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A low-memory Lanczos method with rational Krylov compression for matrix functions

Angelo A. Casulli, Igor Simunec

TL;DR

The paper addresses the challenge of applying a matrix function to a vector when the matrix is large and Hermitian, with strict memory constraints. It introduces RKcompress, a memory-efficient algorithm that couples an outer rational Lanczos method with an inner, small-scale rational Krylov compression to avoid storing the full Krylov basis. Theoretical results show exact equivalence to rational Lanczos for rational functions and provide a bounded error for general analytic functions, with memory scaling like $m+k$ and inner computations on matrices of size at most $(k+m)$. Numerical experiments on exponential and Markov functions, as well as shift-invert Krylov setups, demonstrate competitive performance and substantial memory savings, highlighting practical impact for large-scale matrix functions.

Abstract

In this work we introduce a memory-efficient method for computing the action of a Hermitian matrix function on a vector. Our method consists of a rational Lanczos algorithm combined with a basis compression procedure based on rational Krylov subspaces that only involve small matrices. The cost of the compression procedure is negligible with respect to the cost of the Lanczos algorithm. This enables us to avoid storing the whole Krylov basis, leading to substantial reductions in memory requirements. This method is particularly effective when the rational Lanczos algorithm needs a significant number of iterations to converge and each iteration involves a low computational effort. This scenario often occurs when polynomial Lanczos, as well as extended and shift-and-invert Lanczos are employed. Theoretical results prove that, for a wide variety of functions, the proposed algorithm differs from rational Lanczos by an error term that is usually negligible. The algorithm is compared with other low-memory Krylov methods from the literature on a variety of test problems, showing competitive performance.

A low-memory Lanczos method with rational Krylov compression for matrix functions

TL;DR

The paper addresses the challenge of applying a matrix function to a vector when the matrix is large and Hermitian, with strict memory constraints. It introduces RKcompress, a memory-efficient algorithm that couples an outer rational Lanczos method with an inner, small-scale rational Krylov compression to avoid storing the full Krylov basis. Theoretical results show exact equivalence to rational Lanczos for rational functions and provide a bounded error for general analytic functions, with memory scaling like and inner computations on matrices of size at most . Numerical experiments on exponential and Markov functions, as well as shift-invert Krylov setups, demonstrate competitive performance and substantial memory savings, highlighting practical impact for large-scale matrix functions.

Abstract

In this work we introduce a memory-efficient method for computing the action of a Hermitian matrix function on a vector. Our method consists of a rational Lanczos algorithm combined with a basis compression procedure based on rational Krylov subspaces that only involve small matrices. The cost of the compression procedure is negligible with respect to the cost of the Lanczos algorithm. This enables us to avoid storing the whole Krylov basis, leading to substantial reductions in memory requirements. This method is particularly effective when the rational Lanczos algorithm needs a significant number of iterations to converge and each iteration involves a low computational effort. This scenario often occurs when polynomial Lanczos, as well as extended and shift-and-invert Lanczos are employed. Theoretical results prove that, for a wide variety of functions, the proposed algorithm differs from rational Lanczos by an error term that is usually negligible. The algorithm is compared with other low-memory Krylov methods from the literature on a variety of test problems, showing competitive performance.
Paper Structure (19 sections, 7 theorems, 79 equations, 4 figures, 4 tables, 3 algorithms)

This paper contains 19 sections, 7 theorems, 79 equations, 4 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Notation
  3. Background
  4. Krylov subspaces
  5. Low-rank updates of matrix functions
  6. Algorithm description
  7. First cycle
  8. $i$-th cycle
  9. Final cycle
  10. Analysis and comparison with existing algorithms
  11. Theoretical results
  12. Computational discussion
  13. Comparison with other low-memory Krylov methods
  14. Numerical experiments
  15. Exponential function
  16. ...and 4 more sections

Key Result

Proposition 2.3

Let $A\in \mathbb{C}^{n \times n}$, $\boldsymbol{b}\in \mathbb{C}^{n}$ and let $\boldsymbol{\xi}_k$ be a list of $k$ poles. Denoting by $\mathbf{Q}_k$ an orthonormal basis of $\mathcal{Q}(A,\boldsymbol{b},\boldsymbol{\xi}_k)$, for any matrix $U$ with orthonormal columns such that $\text{span}(\mathb

Figures (4)

  • Figure 1: Time needed for the computation of $e^{-tA}\boldsymbol{1}$ with accuracy of $10^{-10}$, for different values of $t$ and employing different methods.
  • Figure 2: Time needed for the computation of $A^{-1/2}\boldsymbol{1}$ with relative tolerance $10^{-8}$, where $A \in \mathbb{C}^{n^2 \times n^2}$ is a discretization of the 2D Laplace operator for increasing $n$, employing different low-memory methods.
  • Figure 3: Effect of numerical loss of orthogonality in the computation of $e^A \boldsymbol{b}$, where $A \in \mathbb{C}^{2000 \times 2000}$ has logspaced eigenvalues in $[-10^{4}, -10^{-4}]$ and $\boldsymbol{b}$ is a random vector.
  • Figure : Rational Lanczos

Theorems & Definitions (15)

  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Theorem 2.4
  • Proposition 2.5
  • proof
  • Corollary 2.6
  • proof
  • Corollary 2.7
  • ...and 5 more