Compact Rational Krylov for Parametrized Systems with Application to BEM Frequency Sweeping

Jul 16, 2026

Compact Rational Krylov (CORK) is developed to efficiently approximate solutions of parametrized linear systems P(µ)x=b when the system matrix depends nonlinearly on a parameter µ and solutions are required for many µ values. The approach linearizes the parametrized system into a large shifted system (A−µB)y=d, formulates left- and right-preconditioned rational Krylov GMRES, and accelerates these methods via CORK. It also handles parameter-dependent right-hand sides b(µ), shift selection, and inexact solves. The framework is applied to frequency sweeping Helmholtz scattering in the Boundary Element Method using a dense but data-sparse wavenumber-dependent representation.

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CategoryResearch & Report

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Pages26

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arXiv :2607 .07440v1 [math .NA] 8 Jul 2026 COMPACT RATIONAL KRYLOV FOR PARAMETRIZED SYSTEMS WITH APPLICATION TO BEM FREQUENCY SWEEPING ∗ KOBE BRUYNINCKX†, DAAN HUYBRECHS†, AND KARL MEERBERGEN† Abstract. In parametrized linear systems P (µ)x = b the system matrix P depends nonlinearlyon a parameter µ and solutions are sought for many values of this parameter. We show that the compact rational Krylov (CORK) framework, originally introduced to solve nonlinear eigenvalue problems, can be used to efficiently produce approximate solutions to such a system for many values of the parameter at once. In this approach the parametrized system is first linearized, resulting ina large shifted linear system (A − µB)y = d. We formulate a left-and right-preconditioned rational Krylov GMRES method for shifted linear systems and show how the CORK framework can be used to speed up these methods in the setting of parametrized linear systems. Additionally, we show how to incorporate a right-hand side b (µ) that also depends on the parameter, how to choose the shifts to steer convergence and how to allow for inexact solves at these shifts throughout the iterations. As an application we consider the ‘frequency sweeping’ of Helmholtz scattering problems through the Boundary Element Method (BEM), enabled via an efficient representation of the dense but data-sparse wavenumber-dependent system matrix. Key words. rational Krylov, parameterized linear systems, companion linearization, shifted linear systems, inexact Krylov, hierarchical matrices, boundary element method, Helmholtz equation MSC codes. 65F10, 65F55, 65N38, 35J05 1. Introduction. The motivation of this work arises from the field of integral equations. The Boundary Element Method (BEM) is a popular numerical technique for solving scattering problems in the frequency domain, based on an integral equation reformulation of the governing partial differential equation, such as the Helmholtz equation. Among many advantages of this approach, the resulting BEM system matrix is generally dense and nonlinearly dependent on the frequency. The dense matrix can still be approximated in a data-sparse way, using a range of algebraic and analytic techniques [26, 30, 31, 62, 36, 42, 33, 8] and the resulting systems can be solved efficiently with iterative or direct solvers [4, 11, 12, 36, 17, 57] . Still, the issue of nonlinear frequency dependency remains. In a ‘frequency sweep’, the solution to a scattering problem is sought over a range of frequencies. While the above techniques speed up solution at a single frequency, repeating such procedures for all frequencies of interest may still be intractable. Earlier research on frequency sweeping has involved both the Finite Element Method (FEM) and BEM, see [39] for a recent survey. Frequency sweeping requires tackling both the efficient construction and representation of the frequency-dependent system, e.g. [23, 19], as well as efficiently resolving the solutions over the frequency range, e.g. [44, 2, 3] . The wavenumber-dependent system for Helmholtz BEM leads to a parametrized linear system, given in general by (1.1) P (µ)x (µ) = b (µ) with µ ∈ S ⊆ C and P : S → Cn×n, where x(µ) is sought for many values of µ . The study of parametrized linear systems is rich, see [24, 56, 35, 15] and references therein. ∗ To be submitted to the journal’s Software, High-Performance Computing, and Computational Methods in Science and Engineering section 2026 . Funding: This work was supported by FWO-Flanders project G088622N †Department of Computer Science, KU Leuven, Leuven, 3000 (kobe.bruyninckx@kuleuven.be, [daan.huybrechs@kuleuven.be](daan.huybrechs@kuleuven.be), karl.meerbergen@kuleuven.be). 2 K. BRUYNINCKX, D. HUYBRECHS AND K. MEERBERGEN Using companion linearization techniques, we formulate the BEM system as a set of shifted linear systems, linear in the parameter µ, with a matrix pencil (A, B) as (1.2) (A − µB)y = d. This involves the app

FAQs

What problem does CORK address for parametrized linear systems?

CORK targets parametrized linear systems where P(µ)x=b depends nonlinearly on µ and solutions are needed for many parameter values efficiently.

How is the parametrized system transformed in the proposed method?

The method linearizes the system into shifted linear equations (A−µB)y=d using a companion linearization with a rational-function approximation of P(µ).

How is the framework applied to frequency sweeping in boundary element methods?

The approach performs Helmholtz scattering frequency sweeping using the Boundary Element Method, leveraging an efficient dense-but-data-sparse representation of the wavenumber-dependent system matrix and solving the resulting shifted parametrized systems across frequencies.

Jul 16, 2026

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