]  Denis Talay,et al.  Convergence rate of the Sherman and Peskin branching stochastic particle method , 2004, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[22]  G. A. Baker,et al.  THE PADE APPROXIMANT , 1961 .

[23]  P. Zweifel.  Advanced Mathematical Methods for Scientists and Engineers , 1980 .

[24]  Peter Arbenz,et al.  Introduction to parallel computing : a practical guide with examples in C , 2004 .