Skip to main content
SpringerLink
Log in

Efficient sum-of-exponentials approximations for the heat kernel and their applications

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we show that efficient separated sum-of-exponentials approximations can be constructed for the heat kernel in any dimension. In one space dimension, the heat kernel admits an approximation involving a number of terms that is of the order \(O(\log (\frac {T}{\delta }) (\log (\frac {1}{\epsilon })+\log \log (\frac {T}{\delta })))\) for any \(x\in \mathbb R\) and δ≤t≤T, where 𝜖 is the desired precision. In all higher dimensions, the corresponding heat kernel admits an approximation involving only \(O(\log ^{2}(\frac {T}{\delta }))\) terms for fixed accuracy 𝜖. These approximations can be used to accelerate integral equation-based methods for boundary value problems governed by the heat equation in complex geometry. The resulting algorithms are nearly optimal. For N S points in the spatial discretization and N T time steps, the cost is \(O(N_{S} N_{T} \log ^{2} \frac {T}{\delta })\) in terms of both memory and CPU time for fixed accuracy 𝜖. The algorithms can be parallelized in a straightforward manner. Several numerical examples are presented to illustrate the accuracy and stability of these approximations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alpert, B.K.: Hybrid Gauss-trapezoidal quadrature rules. SIAM J. Sci. Comput. 20, 1551–1584 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  2. Alpert, B., Greengard, L., Hagstrom, T.: Rapid evaluation of nonreflecting boundary kernels for time-domain wave propagation. SIAM J. Numer. Anal. 37(4), 1138–1164 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  3. Alpert, B., Greengard, L., Hagstrom, T.: Nonreflecting boundary conditions for the time-dependent wave equation. J. Comput. Phys. 180(1), 270–296 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  4. Beylkin, G., Monzón, L.: On generalized gaussian quadrature for exponentials and their applications. Appl. Comput. Harmon. Anal. 12, 332–373 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  5. Beylkin, G., Monzón, L.: On approximation of functions by exponential sums. Appl. Comput. Harmon. Anal. 19, 17–48 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  6. Beylkin, G., Monzón, L.: Approximation by exponential sums revisited. Appl. Comput. Harmon. Anal. 28, 131–140 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  7. Braess, D., Hackbusch, W.: Approximation of 1/x by exponential sums in [1, ∞). IMA J. Numer. Anal. 25(4), 685–697 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Braess, D., Hackbusch, W.: Multiscale, Nonlinear and Adaptive Approximation. Springer, Berlin (2009)

    Google Scholar 

  9. Brattkus, K., Meiron, D.I.: Numerical simulations of unsteady crystal growth. SIAM J. Appl. Math. 52, 1303–1320 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  10. Brebbia, C.A. (ed.) Topics in Boundary Element Research, vol. 1. Springer-Verlag, Berlin (1981)

  11. Bremer, J., Gimbutas, Z., Rokhlin, V.: A nonlinear optimization procedure for generalized Gaussian quadratures. SIAM J. Sci. Comput. 32(4), 1761–1788 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  12. Chandrasekaran, S., Dewilde, P., Gu, M., Lyons, W., Pals, T.: A fast solver for HSS representations via sparse matrices. SIAM J. Matrix Anal. Appl. 29, 67–81 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  13. Chapko, R., Kress, R.: Rothe’s method for the heat equation and boundary integral equations. J. Integral Equations Appl. 9(1), 47–69 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  14. Cheng, H., Greengard, L., Rokhlin, V.: A fast adaptive multipole algorithm in three dimensions. J. Comput. Phys. 155(2), 468–498 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  15. Costabel, M.: Time-dependent problems with the boundary integral equation method. In: Stein, E., De Borst, R., Hughes T.J.R. (eds.) Chapter 25, Encyclopedia of Computational Mechanics. Wiley (2004)

  16. Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs (1964)

    MATH  Google Scholar 

  17. Gillman, A., Young, P., Martinsson, P.G.: A direct solver with O(N) complexity for integral equations on one-dimensional domains. Front Math. China 7, 217–247 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  18. Greengard, L., Lin, P.: Spectral approximation of the free-space heat kernel. Appl. Comput. Harmon. Anal. 9, 83–97 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  19. Greengard, L., Strain, J.: A fast algorithm for the evaluation of heat potentials. Comm. Pure Appl. Math. 43, 949–963 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  20. Guenther, R.B., Lee, J.W.: Partial Differential Equations of Mathematical Physics and Integral Equations. Prentice-Hall, Englewood Cliffs (1988)

    Google Scholar 

  21. Hackbusch, W., Börm, S.: Data-sparse approximation by adaptive H2-matrices. Comput. 69(1), 1–35 (2002)

    Article  MATH  Google Scholar 

  22. Hagstrom, T.: Radiation boundary conditions for the numerical simulation of waves. Acta Numerica 8, 47–106 (1999)

    Article  MathSciNet  Google Scholar 

  23. Ho, K.L., Greengard, L.: A fast direct solver for structured linear systems by recursive skeletonization. SIAM J. Sci. Comput. 34(5), A2507–A2532 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  24. Ibanez, M.T., Power, H.: An efficient direct BEM numerical scheme for phase change problems using Fourier series. Comput. Methods Appl. Mech. Engrg. 191, 2371–2402 (2002)

    Article  MATH  Google Scholar 

  25. Jiang, S., Greengard, L.: Fast evaluation of nonreflecting boundary conditions for the Schrödinger equation in one dimension. Comput. Math. Appl. 47(6-7), 955–966 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  26. Jiang, S., Greengard, L.: Efficient representation of nonreflecting boundary conditions for the time-dependent Schrödinger equation in two dimensions. Comm. Pure Appl. Math. 61(2), 261–288 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  27. Jiang, S., Veerapaneni, S., Greengard, L.: Integral equation methods for unsteady Stokes flow in two dimensions. SIAM J. Sci. Comput. 34(4), A2197–A2219 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  28. Kloeckner, A., Barnett, A., Greengard, L., ONeil, M.: Quadrature by expansion: a new method for the evaluation of layer potentials. arXiv:1207.4461.2012

  29. Kress, R.: Linear Integral Equations. Second Edition. Springer (1999)

  30. Li, J.R., Greengard, L.: On the numerical solution of the heat equation. I. Fast solvers in free space. J. Comput. Phys. 226(2), 1891–1901 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  31. Li, J., Greengard, L.: High order accurate methods for the evaluation of layer heat potentials. SIAM J. Sci. Comput. 31, 3847–3860 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  32. López-Fernández, M., Palencia, C.: On the numerical inversion of the Laplace transform of certain holomorphic mappings. Appl. Numer. Math. 51, 289–303 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  33. López-Fernández, M., Palencia, C., Schädle, A.: A spectral order method for inverting sectorial Laplace transforms. SIAM J. Numer. Anal. 44(3), 1332–1350 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  34. López-Fernández, M., Lubich, C., Schädle, A.: Adaptive, fast, and oblivious convolution in evolution equations with memory. SIAM J. Sci. Comput. 30(2), 1015–1037 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  35. Lubich, C., Schädle, A.: Fast convolution for nonreflecting boundary conditions. SIAM J. Sci. Comput. 24, 161–182 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  36. Morino, L., Piva, R. (eds.) Boundary Integral Methods: Theory and Applications. Springer-Verlag, Berlin (1990)

  37. Ma, J., Rokhlin, V., Wandzura, S.: Generalized Gaussian quadrature rules for systems of arbitrary functions. SIAM J. Numer. Anal. 33(3), 971–996 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  38. McIntyre, E. A.: Boundary integral solutions of the heat equation. Math. Comp. 46(173), 71–79, S1–S14. (1986)

    Article  MathSciNet  Google Scholar 

  39. Noon, P.J.: The Single Layer Heat Potential and Galerkin Boundary Element Methods for the Heat Equation. Ph.D, Thesis, University of Maryland (1988)

  40. Pogorzelski, W.: Integral Equations and Their Applications. Pergamon Press, Oxford (1966)

    MATH  Google Scholar 

  41. Schädle, A., López-Fernández, M., Lubich, C.: Fast and oblivious convolution quadrature. SIAM J. Sci. Comput. 28(2), 421–438 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  42. Sethian, J.A., Strain, J.: Crystal growth and dendritic solidification. J. Comput. Phys. 98, 231–253 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  43. Stenger, F.: Approximation via Whitaker’s Cardinal Functions. J. Approx. Theory. 17, 222–240 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  44. Stenger, F.: Numerical methods based on Whitaker Cardinal, or sinc functions. SIAM Rev. 23, 165–224 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  45. Tausch, J.: A fast method for solving the heat equation by layer potentials. J. Comput. Phys. 224(2), 956–969 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  46. Talbot, A.: The accurate numerical inversion of Laplace transforms. J. Inst. Math. Appl. 23, 97–120 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  47. Trefethen, L.N., Weideman, J.A.C., Schmelzer, T.: Talbot quadratures and rational approximations. BIT Numer. Math. 46, 653–670 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  48. Xu, K., Jiang, S.: A bootstrap method for sum-of-poles approximations. J. Sci. Comput. 55, 16–39 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  49. Yarvin, N., Rokhlin, V.: An improved fast multipole algorithm for potential fields on the line. SIAM J. Numer. Anal. 36(2), 629–666 (1999)

    Article  MathSciNet  Google Scholar 

  50. Ying, L., Biros, G., Zorin, D.: A kernel-independent adaptive fast multipole algorithm in two and three dimensions. J. Comput. Phys. 196, 591–626 (2004)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ, 07102, USA

    Shidong Jiang & Shaobo Wang

  2. Courant Institute of Mathematical Sciences, New York University, New York, NY, 10012, USA

    Leslie Greengard

Authors
  1. Shidong Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Leslie Greengard
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shaobo Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Leslie Greengard.

Additional information

Communicated by: Zydrunas Gimbutas

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jiang, S., Greengard, L. & Wang, S. Efficient sum-of-exponentials approximations for the heat kernel and their applications. Adv Comput Math 41, 529–551 (2015). https://doi.org/10.1007/s10444-014-9372-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10444-014-9372-1

Keywords

Mathematics Subject Classification (2000)

Search

Navigation