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Solving equations through particle dynamics
Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences.
Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences.ORCID iD: 0000-0002-0529-1009
Mid Sweden University, Faculty of Science, Technology and Media, Department of Natural Sciences.ORCID iD: 0000-0001-7329-3359
2015 (English)In: Computer Physics Communications, ISSN 0010-4655, E-ISSN 1879-2944, Vol. 197, 169-181 p.Article in journal (Refereed) Published
Abstract [en]

The present work evaluates a recently developed particle method (DFPM). The basic idea behind this method is to utilize a Newtonian system of interacting particles that through dissipation solves mathematical problems. We find that this second order dynamical system results in an algorithm that is among the best methods known. The present work studies large systems of linear equations. Of special interest is the wide eigenvalue spectrum. This case is common as the discretization of the continuous problem becomes dense. The convergence rate of DFPM is shown to be in parity with that of the conjugate gradient method, both analytically and through numerical examples. However, an advantage with DFPM is that it is cheaper per iteration. Another advantage is that it is not restricted to symmetric matrices only, as is the case for the conjugate gradient method. The convergence properties of DFPM are shown to be superior to the closely related approach utilizing only a first order dynamical system, and also to several other iterative methods in numerical linear algebra. The performance properties are understood and optimized by taking advantage of critically damped oscillators in classical mechanics. Just as in the case of the conjugate gradient method, a limitation is that all eigenvalues (spring constants) are required to be of the same sign. DFPM has no other limitation such as matrix structure or a spectral radius as is common among iterative methods. Examples are provided to test the particle algorithm’s merits and also various performance comparisons with existent numerical algorithms are provided.

Place, publisher, year, edition, pages
Elsevier, 2015. Vol. 197, 169-181 p.
Keyword [en]
Particle methods, Computational mechanics, Many-particle dynamics, System of linear equations, Dynamical functional particle method
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:miun:diva-26030DOI: 10.1016/j.cpc.2015.08.028ISI: 000362919500018Scopus ID: 2-s2.0-84942990585OAI: oai:DiVA.org:miun-26030DiVA: diva2:858965
Available from: 2015-10-05 Created: 2015-10-05 Last updated: 2016-01-13Bibliographically approved

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Edvardsson, SverkerNeuman, MagnusEdström, PerOlin, Håkan
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CiteExportLink to record
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