Article
| Title: | Algebraic multilevel preconditioning in spectral fractional diffusion (English) |
| Author: | Margenov, Svetozar |
| Language: | English |
| Journal: | Applications of Mathematics |
| ISSN: | 0862-7940 (print) |
| ISSN: | 1572-9109 (online) |
| Volume: | 70 |
| Issue: | 6 |
| Year: | 2025 |
| Pages: | 851-874 |
| Summary lang: | English |
| . | |
| Category: | math |
| . | |
| Summary: | The numerical solution of linear systems obtained as a result of discretization of a spectral fractional diffusion problem is studied. The finite element method is applied to the considered boundary value problem. The system matrix is a fractional power of the product of the inverse of the mass matrix and the stiffness matrix. The matrix thus defined is symmetric and positive definite (SPD) with respect to the inner product associated with the mass matrix, but is dense, which is consistent with the nonlocal nature of fractional diffusion. The presented results are in the spirit of the BURA (Best Uniform Rational Approximation) method. BURA reduces numerical solution of the dense linear system to the solution of $k$ systems with sparse SPD diffusion-reaction matrices, where $k$ is the degree of rational approximation. We prove the existence of algebraic multilevel iteration (AMLI) methods for preconditioning such type of emergent matrices that satisfy the conditions for optimal computational complexity. Both multiplicative and additive AMLI preconditioners have been developed, determining the minimum possible degree $\theta $ of the hierarchical $\theta $-refinement of the mesh. (English) |
| Keyword: | fractional diffusion |
| Keyword: | BURA method |
| Keyword: | AMLI preconditioning |
| Keyword: | strengthened CBS inequality |
| Keyword: | computational complexity |
| MSC: | 35R11 |
| MSC: | 65F10 |
| MSC: | 65N30 |
| DOI: | 10.21136/AM.2025.0101-25 |
| . | |
| Date available: | 2025-12-20T05:40:38Z |
| Last updated: | 2025-12-22 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153226 |
| . | |
| Reference: | [1] Al-Juboury, R. S., Zghair, H. K., Alyaseen, M. A. A.: Dynamics and stability of interconnected systems: A graph-theoretic neuromorphic approach.Int. J. Neutrosophic Sci. 23 (2024), 150-155. 10.54216/ijns.230212 |
| Reference: | [2] Axelsson, A.: Iterative Solution Methods.Cambridge University Press, Cambridge (1994). Zbl 0795.65014, MR 1276069, 10.1017/CBO9780511624100 |
| Reference: | [3] Axelsson, O., Blaheta, R.: Two simple derivations of universal bounds for the C.B.S. inequality constant.Appl. Math., Praha 49 (2004), 57-72. Zbl 1099.65103, MR 2032148, 10.1023/B:APOM.0000024520.06175.8b |
| Reference: | [4] Axelsson, O., Margenov, S.: On multilevel preconditioners which are optimal with respect to both problem and discretization parameters.Comput. Methods Appl. Math. 3 (2003), 6-22. Zbl 1039.65078, MR 2002253, 10.2478/cmam-2003-0002 |
| Reference: | [5] rland, T. Bæ, Kuchta, M., Mardal, K.-A.: Multigrid methods for discrete fractional Sobolev spaces.SIAM J. Sci. Comput. 41 (2019), A948--A972. Zbl 1419.65102, MR 3934111, 10.1137/18M1191488 |
| Reference: | [6] Bonito, A., Pasciak, J. E.: Numerical approximation of fractional powers of elliptic operators.Math. Comput. 84 (2015), 2083-2110. Zbl 1331.65159, MR 3356020, 10.1090/S0025-5718-2015-02937-8 |
| Reference: | [7] Eijkhout, V., Vassilevski, P.: The role of the strengthened Cauchy-Buniakowskii-Schwarz inequality in multilevel methods.SIAM Rev. 33 (1991), 405-419. Zbl 0737.65026, MR 1124360, 10.1137/1033098 |
| Reference: | [8] Harizanov, S., Kosturski, N., Lirkov, I., Margenov, S., Vutov, Y.: Reduced multiplicative (BURA-MR) and additive (BURA-AR) best uniform rational approximation methods and algorithms for fractional elliptic equations.Fractal Fractional 5 (2021), Article ID 61, 22 pages. 10.3390/fractalfract5030061 |
| Reference: | [9] Harizanov, S., Lazarov, R., Margenov, S.: A survey on numerical methods for spectral space-fractional diffusion problems.Frac. Calc. Appl. Anal. 23 (2020), 1605-1646. Zbl 1474.65438, MR 4197087, 10.1515/fca-2020-0080 |
| Reference: | [10] Harizanov, S., Lazarov, R., Margenov, S., Marinov, P., Pasciak, J.: Analysis of numerical methods for spectral fractional elliptic equations based on the best uniform rational approximation.J. Comput. Phys. 408 (2020), Article ID 109285, 21 pages. Zbl 1560.65842, MR 4065429, 10.1016/j.jcp.2020.109285 |
| Reference: | [11] Harizanov, S., Lazarov, R., Margenov, S., Marinov, P., Vutov, Y.: Optimal solvers for linear systems with fractional powers of sparse SPD matrices.Numer. Linear Algebra Appl. 25 (2018), Article ID e2167, 24 pages. Zbl 1513.65132, MR 3859150, 10.1002/nla.2167 |
| Reference: | [12] Harizanov, S., Lirkov, I., Margenov, S.: Rational approximations in robust preconditioning of multiphysics problems.Mathematics 10 (2022), Article ID 780, 18 pages. 10.3390/math10050780 |
| Reference: | [13] Haubner, J., Neumann, F., Ulbrich, M.: A novel density based approach for topology optimization of Stokes flow.SIAM J. Sci. Comput. 45 (2023), A338--A368. Zbl 07673291, MR 4566010, 10.1137/21M143114X |
| Reference: | [14] Hofreither, C.: An algorithm for best rational approximation based on barycentric rational interpolation.Numer. Algorithms 88 (2021), 365-388. Zbl 1473.65015, MR 4297923, 10.1007/s11075-020-01042-0 |
| Reference: | [15] Koh, K. J., Cirak, F.: Stochastic PDE representation of random fields for large-scale Gaussian process regression and statistical finite element analysis.Comput. Methods Appl. Mech. Eng. 417 (2023), Article ID 116358, 41 pages. Zbl 1539.62112, MR 4668256, 10.1016/j.cma.2023.116358 |
| Reference: | [16] Kosturski, N., Margenov, S.: Analysis of BURA and BURA-based approximations of fractional powers of sparse spd matrices.Fract. Calc. Appl. Anal. 27 (2024), 706-724. Zbl 08050546, MR 4736875, 10.1007/s13540-024-00256-6 |
| Reference: | [17] Kraus, J., Margenov, S.: Robust Algebraic Multilevel Methods and Algorithms.Radon Series on Computational and Applied Mathematics 5. De Gruyter, Berlin (2009). Zbl 1184.65113, MR 2574100, 10.1515/9783110214833 |
| Reference: | [18] Kraus, J., Wolfmayr, M.: On the robustness and optimality of algebraic multilevel methods for reaction-diffusion type problems.Comput. Vis. Sci. 16 (2013), 15-32. Zbl 1380.65377, MR 3264809, 10.1007/s00791-014-0221-z |
| Reference: | [19] Kwaśnicki, M.: Ten equivalent definitions of the fractional Laplace operator.Fract. Calc. Appl. Anal. 20 (2017), 7-51. Zbl 1375.47038, MR 3613319, 10.1515/fca-2017-0002 |
| Reference: | [20] Latz, J., Teckentrup, A. L., Urbainczyk, S.: Deep Gaussian process priors for Bayesian image reconstruction.Inverse Probl. 41 (2025), Article ID 065016, 37 pages. Zbl 08056055, MR 4919183, 10.1088/1361-6420/add9be |
| Reference: | [21] A. Lischke, G. Pang, M. Gulian, F. Song, C. Glusa, X. Zheng, Z. Mao, W. Cai, M. M. Meerschaert, M. Ainsworth, G. E. Karniadakis: What is the fractional laplacian? A comparative review with new results.J. Comput. Phys. 404 (2020), Article ID 109009, 62 pages. Zbl 1453.35179, MR 4043885, 10.1016/j.jcp.2019.109009 |
| Reference: | [22] Maitre, J. F., Musy, F.: The contraction number of a class of two-level methods: An exact evaluation for some finite element subspaces and model problems.Multigrid Methods Lecture Notes in Mathematics 960. Springer, Berlin (1982), 535-544. Zbl 0505.65049, MR 0685787, 10.1007/BFb0069942 |
| Reference: | [23] Mazza, M., Serra-Capizzano, S., Sormani, R. L.: Algebra preconditionings for 2D Riesz distributed-order space-fractional diffusion equations on convex domains.Numer. Linear Algebra Appl. 31 (2024), Article ID e2536, 23 pages. Zbl 07893425, MR 4732625, 10.1002/nla.2536 |
| Reference: | [24] Nakatsukasa, Y., Sète, O., Trefethen, L. N.: The AAA algorithm for rational approximation.SIAM J. Sci. Comput. 40 (2018), A1494--A1522. Zbl 1390.41015, MR 3805855, 10.1137/16M1106122 |
| Reference: | [25] : Software BRASIL.Available at https://baryrat.readthedocs.io/en/latest/\#baryrat.brasil. |
| Reference: | [26] Stahl, H. R.: Best uniform rational approximation of $x^\alpha$ on [0,1].Acta Math. 190 (2003), 241-306. Zbl 1077.41012, MR 1998350, 10.1007/BF02392691 |
| . |
Fulltext not available (moving wall 24 months)
Search
Partner of
