Approximate approximations: recent developments in the computation of high dimensional potentials

Lanzara, F.; Schmidt, G.

doi:10.1090/spmj/1601

Approximate approximations: recent developments in the computation of high dimensional potentials
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by F. Lanzara and G. Schmidt

St. Petersburg Math. J. 31 (2020), 355-370

DOI: https://doi.org/10.1090/spmj/1601

Published electronically: February 4, 2020

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Abstract:

The paper is devoted to a fast method of an arbitrary high order for approximating volume potentials that is successful also in the high dimensional case. The cubature formulas have been obtained by using the basis functions introduced in the theory of approximate approximations. As basis functions, we choose products of Gaussians and special polynomials, for which the action of integral operators can be written as one-dimensional integrals with a separable integrand, i.e., a product of functions depending only on one of the variables. Then a separated representation of the density, combined with a suitable quadrature rule, leads to a separable representation of the integral operator. Since only one-dimensional operations are used, the resulting method is efficient also in the high dimensional case. We show how this new approach can be applied to the cubature of polyharmonic potentials, to potentials of elliptic differential operators acting on densities on hyper-rectangular domains, and to parabolic problems.

References

Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 1964. For sale by the Superintendent of Documents. MR 0167642
Gregory Beylkin and Martin J. Mohlenkamp, Numerical operator calculus in higher dimensions, Proc. Natl. Acad. Sci. USA 99 (2002), no. 16, 10246–10251. MR 1918798, DOI 10.1073/pnas.112329799
Gregory Beylkin and Martin J. Mohlenkamp, Algorithms for numerical analysis in high dimensions, SIAM J. Sci. Comput. 26 (2005), no. 6, 2133–2159. MR 2196592, DOI 10.1137/040604959
Lawrence C. Evans, Partial differential equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR 2597943, DOI 10.1090/gsm/019
Ivan P. Gavrilyuk, Wolfgang Hackbusch, and Boris N. Khoromskij, Hierarchical tensor-product approximation to the inverse and related operators for high-dimensional elliptic problems, Computing 74 (2005), no. 2, 131–157. MR 2133692, DOI 10.1007/s00607-004-0086-y
Wolfgang Hackbusch and Boris N. Khoromskij, Tensor-product approximation to multidimensional integral operators and Green’s functions, SIAM J. Matrix Anal. Appl. 30 (2008), no. 3, 1233–1253. MR 2447450, DOI 10.1137/060657017
Filippo Gazzola, Hans-Christoph Grunau, and Guido Sweers, Polyharmonic boundary value problems, Lecture Notes in Mathematics, vol. 1991, Springer-Verlag, Berlin, 2010. Positivity preserving and nonlinear higher order elliptic equations in bounded domains. MR 2667016, DOI 10.1007/978-3-642-12245-3
M. R. Hestenes, Extension of the range of a differentiable function, Duke Math. J. 8 (1941), 183–192. MR 3434
Boris N. Khoromskij, Fast and accurate tensor approximation of a multivariate convolution with linear scaling in dimension, J. Comput. Appl. Math. 234 (2010), no. 11, 3122–3139. MR 2665374, DOI 10.1016/j.cam.2010.02.004
Flavia Lanzara, Vladimir Maz’ya, and Gunther Schmidt, On the fast computation of high dimensional volume potentials, Math. Comp. 80 (2011), no. 274, 887–904. MR 2772100, DOI 10.1090/S0025-5718-2010-02425-1
F. Lanzara, V. G. Maz’ya, and G. Schmidt, Accurate cubature of volume potentials over high-dimensional half-spaces, J. Math. Sci. (N.Y.) 173 (2011), no. 6, 683–700. Problems in mathematical analysis. No. 55. MR 2839852, DOI 10.1007/s10958-011-0267-0
F. Lanzara, V. G. Maz’ya, and G. Schmidt, Computation of volume potentials over bounded domains via approximate approximations, J. Math. Sci. (N.Y.) 189 (2013), no. 3, 508–524. Problems in mathematical analysis. No. 68. MR 3098350, DOI 10.1007/s10958-013-1203-2
F. Lanzara, V. Maz’ya, and G. Schmidt, Fast cubature of volume potentials over rectangular domains by approximate approximations, Appl. Comput. Harmon. Anal. 36 (2014), no. 1, 167–182. MR 3130582, DOI 10.1016/j.acha.2013.06.003
F. Lanzara, V. Maz’ya, and G. Schmidt, Approximation of solutions to multidimensional parabolic equations by approximate approximations, Appl. Comput. Harmon. Anal. 41 (2016), no. 3, 749–767. MR 3546428, DOI 10.1016/j.acha.2015.06.001
F. Lanzara, V. Maz’ya, and G. Schmidt, A fast solution method for time dependent multidimensional Schrödinger equations, Appl. Anal. 98 (2019), no. 1-2, 408–429. MR 3902138, DOI 10.1080/00036811.2017.1359571
Flavia Lanzara, Vladimir Maz’ya, and Gunther Schmidt, Accurate computation of the high dimensional diffraction potential over hyper-rectangles, Bull. TICMI 22 (2018), no. 2, 91–102. MR 3917417
—, Fast cubature of high dimensional biharmonic potential based on “Approximate approximations,” Ann. Univ. Ferrara (2019); https://doi.org/10.1007/s11565-019-00328-z
Flavia Lanzara and Gunther Schmidt, On the computation of high-dimensional potentials of advection-diffusion operators, Mathematika 61 (2015), no. 2, 309–327. MR 3343055, DOI 10.1112/S0025579314000412
V. Maz′ya, A new approximation method and its applications to the calculation of volume potentials, boundary point method, 3. DFG-Kolloqium des DFG-Forschungsschwerpunktes Randelementmethoden, 1991.
V. Maz′ya, Approximate approximations, The mathematics of finite elements and applications (Uxbridge, 1993) Wiley, Chichester, 1994, pp. 77–104. MR 1291219
Vladimir Maz’ya, Sobolev spaces with applications to elliptic partial differential equations, Second, revised and augmented edition, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 342, Springer, Heidelberg, 2011. MR 2777530, DOI 10.1007/978-3-642-15564-2
Vladimir Maz′ya and Gunther Schmidt, “Approximate approximations” and the cubature of potentials, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 6 (1995), no. 3, 161–184 (English, with English and Italian summaries). MR 1363785
Vladimir Maz’ya and Gunther Schmidt, Approximate approximations, Mathematical Surveys and Monographs, vol. 141, American Mathematical Society, Providence, RI, 2007. MR 2331734, DOI 10.1090/surv/141
Dorina Mitrea, Distributions, partial differential equations, and harmonic analysis, Universitext, Springer, New York, 2013. MR 3114783, DOI 10.1007/978-1-4614-8208-6
A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integraly i ryady, “Nauka”, Moscow, 1983 (Russian). Spetsial′nye funktsii. [Special functions]. MR 737562
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
Hidetosi Takahasi and Masatake Mori, Double exponential formulas for numerical integration, Publ. Res. Inst. Math. Sci. 9 (1973/74), 721–741. MR 0347061, DOI 10.2977/prims/1195192451