212
Views
1
CrossRef citations to date
0
Altmetric
Articles

Fast computation of the multidimensional fractional Laplacian

ORCID Icon, &
Pages 4025-4041 | Received 18 Jun 2021, Accepted 14 Sep 2021, Published online: 05 Oct 2021

References

  • Kwaśnicki M. Ten equivalent definitions of the fractional Laplace operator. Fractional Calculus Appl Calculus. 2017;20:7–51.
  • Samko S. Hypersingular integrals and their applications. London: Taylor & Francis; 2002.
  • Stein EM. Singular integrals and differentiability properties of functions. Princeton (NJ): Princeton University Press; 1970.
  • Pozrikidis C. The fractional Laplacian. Boca Raton (FL): CRC Press; 2016.
  • Samko S, Kilbas A, Mariçev O. Fractional integrals and derivatives. Yverdon: Gordon and Breach Science Publ.; 1993.
  • Bucur C, Valdinoci E. Non-local diffusion and applications. Springer; 2016. (Lecture notes of the unione matematica italiana; vol. 20).
  • Landkof NS. Foundations of modern potential theory. New York (NY): Springer-Verlag; 1972. (Die Grundlehren der mathematischen Wissenschaften; vol. 180.)
  • Cont R, Tankov P. Financial modelling with jump processes. Boca Raton, FL: Chapman & Hall/CRC; 2004. (Chapman & hall/crc financial math. ser.; vol. 133).
  • Di Nezza E, Palatucci G, Valdinoci E. Hitchhiker's guide to the fractional Sobolev spaces. Bull Sci Math. 2021;136:521–573.
  • Raible S. Lévy processes in finance: theory, numerics, and empirical facts [Ph.D. thesis]. Freiburg im Breisgau, Germany: Universitat Freiburg i. Br.; 2000.
  • Bonito A, Lei W, Pasciak JE. Numerical approximation of the integral fractional Laplacian. Numer Math. 2019;142:235–278.
  • Duo S, Zhang Y. Accurate numerical methods for two and three dimensional integral fractional Laplacian with applications. Comput Methods Appl Mech Engrg. 2019;355:639–662.
  • Minden V, Ying L. A simple solver for the fractional Laplacian in multiple dimensions. SIAM J Sci Comput. 2020;42(2):A878–A900.
  • Maz'ya V, Schmidt G. Approximate approximations. Providence (RI): AMS; 2007.
  • Maz'ya V. A new approximation method and its applications to the calculation of volume potentials. Boundary point method. 3. DFG-Kolloqium des DFG-Forschungsschwerpunktes Randelementmethoden. Schloss Reisenburg; 30 Sep–5 Oct, 1991.
  • Maz'ya V. Approximate approximations. In: Whiteman JR, editor. The mathematics of finite elements and applications. highlights 1993. Chichester: Wiley & Sons, 1994. p. 77–104.
  • Schmidt G. Approximate Approximations and their applications. The Maz'ya Anniversary Collection, v.1, Operator Theory: Advances and Applications Vol. 109. Basel: Birkhäuser; 1999. p. 111–138.
  • Beylkin G, Mohlenkamp MJ. Numerical-operator calculus in higher dimensions. Proc Natl Acad Sci USA. 2002;99:10246-10251.
  • Hackbusch W, Khoromskij BN. Tensor-product approximation to operators and functions in high dimensions.J Complexity. 2007;23(4–6):697–714.
  • Lanzara F, Maz'ya V, Schmidt G. On the fast computation of high dimensional volume potentials. Math Comput. 2011;80:887–904.
  • Lanzara F, Maz'ya V, Schmidt G. Fast cubature of high dimensional biharmonic potential based on approximate approximations. Annali dell'Università di Ferrara. 2019;65:277–300.
  • Lanzara F, Maz'ya V, Schmidt G. Accurate computation of the high dimensional diffraction potential over hyper-rectangles. Bull TICMI. 2018;22:91–102.
  • Lanzara F, Maz'ya V, Schmidt G. Fast computation of elastic and hydrodynamic potentials using approximate approximations. Anal Math Phys. 2020;10:81.
  • Lanzara F, Schmidt G. On the computation of high-dimensional potentials of advection-diffusion operators. Mathematika. 2015;61:309–327.
  • Lanzara F, Maz'ya V, Schmidt G. Approximation of solutions to multidimensional parabolic equations by approximate approximations. Appl Comput Harmon Anal. 2016;41:749–767.
  • Maz'ya V. Sobolev spaces. Heidelberg: Springer; 2011.
  • Adams R. Sobolev spaces. New York, London: Academic Press; 1975.
  • Abramowitz M, Stegun IA. Handbook of mathematical functions. New York (NY): Dover Publ.; 1968.
  • Prudnikov AP, Brychkov YA, Marichev OI. Integral and series. vol. 2: special functions. New York (NY): Gordon & Breach Science Publishers; 1988.
  • Chinesta F, Ladevèze P. Proper generalized decomposition. In: Benner P, Grivet-Talocia S, Quarteroni A, et al., editors. Snapshot-based methods and algorithms. Vol. 2. Berlin, Boston: De Gruyter; 2020. p. 97–138.
  • Grasedyck L, Kressner D, Tobler C. A literature survey of low-rank tensor approximation techniques. GAMM-Mitt. 2013;36:53–78.
  • Beylkin G, Garcke J, Mohlenkamp MJ. Multivariate regression and machine learning with sums of separable functions. SIAM J Sci Comput. 2009;31(3):1840–1857.
  • Takahasi H, Mori M. Doubly exponential formulas for numerical integration. Publ RIMS, Kyoto Univ. 1974;9:721–741.
  • Tanaka K, Sugihara M, Murota K, et al. Function classes for double exponential integration formulas. Numer Math. 2009;111:631–655.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.