×
Aceto, Lidia; Mazza, Mariarosa; Serra-Capizzano, Stefano

Fractional Laplace operator in two dimensions, approximating matrices, and related spectral analysis. (English) Zbl 07245604

Calcolo 57, No. 3, Paper No. 27, 25 p. (2020).
Summary: In this work we review some proposals to define the fractional Laplace operator in two or more spatial variables and we provide their approximations using finite differences or the so-called Matrix Transfer Technique. We study the structure of the resulting large matrices from the spectral viewpoint. In particular, by considering the matrix-sequences involved, we analyze the extreme eigenvalues, we give estimates on conditioning, and we study the spectral distribution in the Weyl sense using the tools of the theory of Generalized Locally Toeplitz matrix-sequences. Furthermore, we give a concise description of the spectral properties when non-constant coefficients come into play. Several numerical experiments are reported and critically discussed.

Cited in 3 Documents

MSC:

47A58 Linear operator approximation theory
34A08 Fractional ordinary differential equations
15B05 Toeplitz, Cauchy, and related matrices
65F60 Numerical computation of matrix exponential and similar matrix functions
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
65F35 Numerical computation of matrix norms, conditioning, scaling

Keywords:

Riesz fractional derivative operator; Toeplitz matrices; matrix transfer technique; spectral analysis; GLT theory

Software:

ma2dfc
Cite Review PDF
Full Text: DOI
Creative Commons CC license icon Creative Commons BY license icon

References:

[1] Aceto, L.; Bertaccini, D.; Durastante, F.; Novati, P., Rational Krylov methods for functions of matrices with applications to fractional partial differential equations, J. Comput. Phys., 396, 470-482 (2019) · Zbl 1452.65201
[2] Aceto, L.; Novati, P., Efficient implementation of rational approximations to fractional differential operators, J. Sci. Comput., 76, 1, 651-671 (2018) · Zbl 1398.65089
[3] Aceto, L.; Novati, P., Rational approximation to fractional powers of self-adjoint positive operators, Numer. Math., 143, 1, 1-16 (2019) · Zbl 07088062
[4] Aceto, L.; Novati, P., Rational approximation to the fractional Laplacian operator in reaction-diffusion problems, SIAM J. Sci. Comput., 39, A214-A228 (2017) · Zbl 1382.65123
[5] Barbarino, G., Garoni, C., Serra-Capizzano, S.: Block Generalized Locally Toeplitz Sequences: Theory and Applications in the Unidimensional Case, Technical Report, N. 4, July 2019, Department of Information Technology, Uppsala University. http://www.it.uu.se/research/publications/reports/2019-004/ · Zbl 1434.65032
[6] Barbarino, G., Garoni, C., Serra-Capizzano, S.: Block Generalized Locally Toeplitz Sequences: Theory and Applications in the Multiidimensional Case, Technical Report, N. 5, July 2019, Department of Information Technology, Uppsala University. http://www.it.uu.se/research/publications/reports/2019-005/ · Zbl 1434.65032
[7] Bhatia, R., Matrix Analysis (1997), New York: Springer, New York
[8] Bonito, A.; Pasciak, JE, Numerical approximation of fractional powers of elliptic operators, Math. Comput., 84, 295, 2083-2110 (2015) · Zbl 1331.65159
[9] Böttcher, A.; Grudsky, S., On the condition numbers of large semi-definite Toeplitz matrices, Linear Algebra Appl., 279, 1-3, 285-301 (1998) · Zbl 0934.15005
[10] Böttcher, A.; Silbermann, B., Introduction to Large Truncated Toeplitz Matrices (1999), New York: Springer, New York · Zbl 0916.15012
[11] Çelik, C.; Duman, M., Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative, J. Comput. Phys., 231, 1743-1750 (2012) · Zbl 1242.65157
[12] Defterli, O.; D’Elia, M.; Du, Q.; Gunzburger, M.; Lehoucq, R.; Meerschaert, MM, Fractional diffusion on bounded domains, Fract. Calc. Appl. Anal., 18, 2, 342-360 (2015) · Zbl 1488.35557
[13] Donatelli, M.; Mazza, M.; Serra-Capizzano, S., Spectral analysis and structure preserving preconditioners for fractional diffusion equations, J. Comput. Phys., 307, 262-279 (2016) · Zbl 1352.65305
[14] Ekström, SE; Furci, I.; Garoni, C.; Manni, C.; Serra-Capizzano, S.; Speleers, H., Are the eigenvalues of the B-spline isogeometric analysis approximation of \(-\Delta u=\lambda u\) known in almost closed form?, Numer. Linear Algebra Appl., 25, 5, e2198 (2018) · Zbl 1513.65454
[15] Garoni, C.; Manni, C.; Pelosi, F.; Serra-Capizzano, S.; Speleers, H., On the spectrum of stiffness matrices arising from isogeometric analysis, Numer. Math., 127, 751-799 (2014) · Zbl 1298.65172
[16] Garoni, C.; Mazza, M.; Serra-Capizzano, S., Block generalized locally Toeplitz sequences: from the theory to the applications, Axioms, 7, 3, 49 (2018) · Zbl 1434.65034
[17] Garoni, C.; Serra-Capizzano, S., Generalized Locally Toeplitz Sequences: Theory and Applications (2017), Cham: Springer, Cham · Zbl 1376.15002
[18] Garoni, C.; Serra-Capizzano, S., Generalized Locally Toeplitz Sequences: Theory and Applications (2018), Cham: Springer, Cham · Zbl 1448.47004
[19] Grenander, U.; Szegö, G., Toeplitz Forms and their Applications (1984), New York: Chelsea, New York · Zbl 0611.47018
[20] Harizanov, S., Lazarov, R., Marinov, P., Margenov, S., Pasciak, J.: Comparison analysis on two numerical methods for fractional diffusion problems based on rational approximations of \(t^\gamma , 0 \le t \le 1,\) arXiv:1805.00711v1 · Zbl 1429.65064
[21] Ilić, M.; Liu, F.; Turner, I.; Anh, V., Numerical approximation of a fractional-in-space diffusion equation I, Frac. Calc. App. Anal., 8, 323-341 (2005) · Zbl 1126.26009
[22] Ilić, M.; Liu, F.; Turner, I.; Anh, V., Numerical approximation of a fractional-in-space diffusion equation (II)-with nonhomogeneous boundary conditions, Fract. Calc. Appl. Anal., 9, 333-349 (2006) · Zbl 1132.35507
[23] Jin, X-Q; Lin, F-R; Zhao, Z., Preconditioned iterative methods for two-dimensional space-fractional diffusion equations, Commun. Comput. Phys., 18, 2, 469-488 (2015) · Zbl 1388.65057
[24] Kwaśnicki, M., Ten equivalent definitions of the fractional Laplace operator, Fract. Calc. Appl. Anal., 20, 7-51 (2017) · Zbl 1375.47038
[25] Lei, S-L; Sun, H-W, A circulant preconditioner for fractional diffusion equations, J. Comput. Phys., 242, 715-725 (2013) · Zbl 1297.65095
[26] Liu, F.; Zhuang, P.; Turner, I.; Anh, V.; Burrage, K., A semi-alternating direction method for a 2-D fractional FitzHugh-Nagumo monodomain model on an approximate irregular domain, J. Comput. Phys., 293, 252-263 (2015) · Zbl 1349.65316
[27] Lubich, C., Fractional linear multistep methods for Abel-Volterra integral equations of the second kind, Math. Comput., 45, 463-469 (1985) · Zbl 0584.65090
[28] Lubich, C., Discretized fractional calculus, SIAM J. Math. Anal., 17, 3, 704-719 (1986) · Zbl 0624.65015
[29] Moghaderi, H.; Dehghan, M.; Donatelli, M.; Mazza, M., Spectral analysis and multigrid preconditioners for two-dimensional space-fractional diffusion equations, J. Comput. Phys., 350, 992-1011 (2017) · Zbl 1380.65240
[30] Ortigueira, M.D.: Riesz potential operators and inverses via fractional centred derivatives. Int. J. Math. Math. Sci. (2006) · Zbl 1122.26007
[31] Ortigueira, MD, Fractional central differences and derivatives, J. Vib. Control, 14, 1255-1266 (2008) · Zbl 1229.26015
[32] Pang, H-K; Sun, H-W, Multigrid method for fractional diffusion equations, J. Comput. Phys., 231, 2, 693-703 (2012) · Zbl 1243.65117
[33] Podlubny, I., Fractional Differential Equations: Mathematics in Science and Engineering (1999), San Diego, CA: Academic Press Inc, San Diego, CA · Zbl 0924.34008
[34] Podlubny, I., Matrix approach to discrete fractional calculus, Fract. Calc. Appl. Anal., 3, 359-386 (2000) · Zbl 1030.26011
[35] Podlubny, I.; Chechkin, A.; Skovranek, T.; Chen, YQ; Vinagre Jara, BM, Partial fractional differential equations: Matrix approach to discrete fractional calculus II, J. Comput. Phys., 228, 3137-3153 (2009) · Zbl 1160.65308
[36] Popolizio, M., A matrix approach for partial differential equations with Riesz space fractional derivatives, Eur. Phys. J. Spec. Top., 222, 1975-1985 (2013)
[37] Riesz, M., Intégrales de Riemann-Liouville et potentiels, Acta Sci. Math. Szeged, 9, 1-42 (1938) · JFM 64.0476.03
[38] Samko, SG; Kilbas, AA; Marichev, OI, Fractional Integral and Derivatives-Theory and Applications (1987), New York: Gordon and Breach Science Publishers, New York · Zbl 0617.26004
[39] Serra, S., On the extreme eigenvalues of Hermitian (block) Toeplitz matrices, Linear Algebra Appl., 270, 109-129 (1998) · Zbl 0892.15014
[40] Serra, S., On the extreme eigenvalues of Hermitian (block) Toeplitz matrices, SIAM J. Matrix Anal. Appl., 20, 1, 31-44 (1998) · Zbl 0932.65037
[41] Tilli, P., A note on the spectral distribution of Toeplitz matrices, Linear Multilin. Algebra, 45, 2-3, 147-159 (1998) · Zbl 0951.65033
[42] Tyrtyshnikov, E., A unifying approach to some old and new theorems on distribution and clustering, Linear Algebra Appl., 232, 1-43 (1996) · Zbl 0841.15006
[43] Zamarashkin, NL; Tyrtyshnikov, E., Distribution of eigenvalues and singular values of Toeplitz matrices under weakened conditions on the generating function, Sbornik: Math., 188, 8, 1191 (1997) · Zbl 0898.15007
[44] Yang, Q.; Turner, I.; Liu, F.; Ilić, M., Novel numerical methods for solving the time-space fractional diffusion equation in two dimensions, SIAM J. Sci. Comput., 33, 3, 1159-1180 (2011) · Zbl 1229.35315
[45] Yu, Q.; Liu, F.; Turner, I.; Burrage, K., Numerical investigation of three types of space and time fractional Bloch-Torrey equations in 2D, Cent. Eur. J. Phys., 11, 6, 646-665 (2013)
[46] Zoia, A.; Rosso, A.; Kardar, M., Fractional Laplacian in bounded domains, Phys. Rev. E, 76, 021116 (2007)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.