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On the fast computation of high dimensional volume potentials

Authors: Flavia Lanzara, Vladimir Maz’ya and Gunther Schmidt
Journal: Math. Comp. 80 (2011), 887-904
MSC (2010): Primary 65D32; Secondary 65-05
Published electronically: September 22, 2010
MathSciNet review: 2772100
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Abstract: A fast method of an arbitrary high order for approximating volume potentials is proposed, which is effective also in high dimensional cases. Basis functions introduced in the theory of approximate approximations are used. Results of numerical experiments, which show approximation order $ O(h^8)$ for the Newton potential in high dimensions, for example, for $ n= 200 000$, are provided. The computation time scales linearly in the space dimension. New one-dimensional integral representations with separable integrands of the potentials of advection-diffusion and heat equations are obtained.

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Additional Information

Flavia Lanzara
Affiliation: Dipartimento di Matematica, Università “La Sapienza”, Piazzale Aldo Moro 2, 00185 Rome, Italy

Vladimir Maz’ya
Affiliation: Department of Mathematical Sciences, M&O Building, University of Liverpool, Liverpool L69 3BX, United Kingdom –and– Department of Mathematics, University of Linköping, 581 83 Linköping, Sweden

Gunther Schmidt
Affiliation: Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany

Received by editor(s): November 2, 2009
Received by editor(s) in revised form: February 20, 2010
Published electronically: September 22, 2010
Additional Notes: This research was partially supported by the UK and Engineering and Physical Sciences Research Council via the grant EP/F005563/1.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.