- Author:
- Mario Bebendorf
Hierarchical matrices are an efficient framework for large-scale fully populated matrices arising, e.g., from the finite element discretization of solution operators of elliptic boundary value problems. In addition to storing such matrices, approximations of the usual matrix operations can be computed with logarithmic-linear complexity, which can be exploited to setup approximate preconditioners in an efficient and convenient way. Besides the algorithmic aspects of hierarchical matrices, the main aim of this book is to present their theoretical background. The book contains the existing approximation theory for elliptic problems including partial differential operators with nonsmooth coefficients. Furthermore, it presents in full detail the adaptive cross approximation method for the efficient treatment of integral operators with non-local kernel functions. The theory is supported by many numerical experiments from real applications.
Cited By
Börm S (2023). Distributed ℋ2-Matrices for Boundary Element Methods, ACM Transactions on Mathematical Software, 49:2, (1-21), Online publication date: 30-Jun-2023.- Cohen M, Daulton S and Osborne M Log-linear-time Gaussian processes using binary tree kernels Proceedings of the 36th International Conference on Neural Information Processing Systems, (8118-8129)
- Apriansyah M and Yokota R (2022). Parallel QR Factorization of Block Low-rank Matrices, ACM Transactions on Mathematical Software, 48:3, (1-28), Online publication date: 30-Sep-2022.
- Massei S (2022). Some algorithms for maximum volume and cross approximation of symmetric semidefinite matrices, BIT, 62:1, (195-220), Online publication date: 1-Mar-2022.
- Matsushima K, Isakari H, Takahashi T and Matsumoto T (2020). A topology optimisation of composite elastic metamaterial slabs based on the manipulation of far-field behaviours, Structural and Multidisciplinary Optimization, 63:1, (231-243), Online publication date: 1-Jan-2021.
- Cao Q, Pei Y, Akbudak K, Mikhalev A, Bosilca G, Ltaief H, Keyes D and Dongarra J Extreme-Scale Task-Based Cholesky Factorization Toward Climate and Weather Prediction Applications Proceedings of the Platform for Advanced Scientific Computing Conference, (1-11)
- Balabanov O and Nouy A (2019). Randomized linear algebra for model reduction. Part I: Galerkin methods and error estimation, Advances in Computational Mathematics, 45:5-6, (2969-3019), Online publication date: 1-Dec-2019.
- Wang J and James D (2019). KleinPAT, ACM Transactions on Graphics, 38:4, (1-12), Online publication date: 31-Aug-2019.
- Takahashi T, Yamamoto T, Shimba Y, Isakari H and Matsumoto T (2019). A framework of shape optimisation based on the isogeometric boundary element method toward designing thin-silicon photovoltaic devices, Engineering with Computers, 35:2, (423-449), Online publication date: 1-Apr-2019.
- Amestoy P, Buttari A, L'Excellent J and Mary T (2019). Performance and Scalability of the Block Low-Rank Multifrontal Factorization on Multicore Architectures, ACM Transactions on Mathematical Software, 45:1, (1-26), Online publication date: 28-Mar-2019.
- Zaspel P (2019). Algorithmic Patterns for $$\mathcal {H}$$H-Matrices on Many-Core Processors, Journal of Scientific Computing, 78:2, (1174-1206), Online publication date: 1-Feb-2019.
- Claeys X, Hiptmair R and Spindler E (2017). Second-kind boundary integral equations for electromagnetic scattering at composite objects, Computers & Mathematics with Applications, 74:11, (2650-2670), Online publication date: 1-Dec-2017.
- Zapletal J, Merta M and Mal L (2017). Boundary element quadrature schemes for multi- and many-core architectures, Computers & Mathematics with Applications, 74:1, (157-173), Online publication date: 1-Jul-2017.
- Gatto P and Hesthaven J (2017). Efficient Preconditioning of hp-FEM Matrices by Hierarchical Low-Rank Approximations, Journal of Scientific Computing, 72:1, (49-80), Online publication date: 1-Jul-2017.
- Georgieva I and Hofreither C (2017). An algorithm for low-rank approximation of bivariate functions using splines, Journal of Computational and Applied Mathematics, 310:C, (80-91), Online publication date: 15-Jan-2017.
- Hao S and Martinsson P (2016). A direct solver for elliptic PDEs in three dimensions based on hierarchical merging of Poincaré-Steklov operators, Journal of Computational and Applied Mathematics, 308:C, (419-434), Online publication date: 15-Dec-2016.
- Zepeda-Núñez L and Demanet L (2016). The method of polarized traces for the 2D Helmholtz equation, Journal of Computational Physics, 308:C, (347-388), Online publication date: 1-Mar-2016.
- Aminfar A, Ambikasaran S and Darve E (2016). A fast block low-rank dense solver with applications to finite-element matrices, Journal of Computational Physics, 304:C, (170-188), Online publication date: 1-Jan-2016.
- Gillman A, Barnett A and Martinsson P (2015). A spectrally accurate direct solution technique for frequency-domain scattering problems with variable media, BIT, 55:1, (141-170), Online publication date: 1-Mar-2015.
- Śmigaj W, Betcke T, Arridge S, Phillips J and Schweiger M (2015). Solving Boundary Integral Problems with BEM++, ACM Transactions on Mathematical Software, 41:2, (1-40), Online publication date: 4-Feb-2015.
- Banjai L and Kachanovska M (2014). Fast convolution quadrature for the wave equation in three dimensions, Journal of Computational Physics, 279:C, (103-126), Online publication date: 15-Dec-2014.
- (2014). iBem3D, a three-dimensional iterative boundary element method using angular dislocations for modeling geologic structures, Computers & Geosciences, 72:C, (1-17), Online publication date: 1-Nov-2014.
- Bebendorf M, Bollhöfer M and Bratsch M (2013). Hierarchical matrix approximation with blockwise constraints, BIT, 53:2, (311-339), Online publication date: 1-Jun-2013.
- Zechner J and Beer G (2013). A fast elasto-plastic formulation with hierarchical matrices and the boundary element method, Computational Mechanics, 51:4, (443-453), Online publication date: 1-Apr-2013.
Index Terms
- Hierarchical Matrices: A Means to Efficiently Solve Elliptic Boundary Value Problems
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