Computing Functions of Symmetric Hierarchically Semiseparable Matrices (2402.17369v1)
Abstract: The aim of this work is to develop a fast algorithm for approximating the matrix function $f(A)$ of a square matrix $A$ that is symmetric and has hierarchically semiseparable (HSS) structure. Appearing in a wide variety of applications, often in the context of discretized (fractional) differential and integral operators, HSS matrices have a number of attractive properties facilitating the development of fast algorithms. In this work, we use an unconventional telescopic decomposition of $A$, inspired by recent work of Levitt and Martinsson on approximating an HSS matrix from matrix-vector products with a few random vectors. This telescopic decomposition allows us to approximate $f(A)$ by recursively performing low-rank updates with rational Krylov subspaces while keeping the size of the matrices involved in the rational Krylov subspaces small. In particular, no large-scale linear system needs to be solved, which yields favorable complexity estimates and reduced execution times compared to existing methods, including an existing divide-and-conquer strategy. The advantages of our newly proposed algorithms are demonstrated for a number of examples from the literature, featuring the exponential, the inverse square root, and the sign function of a matrix. Even for matrix inversion, our algorithm exhibits superior performance, even if not specifically designed for this task.
- Low-rank updates of matrix functions II: rational Krylov methods. SIAM J. Numer. Anal., 59(3):1325–1347, 2021.
- Error estimates and evaluation of matrix functions via the Faber transform. SIAM J. Numer. Anal., 47(5):3849–3883, 2009.
- Decay bounds for functions of Hermitian matrices with banded or Kronecker structure. SIAM J. Matrix Anal. Appl., 36(3):1263–1282, 2015.
- A fast ULV decomposition solver for hierarchically semiseparable representations. SIAM J. Matrix Anal. Appl., 28(3):603–622, 2006.
- Divide-and-conquer methods for functions of matrices with banded or hierarchical low-rank structure. SIAM J. Matrix Anal. Appl., 43(1):151–177, 2022.
- The block rational Arnoldi method. SIAM J. Matrix Anal. Appl., 41(2):365–388, 2020.
- Analysis of probing techniques for sparse approximation and trace estimation of decaying matrix functions. SIAM J. Matrix Anal. Appl., 42(3):1290–1318, 2021.
- Equilibrium distributions and the rate of rational approximation of analytic functions. Mat. Sb. (N.S.), 134(176)(3):306–352, 447, 1987.
- Solution of large scale algebraic matrix Riccati equations by use of hierarchical matrices. Computing, 70(2):121–165, 2003.
- Stefan Güttel. Rational Krylov methods for operator functions. PhD thesis, Technische Universität Bergakademie Freiberg, 2010.
- Stefan Güttel. Rational Krylov approximation of matrix functions: Numerical methods and optimal pole selection. GAMM-Mitteilungen, 36(1):8–31, 2013.
- Wolfgang Hackbusch. Hierarchical matrices: algorithms and analysis, volume 49 of Springer Series in Computational Mathematics. Springer, Heidelberg, 2015.
- Structured matrix recovery from matrix-vector products. Numer. Linear Algebra Appl., 31(1):Paper No. e2531, 27, 2024.
- Nicholas J. Higham. Functions of matrices. SIAM, Philadelphia, PA, 2008.
- Nicholas J. Higham. The scaling and squaring method for the matrix exponential revisited. SIAM Rev., 51(4):747–764, 2009.
- On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal., 34(5):1911–1925, 1997.
- Fast computation of spectral projectors of banded matrices. SIAM J. Matrix Anal. Appl., 38(3):984–1009, 2017.
- Linear-complexity black-box randomized compression of hierarchically block separable matrices. arXiv preprint arXiv:2205.02990, 2022.
- Per-Gunnar Martinsson. A fast randomized algorithm for computing a hierarchically semiseparable representation of a matrix. SIAM J. Matrix Anal. Appl., 32(4):1251–1274, 2011.
- Fast solvers for two-dimensional fractional diffusion equations using rank structured matrices. SIAM J. Sci. Comput., 41(4):A2627–A2656, 2019.
- hm-toolbox: MATLAB software for HODLR and HSS matrices. SIAM J. Sci. Comput., 42(2):C43–C68, 2020.
- Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math., 172(1):65–77, 2004.
- The AAA algorithm for rational approximation. SIAM J. Sci. Comput., 40(3):A1494–A1522, 2018.
- Approximating sparse matrices and their functions using matrix-vector products. arXiv preprint arXiv:2310.05625, 2023.
- Rational approximation of real functions, volume 28 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1987.
- Fast algorithms for hierarchically semiseparable matrices. Numer. Linear Algebra Appl., 17(6):953–976, 2010.