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A framework of the harmonic Arnoldi method for evaluating φ-functions with applications to exponential integrators

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Abstract

In recent years, a great deal of attention has been focused on exponential integrators. The important ingredient to the implementation of exponential integrators is the efficient and accurate evaluation of the so called φ-functions on a given vector. The Krylov subspace method is an important technique for this problem. For this type of method, however, restarts become essential for the sake of storage requirements or due to computational complexities of evaluating matrix function on a reduced matrix of growing size. Another problem in computing φ-functions is the lack of a clear residual notion. The contribution of this work is threefold. First, we introduce a framework of the harmonic Arnoldi method for φ-functions, which is based on the residual and the oblique projection technique. Second, we establish the relationship between the harmonic Arnoldi approximation and the Arnoldi approximation, and compare the harmonic Arnoldi method and the Arnoldi method from a theoretical point of view. Third, we apply the thick-restarting strategy to the harmonic Arnoldi method, and propose a thick-restarted harmonic Arnoldi algorithm for evaluating φ-functions. An advantage of the new algorithm is that we can compute several φ-functions simultaneously in the same search subspace after restarting. The relationship between the error and the residual of the harmonic Arnoldi approximation is also investigated. Numerical experiments show the superiority of our new algorithm over many state-of-the-art algorithms for computing φ-functions.

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References

  1. Afanasjew, M., Eiermann, M., Ernst, O., Güttel, S.: A generalization of the steepest descent method for matrix functions. Electron. Trans. Numer. Anal. 28, 206–222 (2008)

    MathSciNet  MATH  Google Scholar 

  2. Afanasjew, M., Eiermann, M., Ernst, O., Güttel, S.: Implementation of a restarted Krylov subspace method for the evaluation of matrix functions. Linear Algebra Appl. 429, 2293–2314 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. AL-Mohy, A., Higham, N.J.: Computing the action of matrix exponential, with an application to exponential integrators. SIAM J. Sci. Comput. 33, 488–511 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Beckermann, B., Reichel, L.: Error estimation and evaluation of matrix functions via the Faber transform. SIAM J. Numer. Anal. 47, 3849–3883 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Berland, H., Owren, B., Skaflestad, B.: Solving the nonlinear Schrödinger equation using exponential integrators. Model. Identif. Control 27, 201–217 (2006)

    Article  Google Scholar 

  6. Berland, H., Skaflestad, B., Wright, W.: Expint–A matlab package for exponential integrators. ACM Tran. Math. Soft. 33(4) (2007)

  7. Botchev, M., Harutyunyan, D., van der Vegt, J.: The Gautschi time stepping scheme for edge finite element discretizations of the Maxwell equations. J. Comput. Phys. 216, 654–686 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Botchev, M., Grimm, V., Hochbruck, M.: Residual, restarting and Richardson iteration for the matrix exponential. SIAM J. Sci. Comput. 35, A1376–A1397 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Caliari, M., Kandolf, P., Ostermann, A., Rainer, S.: Comparison of methods for computing the action of the matrix exponential. BIT 52, 113–128 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Carr, E., Turner, I., Ilić, M.: Krylov subspace approximations for the exponential Euler method: error estimates and the harmonic Ritz approximant. ANZIAM J. 52, C612–C627 (2011)

    MathSciNet  Google Scholar 

  11. Celledoni, E., Moret, I.: A Krylov projection method for systems of ODEs. Appl. Numer, Math. 24, 365–378 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  12. Druskin, V., Greenbaum, A., Knizhnerman, L.: Using nonorthogonal Lanczos vectors in the computation of matrix functions. SIAM J. Sci. Comput. 19, 38–54 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  13. Druskin, V., Knizhnerman, L.: Extended Krylov subspaces: approximation of the matrix square root and related functions. SIAM J. Matrix Anal. Appl. 19, 775–771 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  14. Eiermann, M., Ernst, O.: A restarted Krylov subspace method for the evaluation of matrix functions. SIAM J. Numer. Anal. 44, 2481–2504 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Eiermann, M., Ernst, O., Güttel, S.: Deflated restarting for matrix functions. SIAM J. Matrix Anal. Appl. 32, 621–641 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Friesner, R., Tuckerman, L., Dornblaser, B., Russo, T.: A method for exponential propagation of large systems of stiff nonlinear differential equations. J. Sci. Comput. 4, 327–354 (1989)

    Article  MathSciNet  Google Scholar 

  17. Frommer, A., Güttel, S., Schweitzer, M.: Efficient and stable Arnoldi restarts for matrix functions based on quadrature. SIAM J. Matrix Anal. Appl. 35, 661–683 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Frommer, A., Güttel, S., Schweitzer, M.: Convergence of restarted Krylov subspace methods for Stieltjes functions of matrices. SIAM J. Matrix Anal. Appl. 35, 1602–1624 (2014)

    Article  MATH  Google Scholar 

  19. Gallopoulos, E., Saad, Y.: Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Statist. Comput. 13, 1236–1264 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  20. Grimm, V.: Resolvent Krylov subspace approximation to operator functions. BIT 52, 639–659 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Göckler, T., Grimm, V.: Convergence analysis of an extented Krylov subspace method for the approximation of operator function in exponential integrators. SIAM J. Numer. Anal. 51, 2189–2213 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Göckler, T., Grimm, V.: Uniform approximation of φ functions in exponential integerators by a rational Krylov subspace method with simple poles. SIAM J. Matrix Anal. Appl. 35, 1467–1489 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Golub, G.H., Van Loan, C.F.: Matrix Computations, 4th edn. John Hopkins University Press, Baltimore (2012)

    MATH  Google Scholar 

  24. Güttel, S.: Rational Krylov approximation of matrix functions: numerical methods and optimal pole selection. GAMM 36, 8–31 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Güttel, S., Knizhnerman, L.: A black-box rational Arnoldi variant for Cauchy-Stieltjes matrix functions. BIT 24, 595–616 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, 2nd edn. Springer, Berlin (2004)

    MATH  Google Scholar 

  27. Higham, N.J.: Functions of Matrices: Theory and Computation. SIAM, Philadelphia (2008)

    Book  MATH  Google Scholar 

  28. Higham, N.J.: The scaling and squaring method for the matrix exponential revisited. SIAM Rev. 51, 747–764 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Hochbruck, M., Hochstenbach, M.: Subspace extraction for matrix functions. Submitted for publication, available from http://www.win.tue.nl/~hochsten/publications.html (2005)

  30. Hochbruck, M., Hönig, M., Ostermann, A.: Regularization of nonlinear ill-posed problems by exponential integrators. Math. Model. Numer. Anal. 43, 709–720 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. Hochbruck, M., Lubich, C.: On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34, 1911–1925 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  32. Hochbruck, M., Lubich, C., Selhofer, H.: Exponential integrators for large systems of differential equations. SIAM J. Sci. Comput. 19, 1552–1574 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  33. Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numer. 19, 209–286 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. Ilić, M., Turner, I., Simpson, D.: A restarted Lanczos approximation to functions of a symmetric matrix. IMA J. Numer. Anal. 30, 1044–1061 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  35. Jiang, W., Wu, G.: A thick-restarted block Arnoldi algorithm with modified Ritz vectors for large eigenproblems. Comput. Math. Appl. 60, 873–889 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  36. Kandolf, P., Ostermann, A., Rainer, S.: A residual based error estimate for Leja interpolation of matrix functions. Linear Algebra Appl. 456, 157–173 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  37. Knizhnerman, L., Simoncini, V.: A new investigation of the extended Krylov subspace method for matrix function evaluations. Numer. Linear Algebra Appl. 17, 615–638 (2010)

    MathSciNet  MATH  Google Scholar 

  38. Krogstad, S.: Generalized integrating factor methods for stiff PDEs. J. Comput. Phys. 203, 72–88 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  39. Lee, S., Pang, H., Sun, H.: Shift-invert Arnoldi approximation to the Toeplitz matrix exponential. SIAM J. Sci. Comput. 32, 774–792 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  40. Meerschaert, M., Tadjeran, C.: Finite difference approximations for fractional advection dispersion flow equations. J. Comput. Appl. Math. 172, 65–77 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  41. Merton, R.: Option pricing when underlying stock returns are discontinuous. J. Financ. Econ. 3, 125–144 (1976)

    Article  MATH  Google Scholar 

  42. Moler, C., Van Loan, C.F.: Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev. 45, 3–49 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  43. Moret, I., Novati, P.: RD ratianal approximations of the matrix exponential. BIT 44, 595–615 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  44. Moret, I., Popolizio, M.: The restarted shift-and-invert Krylov method for matrix functions. Numer. Linear Algebra Appl. 21, 68–80 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  45. Morgan, R.: GMRES with deflated restarting. SIAM J. Sci. Comput. 24, 20–37 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  46. Morgan, R., Zeng, M.: A Harmonic restarted Arnoldi algorithm for calculating eigenvalues and determining multiplicity. Linear Algebra Appl. 415, 96–113 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  47. Niesen, J., Wright, W.: A Krylov subspace method for option pricing. Submitted for publication, available from www.maths.leeds.ac.uk/~jitse/options.pdf(2013)

  48. Niesen, J., Wright, W.: Algorithm 919: a Krylov subspace algorithm for evaluating the φ-functions appearing in exponential integrators. ACM Trans. Math. Soft. 38(22) (2012)

  49. Novati, P.: Using the restricted-denominator rational Arnoldi method for exponential integrators. SIAM J. Matrix Anal. Appl. 32, 1537–1558 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  50. Paige, C., Parlett, B., van der Vorst, H.: Approximate solutions and eigenvalue bounds from Krylov subspaces. Numer. Linear Algebra Appl. 2, 115–133 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  51. Pang, H., Sun, H.: Multigrid method for fractional diffusion equations. J. Comput. Phys. 231, 693–703 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  52. Podlubny, I.: Fractional Differential Equations. Academic, New York (1999)

    MATH  Google Scholar 

  53. Ragni, S.: Rational Krylov methods in exponential integrators for European option pricing. Numer. Linear Algebra Appl. 21, 494–512 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  54. Saad, Y.: Analysis of some Krylov subapace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 29, 209–228 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  55. Schlick, T., Skeel, R., Brunger, A., Kalé, L., Board, J., Hermans, J., Schulten, K.: Algorithmic challenges in computational molecular biophysics. J. Comput. Phys. 151, 9–48 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  56. Schmelzer, T., Trefethen, L.N.: Evaluating matrix functions for exponential integrators via Carathéodory-Fejér approximation and contour integrals. Electron. Trans. Numer. Anal. 29, 1–18 (2007)

    MathSciNet  MATH  Google Scholar 

  57. Sidje, R.: EXPOKIT: a software package for computing matrix exponentials. ACM Tran. Math. Soft. 24, 130–156 (1998)

    Article  MATH  Google Scholar 

  58. Skaflestad, B., Wright, W.: The scaling and modified squaring method for matrix functions related to the exponential. Appl. Numer. Math. 59, 783–799 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  59. Stewart, G.W.: Matrix Algorithms II: Eigensystems. SIAM, Philadelphia (2001)

    Book  MATH  Google Scholar 

  60. Tal-Ezer, H.: On restart and error estimation for Krylov approximation of w = f(A)v. SIAM J. Sci. Comput. 29, 2426–2441 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  61. Tokman, M., Loffeld, J., Tranquilli, P.: New adaptive exponential propagation iterative methods of Runge-Kutta type. SIAM J. Sci. Comput. 34, A2650–A2669 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  62. van den Eshof, J., Hochbruck, M.: Preconditioning Lanczos approximations to the matrix exponential. SIAM J. Sci. Comput. 27, 1438–1457 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  63. Wang, H., Wang, K., Sircar, T.: A direct O(N log2 N) finite difference method for fractional diffusion equations. J. Comput. Phys. 229, 8095–8104 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  64. Wu, G., Wei, Y.: On analysis of projection methods for rational function approximation to the matrix exponential. SIAM J. Numer. Anal. 48, 191–197 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  65. Wu, K., Simon, H.: Thick-restart Lanczos method for large symmetric eigenvalue problems. SIAM J. Matrix Anal. Appl. 22, 602–616 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, China University of Mining and Technology, Xuzhou, 221116, Jiangsu, People’s Republic of China

    Gang Wu

  2. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, Jiangsu, People’s Republic of China

    Gang Wu

  3. School of Mathematics and Physical Sciences Technology, Xuzhou Institute of Technology, Xuzhou, 221111, Jiangsu, People’s Republic of China

    Lu Zhang

  4. Jingjiang College, Jiangsu University, Zhenjiang, 212013, Jiangsu, People’s Republic of China

    Ting-ting Xu

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  1. Gang Wu
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  2. Lu Zhang
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  3. Ting-ting Xu
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Correspondence to Gang Wu.

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Communicated by: Helge Holden

Gang Wu is supported by the National Science Foundation of China under grant 11371176, the Natural Science Foundation of Jiangsu Province under grant BK20131126, the 333 Project of Jiangsu Province, and the Talent Introduction Program of China University of Mining and Technology.

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Wu, G., Zhang, L. & Xu, Tt. A framework of the harmonic Arnoldi method for evaluating φ-functions with applications to exponential integrators. Adv Comput Math 42, 505–541 (2016). https://doi.org/10.1007/s10444-015-9433-0

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