Parallel Algorithms for Parabolic Problems on Graphs

In this paper two parallel numerical algorithms for solution of parabolic problems on graphs are investigated. The fully implicit and predictor-corrector finite difference schemes are proposed to approximate the differential equations on the given graph. The parallelization of the discrete algorithm is based on the domain decomposition method. Scalability analysis of the parallel algorithms is done. Some results of numerical simulations are presented and the efficiency of the proposed parallel algorithms is investigated.

[1]  Xian-He Sun,et al.  Stabilized Explicit-Implicit Domain Decomposition Methods for the Numerical Solution of Parabolic Equations , 2002, SIAM J. Sci. Comput..

[2]  Michael L. Hines,et al.  Independent Variable Time-Step Integration of Individual Neurons for Network Simulations , 2005, Neural Computation.

[3]  David L. Chopp,et al.  A Predictor-Corrector Algorithm for Reaction-Diffusion Equations Associated with Neural Activity on Branched Structures , 2006, SIAM J. Sci. Comput..

[4]  Hong-Lin Liao,et al.  Unconditional Stability of Corrected Explicit-Implicit Domain Decomposition Algorithms for Parallel Approximation of Heat Equations , 2006, SIAM J. Numer. Anal..

[5]  R. Čiegis,et al.  Finite-difference schemes for parabolic problems on graphs , 2010 .

[6]  Michael L. Hines,et al.  The NEURON Book , 2006 .

[7]  M L Hines,et al.  Neuron: A Tool for Neuroscientists , 2001, The Neuroscientist : a review journal bringing neurobiology, neurology and psychiatry.

[8]  V. Starikovièius,et al.  Parallel numerical algorithms for optimization of electrical cables , 2008 .

[9]  Raimondas Ciegis,et al.  Paralel Predictor-Corrector Schemes for Parabolic Problems on Graphs , 2010, Comput. Methods Appl. Math..

[10]  Michael Mascagni,et al.  The backward Euler method for numerical solution of the Hodgkin-Huxley equations of nerve conduction , 1990 .

[11]  Raimondas Čiegis,et al.  A Parallel Solver for the Design of Oil Filters , 2011 .

[12]  Michael L. Hines,et al.  Parallel network simulations with NEURON , 2006, Journal of Computational Neuroscience.