A numerical solution using an adaptively preconditioned Lanczos method for a class of linear systems related with the fractional poisson equation

Journal of Applied Mathematics and Stochastic Analysis

Volumn 2008, Issue , 2008, Pages

  Ilic M ^a   Turner, I W ^a   Anh, V ^a  

^a QUEENSLAND UNIVERSITY OF TECHNOLOGY   (Australia)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 64249158296     PISSN: 10489533     EISSN: 16872177     Source Type: Journal    
DOI: 10.1155/2008/104525     Document Type: Article

References (35)

1. F. Lorenzo and T.T. Hartley, "Initialization, conceptualization, and application in the generalized fractional calculus," Tech. Rep. NASA/TP - 208415, NASA Center for Aerospace Information, Hanover, Md, USA, 1998.
2. R. Metzler and J. Klafter, "The random walk's guide to anomalous diffusion: A fractional dynamics approach," Physics Reports, vol. 339, no. 1, pp. 1-77, 2000.
3. R. Metzler and J. Klafter, "The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics," Journal of Physics A, vol. 37, no. 31, pp. R161-R208, 2004.
4. D. A. Benson, S.W. Wheatcraft, and M. M. Meerschaert, "Application of a fractional advectiondispersion equation," Water Resources Research, vol. 36, no. 6, pp. 1403-1412, 2000.
5. D. A. Benson, S. W. Wheatcraft, and M. M. Meerschaert, "The fractional-order governing equation of levy motion," Water Resources Research, vol. 36, no. 6, pp. 1413-1423, 2000.
6. M. M. Meerschaert, J. Mortensen, and S. W. Wheatcraft, "Fractional vector calculus for fractional advection-dispersion," Physica A, vol. 367, pp. 181-190, 2006.
7. B. Friedman, Principles and Techniques of AppliedMathematics, JohnWiley & Sons, New York, NY, USA, 1966.
8. M. Ilić, F. Liu, I. W. Turner, and V. Anh, "Numerical approximation of a fractional-in-space diffusion equation - II-with nonhomogeneous boundary conditions," Fractional Calculus & Applied Analysis, vol. 9, no. 4, pp. 333-349, 2006.
9. B. J. West and V. Seshadri, "Linear systems with Lévy fluctuations," Physica A, vol. 113, no. 1-2, pp. 203-216, 1982.
10. R. Gorenflo and F. Mainardi, "Random walk models for space-fractional diffusion processes," Fractional Calculus & Applied Analysis, vol. 1, no. 2, pp. 167-191, 1998.
11. R. Gorenflo and F. Mainardi, "Approximation of Lévy-Feller diffusion by random walk," Journal for Analysis and Its Applications, vol. 18, no. 2, pp. 231-246, 1999.
12. I. P. Gavrilyuk, W. Hackbusch, and B. N. Khoromskij, "Data-sparse approximation to the operatorvalued functions of elliptic operator," Mathematics of Computation, vol. 73, no. 247, pp. 1297-1324, 2004.
13. W. Hackbusch and B. N. Khoromskij, "Low-rank Kronecker-product approximation to multidimensional nonlocal operators - part I. Separable approximation of multi-variate functions," Computing, vol. 76, no. 3, pp. 177-202, 2006.
14. W. Hackbusch and B. N. Khoromskij, "Low-rank Kronecker-product approximation to multidimensional nonlocal operators - part II. HKT representation of certain operators," Computing, vol. 76, no. 3, pp. 203-225, 2006.
15. M. Ilić, F. Liu, I. W. Turner, and V. Anh, "Numerical approximation of a fractional-in-space diffusion equation - I," Fractional Calculus & Applied Analysis, vol. 8, no. 3, pp. 323-341, 2005.
16. G. Gripenberg and I. Norros, "On the prediction of fractional Brownian motion," Journal of Applied Probability, vol. 33, no. 2, pp. 400-410, 1996.
17. Y. Hu, "Prediction and translation of fractional Brownian motions," in Stochastics in Finite and Infinite Dimensions, T. Hida, R. L. Karandikar, H. Kunita, B. S. Rajput, S.Watanabe, and J. Xiong, Eds., Trends in Mathematics, pp. 153-171, Birkhauser, Boston, Mass, USA, 2001.
18. G. H. Golub and C. F. Van Loan, Matrix Computations, Johns Hopkins Studies in the Mathematical Sciences, The Johns Hopkins University Press, Baltimore, Md, USA, 3rd edition, 1996.
19. Y. Saad, Iterative Methods for Sparse Linear Systems, SIAM, Philadelphia, Pa, USA, 2nd edition, 2003.
20. H. A. van der Vorst, Iterative Krylov Methods for Large Linear Systems, vol. 13 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, UK, 2003.
21. N. J. Higham, Functions of Matrices: Theory and Computation, SIAM, Philadelphia, Pa, USA, 2008.
22. V. Druskin and L. Knizhnerman, "Krylov subspace approximation of eigenpairs and matrix functions in exact and computer arithmetic," Numerical Linear Algebra with Applications, vol. 2, no. 3, pp. 205-217, 1995.
23. M. Hochbruck and C. Lubich, "On Krylov subspace approximations to the matrix exponential operator," SIAM Journal on Numerical Analysis, vol. 34, no. 5, pp. 1911-1925, 1997.
24. M. Eiermann and O. G. Ernst, "A restarted Krylov subspace method for the evaluation of matrix functions," SIAM Journal on Numerical Analysis, vol. 44, no. 6, pp. 2481-2504, 2006.
25. L. Lopez and V. Simoncini, "Analysis of projection methods for rational function approximation to the matrix exponential," SIAM Journal on Numerical Analysis, vol. 44, no. 2, pp. 613-635, 2006.
26. J. van den Eshof, A. Frommer, T. Lippert, K. Schilling, and H. A. van der Vorst, "Numerical methods for the QCD overlap operator - I. Sign-function and error bounds," Computer Physics Communications, vol. 146, pp. 203-224, 2002.
27. M. Ilić and I. W. Turner, "Approximating functions of a large sparse positive definite matrix using a spectral splitting method," The ANZIAM Journal, vol. 46(E), pp. C472-C487, 2005.
28. M. Ilić, I.W. Turner, and A. N. Pettitt, "Bayesian computations and efficient algorithms for computing functions of large, sparse matrices," The ANZIAM Journal, vol. 45(E), pp. C504-C518, 2004.
29. R. B. Lehoucq and D. C. Sorensen, "Deflation techniques for an implicitly restarted Arnoldi iteration," SIAM Journal on Matrix Analysis and Applications, vol. 17, no. 4, pp. 789-821, 1996.
30. A. Stathopoulos, Y. Saad, and K. Wu, "Dynamic thick restarting of the Davidson, and the implicitly restarted Arnoldi methods," SIAM Journal on Scientific Computing, vol. 19, no. 1, pp. 227-245, 1998.
31. M. Ilić and I. W. Turner, "Krylov subspaces and the analytic grade," Numerical Linear Algebra with Applications, vol. 12, no. 1, pp. 55-76, 2005.
32. R. B. Morgan, "A restarted GMRES method augmented with eigenvectors," SIAM Journal on Matrix Analysis and Applications, vol. 16, no. 4, pp. 1154-1171, 1995.
33. J. Baglama, D. Calvetti,G.H. Golub, and L. Reichel, "Adaptively preconditioned GMRES algorithms," SIAM Journal on Scientific Computing, vol. 20, no. 1, pp. 243-269, 1998.
34. J. Erhel, K. Burrage, and B. Pohl, "Restarted GMRES preconditioned by deflation," Journal of Computational and Applied Mathematics, vol. 69, no. 2, pp. 303-318, 1996.
35. R. V. Churchill, Complex Variables and Applications, McGraw-Hill, New York, NY, USA, 2nd edition, 1960.