Computing Functions of Symmetric Hierarchically Semiseparable Matrices

Angelo A. Casulli; Daniel Kressner; Leonardo Robol

Home/Archive/arXiv

arXiv · 2402.17369

Computing Functions of Symmetric Hierarchically Semiseparable Matrices

Angelo A. Casulli·Daniel Kressner·Leonardo Robol

View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.NA cs.NA

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Angelo A. Casulli, Daniel Kressner, Leonardo Robol. 2024-02-27. Computing Functions of Symmetric Hierarchically Semiseparable Matrices. https://arxiv.org/abs/2402.17369

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Computational Models for Magnetic Drug Targeting, Part 1: Derivation and Numerical Simulation of a Thermodynamically Consistent Magneto Two-phase Flow Model

Motivated by Magnetic Drug Targeting (MDT), we derive a thermodynamically consistent two-phase model for the coupled dynamics of superparamagnetic iron oxide nanoparticles (SPIONs), a carrier fluid, and the magnetic field. The model includes a convection-diffusion equation for the SPION dispersion volume density and a modified Navier-Stokes system for the mixture velocity with SPION-dependent density and viscosity, an additional momentum flux due to relative SPION motion, and a magnetic body force. Additionally, the SPIONs interact with a quasi-static Maxwell system through their magnetization which is of Langevin type. The coupled system satisfies a global energy-dissipation balance. Moreover, we develop a semi-implicit finite element scheme with explicit flux-corrected transport stabilization and simulate the attraction of an initially localized SPION dispersion towards an external magnet in a system closed to mass exchange, using parameter values representative of MDT settings. The resulting SPION redistribution and induced flow are analyzed through physical observables, including energy and dissipation contributions and iron mass within a target region. Temporal convergence studies assess the ability of the numerical scheme to reproduce the energy-dissipation balance. Finally, we study the influence of individual coupling mechanisms on particle accumulation, maximum iron density, and peak flow velocity.

math.NA↗

arXiv2026

Accelerating an ensemble of variational data assimilations with randomized preconditioning

Ensembles of variational data assimilations (EDA) require solving systems of linear equations with iterative methods. The solution process can be accelerated using a limited memory preconditioner constructed with approximations of the leading eigenpairs of the Hessian matrix. Randomized methods for low-rank matrix approximations provide a feasible approach for computing these eigenpairs. These methods use a random sketching matrix to obtain a low-rank representation of the Hessian matrix, which is then used for computing the eigendecomposition. The sketching matrix highly influences the quality of the approximation. In this paper, we show how the structure of the EDA can be exploited to construct a suitable sketching matrix, i.e., using the differences of the right-hand sides of the linear systems of equations. Idealised numerical experiments with the Lorenz-96 model show that the resulting preconditioner is able to accelerate the EDA solution process for all ensemble members, even if constructed from the control member only.

math.NA↗

arXiv2026

LGNO: A Local-Global Neural Operator for Hyperbolic Conservation Laws

Solutions of hyperbolic conservation laws exhibit both smooth structures across large scales and sharp localized features such as shocks and contact discontinuities, making them difficult to approximate accurately with existing neural operators. The Fourier Neural Operator (FNO) captures long-range interactions well but tends to smear localized structures through excessive numerical dissipation. To address this, we propose a Local-Global Neural Operator (LGNO) that learns a one-step discrete flow map by combining a global FNO branch for representing smooth dynamics at large scales with a local multiresolution branch for enhancing localized discontinuities and nonsmooth features. The model is trained with a one-step loss that combines a physical space prediction term and a spectral penalty on high frequencies to suppress spurious oscillations near steep fronts. On a large collection of benchmarks in one and two dimensions, LGNO consistently outperforms FNO baselines with matched parameter counts, reducing one-step errors by factors of 2-5 and remaining significantly more accurate over long autoregressive rollouts. Most strikingly, although it is trained only on short-time data from a high-order WENO-Z scheme, the long-time rollout of LGNO on a coarse $256^2$ grid exhibits lower numerical dissipation than the same WENO-Z scheme run on a finer $512^2$ grid, while being orders of magnitude cheaper to evaluate. These results suggest that, with an appropriate architecture and training objective, learned operators can effectively learn discrete flow maps. They further suggest that such learned operators have the potential to control long-time numerical dissipation better than the conventional shock-capturing schemes that generate the training data.

math.NA↗