Abstract
In this article, we extend the adaptive cross approximation (ACA) method known for the efficient approximation of discretisations of integral operators to a block-adaptive version. While ACA is usually employed to assemble hierarchical matrix approximations having the same prescribed accuracy on all blocks of the partition, for the solution of linear systems, it may be more efficient to adapt the accuracy of each block to the actual error of the solution as some blocks may be more important for the solution error than others. To this end, error estimation techniques known from adaptive mesh refinement are applied to automatically improve the blockwise matrix approximation. This allows to interlace the assembling of the coefficient matrix with the iterative solution.
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: BE2626/4-1
Funding statement: This work was supported by the DFG project BE2626/4-1.
References
[1] M. Ainsworth and C. Glusa, Aspects of an adaptive finite element method for the fractional Laplacian: A priori and a posteriori error estimates, efficient implementation and multigrid solver, Comput. Methods Appl. Mech. Engrg. 327 (2017), 4–35. 10.1016/j.cma.2017.08.019Search in Google Scholar
[2] M. Aurada, M. Feischl, T. Führer, M. Karkulik and D. Praetorius, Efficiency and optimality of some weighted-residual error estimator for adaptive 2D boundary element methods, Comput. Methods Appl. Math. 13 (2013), no. 3, 305–332. 10.1515/cmam-2013-0010Search in Google Scholar
[3] M. Aurada, S. Ferraz-Leite and D. Praetorius, Estimator reduction and convergence of adaptive BEM, Appl. Numer. Math. 62 (2012), no. 6, 787–801. 10.1016/j.apnum.2011.06.014Search in Google Scholar PubMed PubMed Central
[4] M. Bebendorf, Approximation of boundary element matrices, Numer. Math. 86 (2000), no. 4, 565–589. 10.1007/PL00005410Search in Google Scholar
[5] M. Bebendorf, Efficient inversion of the Galerkin matrix of general second-order elliptic operators with nonsmooth coefficients, Math. Comp. 74 (2005), no. 251, 1179–1199. 10.1090/S0025-5718-04-01716-8Search in Google Scholar
[6] M. Bebendorf, Approximate inverse preconditioning of finite element discretizations of elliptic operators with nonsmooth coefficients, SIAM J. Matrix Anal. Appl. 27 (2006), no. 4, 909–929. 10.1137/S0895479803437621Search in Google Scholar
[7] M. Bebendorf, Hierarchical Matrices, Lect. Notes Comput. Sci. Eng. 63, Springer, Berlin, 2008. Search in Google Scholar
[8] M. Bebendorf, M. Bollhöfer and M. Bratsch, On the spectral equivalence of hierarchical matrix preconditioners for elliptic problems, Math. Comp. 85 (2016), no. 302, 2839–2861. 10.1090/mcom/3086Search in Google Scholar
[9] M. Bebendorf and R. Grzhibovskis, Accelerating Galerkin BEM for linear elasticity using adaptive cross approximation, Math. Methods Appl. Sci. 29 (2006), no. 14, 1721–1747. 10.1002/mma.759Search in Google Scholar
[10] M. Bebendorf and W. Hackbusch, Stabilized rounded addition of hierarchical matrices, Numer. Linear Algebra Appl. 14 (2007), no. 5, 407–423. 10.1002/nla.525Search in Google Scholar
[11] M. Bebendorf and R. Kriemann, Fast parallel solution of boundary integral equations and related problems, Comput. Vis. Sci. 8 (2005), no. 3–4, 121–135. 10.1007/s00791-005-0001-xSearch in Google Scholar
[12] M. Bebendorf and S. Rjasanow, Adaptive low-rank approximation of collocation matrices, Computing 70 (2003), no. 1, 1–24. 10.1007/s00607-002-1469-6Search in Google Scholar
[13] Y. Chen, A fast, direct algorithm for the Lippmann–Schwinger integral equation in two dimensions, Adv. Comput. Math. 16 (2002), no. 2–3, 175–190. 10.1023/A:1014450116300Search in Google Scholar
[14] M. D’Elia and M. Gunzburger, The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator, Comput. Math. Appl. 66 (2013), no. 7, 1245–1260. 10.1016/j.camwa.2013.07.022Search in Google Scholar
[15] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), no. 5, 521–573. 10.1016/j.bulsci.2011.12.004Search in Google Scholar
[16] W. Dörfler, A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal. 33 (1996), no. 3, 1106–1124. 10.1137/0733054Search in Google Scholar
[17] S. Ferraz-Leite and D. Praetorius, Simple a posteriori error estimators for the h-version of the boundary element method, Computing 83 (2008), no. 4, 135–162. 10.1007/s00607-008-0017-4Search in Google Scholar
[18] T. Gantumur, Adaptive boundary element methods with convergence rates, Numer. Math. 124 (2013), no. 3, 471–516. 10.1007/s00211-013-0524-xSearch in Google Scholar
[19] S. A. Goreinov, E. E. Tyrtyshnikov and N. L. Zamarashkin, A theory of pseudoskeleton approximations, Linear Algebra Appl. 261 (1997), 1–21. 10.1016/S0024-3795(96)00301-1Search in Google Scholar
[20] L. Greengard and V. Rokhlin, A fast algorithm for particle simulations, J. Comput. Phys. 73 (1987), no. 2, 325–348. 10.1016/0021-9991(87)90140-9Search in Google Scholar
[21] L. Greengard and J. Strain, The fast Gauss transform, SIAM J. Sci. Statist. Comput. 12 (1991), no. 1, 79–94. 10.21236/ADA211287Search in Google Scholar
[22]
W. Hackbusch,
A sparse matrix arithmetic based on
[23] W. Hackbusch, Hierarchical Matrices: Algorithms and Analysis, Springer Ser. Comput. Math. 49, Springer, Heidelberg, 2015. 10.1007/978-3-662-47324-5Search in Google Scholar
[24]
W. Hackbusch and B. N. Khoromskij,
A sparse
[25] M. Karkulik, G. Of and D. Praetorius, Convergence of adaptive 3D BEM for weakly singular integral equations based on isotropic mesh-refinement, Numer. Methods Partial Differential Equations 29 (2013), no. 6, 2081–2106. 10.1002/num.21792Search in Google Scholar
[26] F. Lanzara, V. Maz’ya and G. Schmidt, Numerical solution of the Lippmann–Schwinger equation by approximate approximations, J. Fourier Anal. Appl. 10 (2004), no. 6, 645–660. 10.1007/s00041-004-3080-zSearch in Google Scholar
[27] E. Liberty, F. Woolfe, P.-G. Martinsson, V. Rokhlin and M. Tygert, Randomized algorithms for the low-rank approximation of matrices, Proc. Natl. Acad. Sci. USA 104 (2007), no. 51, 20167–20172. 10.1073/pnas.0709640104Search in Google Scholar PubMed PubMed Central
[28] Y. Saad, Iterative Methods for Sparse Linear Systems, PWS Publishing, Boston, 1996. Search in Google Scholar
[29] S. A. Sauter and C. Schwab, Boundary Element Methods, Springer Ser. Comput. Math. 39, Springer, Berlin, 2011. 10.1007/978-3-540-68093-2Search in Google Scholar
[30] O. Steinbach, Numerical Approximation Methods for Elliptic Boundary Value Problems, Springer, New York, 2008. 10.1007/978-0-387-68805-3Search in Google Scholar
[31] E. Tyrtyshnikov, Mosaic-skeleton approximations. Toeplitz matrices: Structures, algorithms and applications, Calcolo 33 (1996), no. 1–2, 47–57. 10.1007/BF02575706Search in Google Scholar
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