Krylov subspace methods for functions of fractional differential operators

Moret, Igor; Novati, Paolo

doi:10.1090/mcom/3332

Krylov subspace methods for functions of fractional differential operators
HTML articles powered by AMS MathViewer

by Igor Moret and Paolo Novati HTML | PDF

Math. Comp. 88 (2019), 293-312 Request permission

Abstract:

The paper deals with the computation of functions of fractional powers of differential operators. The spectral properties of these operators naturally suggest the use of rational approximations. In this view we analyze the convergence properties of the shift-and-invert Krylov method applied to operator functions arising from the numerical solution of differential equations involving fractional diffusion.

References

Bernhard Beckermann and Stefan Güttel, Superlinear convergence of the rational Arnoldi method for the approximation of matrix functions, Numer. Math. 121 (2012), no. 2, 205–236. MR 2917161, DOI 10.1007/s00211-011-0434-8
Bernhard Beckermann and Lothar Reichel, Error estimates and evaluation of matrix functions via the Faber transform, SIAM J. Numer. Anal. 47 (2009), no. 5, 3849–3883. MR 2576523, DOI 10.1137/080741744
Peter N. Brown, A theoretical comparison of the Arnoldi and GMRES algorithms, SIAM J. Sci. Statist. Comput. 12 (1991), no. 1, 58–78. MR 1078796, DOI 10.1137/0912003
Kevin Burrage, Nicholas Hale, and David Kay, An efficient implicit FEM scheme for fractional-in-space reaction-diffusion equations, SIAM J. Sci. Comput. 34 (2012), no. 4, A2145–A2172. MR 2970400, DOI 10.1137/110847007
V. L. Druskin and L. A. Knizhnerman, Two polynomial methods for calculating functions of symmetric matrices, Zh. Vychisl. Mat. i Mat. Fiz. 29 (1989), no. 12, 1763–1775 (Russian); English transl., U.S.S.R. Comput. Math. and Math. Phys. 29 (1989), no. 6, 112–121 (1991). MR 1035689, DOI 10.1016/S0041-5553(89)80020-5
Vladimir Druskin and Leonid Knizhnerman, Extended Krylov subspaces: approximation of the matrix square root and related functions, SIAM J. Matrix Anal. Appl. 19 (1998), no. 3, 755–771. MR 1616584, DOI 10.1137/S0895479895292400
Vladimir Druskin, Leonid Knizhnerman, and Mikhail Zaslavsky, Solution of large scale evolutionary problems using rational Krylov subspaces with optimized shifts, SIAM J. Sci. Comput. 31 (2009), no. 5, 3760–3780. MR 2556561, DOI 10.1137/080742403
Vladimir Druskin, Chad Lieberman, and Mikhail Zaslavsky, On adaptive choice of shifts in rational Krylov subspace reduction of evolutionary problems, SIAM J. Sci. Comput. 32 (2010), no. 5, 2485–2496. MR 2684724, DOI 10.1137/090774082
S. W. Ellacott, Computation of Faber series with application to numerical polynomial approximation in the complex plane, Math. Comp. 40 (1983), no. 162, 575–587. MR 689474, DOI 10.1090/S0025-5718-1983-0689474-7
Klaus-Jochen Engel and Rainer Nagel, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, vol. 194, Springer-Verlag, New York, 2000. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt. MR 1721989
A. Erdelyi, ed., Tables of Integral Transforms, McGraw–Hill, New York, 1954.
Roland Freund, On conjugate gradient type methods and polynomial preconditioners for a class of complex non-Hermitian matrices, Numer. Math. 57 (1990), no. 3, 285–312. MR 1057126, DOI 10.1007/BF01386412
Andreas Frommer, Stefan Güttel, and Marcel Schweitzer, Convergence of restarted Krylov subspace methods for Stieltjes functions of matrices, SIAM J. Matrix Anal. Appl. 35 (2014), no. 4, 1602–1624. MR 3505178, DOI 10.1137/140973463
Roberto Garrappa, Numerical evaluation of two and three parameter Mittag-Leffler functions, SIAM J. Numer. Anal. 53 (2015), no. 3, 1350–1369. MR 3350038, DOI 10.1137/140971191
Roberto Garrappa and Marina Popolizio, Evaluation of generalized Mittag-Leffler functions on the real line, Adv. Comput. Math. 39 (2013), no. 1, 205–225. MR 3068601, DOI 10.1007/s10444-012-9274-z
Volker Grimm and Martin Gugat, Approximation of semigroups and related operator functions by resolvent series, SIAM J. Numer. Anal. 48 (2010), no. 5, 1826–1845. MR 2733100, DOI 10.1137/090768084
Stefan Güttel, Rational Krylov approximation of matrix functions: numerical methods and optimal pole selection, GAMM-Mitt. 36 (2013), no. 1, 8–31. MR 3095912, DOI 10.1002/gamm.201310002
S. Güttel and L. Knizhnerman, Automated parameter selection for rational Arnoldi approximation of Markov functions, Proc. Appl. Math. Mech. 11 (2011), 15–18.
Stefan Güttel and Leonid Knizhnerman, A black-box rational Arnoldi variant for Cauchy-Stieltjes matrix functions, BIT 53 (2013), no. 3, 595–616. MR 3095257, DOI 10.1007/s10543-013-0420-x
Nicholas Hale, Nicholas J. Higham, and Lloyd N. Trefethen, Computing $\textbf {A}^\alpha ,\ \log (\textbf {A})$, and related matrix functions by contour integrals, SIAM J. Numer. Anal. 46 (2008), no. 5, 2505–2523. MR 2421045, DOI 10.1137/070700607
Marlis Hochbruck and Alexander Ostermann, Exponential integrators, Acta Numer. 19 (2010), 209–286. MR 2652783, DOI 10.1017/S0962492910000048
M. Ilic, F. Liu, I. Turner, and V. Anh, Numerical approximation of a fractional-in-space diffusion equation. I, Fract. Calc. Appl. Anal. 8 (2005), no. 3, 323–341. MR 2252038
M. Ilic, F. Liu, I. Turner, and V. Anh, Numerical approximation of a fractional-in-space diffusion equation. II. With nonhomogeneous boundary conditions, Fract. Calc. Appl. Anal. 9 (2006), no. 4, 333–349. MR 2300467
L. Knizhnerman, On adaptation of the Lanczos method to the spectrum, Research note, Ridgefield: Schlumberger-Doll Research, 1995.
L. A. Knizhnerman, Adaptation of the Lanczos and Arnoldi methods to the spectrum, or why the two Krylov subspace methods are powerful, Chebyshevskiĭ Sb. 3 (2002), no. 2(4), 141–164. Dedicated to the 85th birthday of Nikolaĭ Mikhaĭlovich Korobov (Russian). MR 2035623
L. Knizhnerman and V. Simoncini, A new investigation of the extended Krylov subspace method for matrix function evaluations, Numer. Linear Algebra Appl. 17 (2010), no. 4, 615–638. MR 2663662, DOI 10.1002/nla.652
Celso Martínez, Miguel Sanz, and Javier Pastor, A functional calculus and fractional powers for multivalued linear operators, Osaka J. Math. 37 (2000), no. 3, 551–576. MR 1789437
Igor Moret, Rational Lanczos approximations to the matrix square root and related functions, Numer. Linear Algebra Appl. 16 (2009), no. 6, 431–445. MR 2522957, DOI 10.1002/nla.625
I. Moret and P. Novati, RD-rational approximations of the matrix exponential, BIT 44 (2004), no. 3, 595–615. MR 2106019, DOI 10.1023/B:BITN.0000046805.27551.3b
Igor Moret and Paolo Novati, On the convergence of Krylov subspace methods for matrix Mittag-Leffler functions, SIAM J. Numer. Anal. 49 (2011), no. 5, 2144–2164. MR 2861713, DOI 10.1137/080738374
Igor Moret and Marina Popolizio, The restarted shift-and-invert Krylov method for matrix functions, Numer. Linear Algebra Appl. 21 (2014), no. 1, 68–80. MR 3150610, DOI 10.1002/nla.1862
Paolo Novati, Using the restricted-denominator rational Arnoldi method for exponential integrators, SIAM J. Matrix Anal. Appl. 32 (2011), no. 4, 1537–1558. MR 2869498, DOI 10.1137/100814202
Manuel Duarte Ortigueira, Riesz potential operators and inverses via fractional centred derivatives, Int. J. Math. Math. Sci. , posted on (2006), Art. ID 48391, 12. MR 2251718, DOI 10.1155/IJMMS/2006/48391
Igor Podlubny, Fractional differential equations, Mathematics in Science and Engineering, vol. 198, Academic Press, Inc., San Diego, CA, 1999. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. MR 1658022
Stefan G. Samko, Anatoly A. Kilbas, and Oleg I. Marichev, Fractional integrals and derivatives, Gordon and Breach Science Publishers, Yverdon, 1993. Theory and applications; Edited and with a foreword by S. M. Nikol′skiĭ; Translated from the 1987 Russian original; Revised by the authors. MR 1347689
M. Schweitzer, Restarting and error estimation in polynomial and extended Krylov subspace methods for the approximation of matrix functions, PhD thesis (2016).
Jasper van den Eshof and Marlis Hochbruck, Preconditioning Lanczos approximations to the matrix exponential, SIAM J. Sci. Comput. 27 (2006), no. 4, 1438–1457. MR 2199756, DOI 10.1137/040605461
Q. Yang, F. Liu, and I. Turner, Numerical methods for fractional partial differential equations with Riesz space fractional derivatives, Appl. Math. Model. 34 (2010), no. 1, 200–218. MR 2566688, DOI 10.1016/j.apm.2009.04.006
Qianqian Yang, Ian Turner, Fawang Liu, and Milos Ilić, Novel numerical methods for solving the time-space fractional diffusion equation in two dimensions, SIAM J. Sci. Comput. 33 (2011), no. 3, 1159–1180. MR 2800568, DOI 10.1137/100800634
Q. Yu, F. Liu, I. Turner and K. Burrage, Numerical investigation on three types of space and time fractional Bloch-Torrey equations in 2D, Cent. Eur. J. Phys. 11 (2013), 646–665.