arXiv subjects

Karl Meerbergen

Publications and source records attributed to Karl Meerbergen.

At least 19 recordsLinked to original sources

arXiv2026

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

In parametrized linear systems $\mathsf{P}(\mu)\mathsf{x}=\mathsf{b}$ the system matrix $\mathsf{P}$ depends nonlinearly on a parameter $\mu$ 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 in a large shifted linear system $(\boldsymbol{\mathsf{A}}-\mu\boldsymbol{\mathsf{B}})\boldsymbol{\mathsf{y}}=\boldsymbol{\mathsf{d}}$. We formulate a left- and right-preconditioned rational Krylov GMRES method for shifted linear systems. In the setting of parametrized linear systems, these can exploit the structure in the linearization, and in combination with the CORK framework, computational and memory complexity mainly depend on the problem size, less the degree of the linearization. Additionally, we show how to incorporate a right-hand side $\mathsf{b}(\mu)$ 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.

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arXiv2026

Conditioning and backward errors for nonlinear eigenvalue problems with eigenvector nonlinearities

We consider eigenvalue condition numbers and backward errors for a class of symmetric nonlinear eigenvalue problems with eigenvector nonlinearities. For both of these quantities, we derive explicit and computable expressions that can be evaluated with little computational effort for a given eigenpair, assuming the matrix perturbations are measured by the spectral or Frobenius norm. We also show how symmetric perturbations can be exploited in the analysis. By means of two numerical experiments we demonstrate that problems incorporating eigenvector nonlinearities potentially need to be treated with additional care, when compared to the linear or eigenvalue-nonlinear theory.

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arXiv2026

Comparison of model order reduction techniques with one-shot procedure for topology optimization for thermal applications

Density-based topology optimization has become a powerful method for automatically generating optimized designs in a wide variety of applications. However, it comes with a large computational cost when solving the physical model requires large-scale simulations. Here, we investigate the use of model order reduction (MOR) techniques to accelerate the simulations in the context of thermal design applications. We project the governing and the adjoint equations onto a low-dimensional subspace by constructing two distinct reduced bases -- one for the forward state and one for the adjoint system -- using solution snapshots from previous design iterations. These snapshots are generated using either the high-fidelity solver or inaccurate fast solvers, such as the one-shot method \citep{amir2024one}. Additionally, we demonstrate that properly selecting the stopping criterion for the iterative linear solver is crucial for the effective use of reduced models. In our 3D example, the proposed framework reduces the overall total simulation time relative to the high-fidelity workflow by a factor up to $3$ when combined with high-fidelity solves and a factor up to $16$ when combined with the one-shot method. Moreover, we find that the reduced order model approach is able to achieve a speed up of $1.54$ with respect to the one-shot method.

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arXiv2025

ParaSLRF: A High Performance Rational Filter Method for Solving Large Scale Eigenvalue Problems

In \emph{Wang et al., A Shifted Laplace Rational Filter for Large-Scale Eigenvalue Problems}, the SLRF method was proposed to compute all eigenvalues of a symmetric definite generalized eigenvalue problem lying in an interval on the real positive axis. The current paper discusses a parallel implementation of this method, abbreviated as ParaSLRF. The parallelization consists of two levels: (1) on the highest level, the application of the rational filter to the various vectors is partitioned among groups of processors; (2) within each group, every linear system is solved in parallel. In ParaSLRF, the linear systems are solved by iterative methods instead of direct ones, in contrast to other rational filter methods, such as, PFEAST. Because of the specific selection of poles in ParaSLRF, the computational cost of solving the associated linear systems for each pole, is almost the same. This intrinsically leads to a better load balance between each group of resources, and reduces waiting times of processes. We show numerical experiments from finite element models of mechanical vibrations, and show a detailed parallel performance analysis. ParaSLRF shows the best parallel efficiency, compared to other rational filter methods based on quadrature rules for contour integration. To further improve performance, the converged eigenpairs are locked, and a good initial guess of iterative linear solver is proposed. These enhancements of ParaSLRF show good two-level strong scalability and excellent load balance in our experiments.

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arXiv2025

Linearizing a nonlinear eigenvalue problem with quadratic rational eigenvector nonlinearities

Nonlinear eigenvalue problems with eigenvector nonlinearities (NEPv) are algebraic eigenvalue problems whose matrix depends on the eigenvector. Applications range from computational quantum mechanics to machine learning. Due to its nonlinear behavior, existing methods almost exclusively rely on fixed-point iterations, the global convergence properties of which are only understood in specific cases. Recently, a certain class of NEPv with linear rational eigenvector nonlinearities has been linearized, i.e., the spectrum of the linear eigenvalue problem contains the eigenvalues of the NEPv. This linear problem is solved using structure exploiting algorithms to improve both convergence and reliability. We propose a linearization for a different class of NEPv with quadratic rational nonlinearities, inspired by the discretized Gross-Pitaevskii equation. The eigenvalues of this NEPv form a subset of the spectrum of a linear multiparameter eigenvalue problem which is equivalent to a system of generalized eigenvalue problems expressed in terms of operator determinants. A structure exploiting Arnoldi algorithm is used to filter a large portion of spurious solutions and to accelerate convergence.

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arXiv2025

Uniform H-matrix Compression with Applications to Boundary Integral Equations

Boundary integral equations lead to dense system matrices when discretized, yet they are data-sparse. Using the $\mathcal{H}$-matrix format, this sparsity is exploited to achieve $\mathcal{O}(N\log N)$ complexity for storage and multiplication by a vector. This is achieved purely algebraically, based on low-rank approximations of subblocks, and hence the format is also applicable to a wider range of problems. The $\mathcal{H}^2$-matrix format improves the complexity to $\mathcal{O}(N)$ by introducing a recursive structure onto subblocks on multiple levels. However, in many cases this comes with a large proportionality constant, making the $\mathcal{H}^2$-matrix format advantageous mostly for large problems. In this paper we investigate the usefulness of a matrix format that lies in between these two: Uniform $\mathcal{H}$-matrices. An algebraic compression algorithm is introduced to transform a regular $\mathcal{H}$-matrix into a uniform $\mathcal{H}$-matrix, which maintains the asymptotic complexity. Using examples of the BEM formulation of the Helmholtz equation, we show that this scheme lowers the storage requirement and execution time of the matrix-vector product without significantly impacting the construction time.

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arXiv2025

A shifted Laplace rational filter for large-scale eigenvalue problems

We present a rational filter for computing all eigenvalues of a symmetric definite eigenvalue problem lying in an interval on the real axis. The linear systems arising from the filter embedded in the subspace iteration framework, are solved via a preconditioned Krylov method. The choice of the poles of the filter is based on two criteria. On the one hand, the filter should enhance the eigenvalues in the interval of interest, which suggests that the poles should be chosen close to or in the interval. On the other hand, the choice of poles has an important impact on the convergence speed of the iterative method. For the solution of problems arising from vibrations, the two criteria contradict each other, since fast convergence of the eigensolver requires poles to be in or close to the interval, whereas the iterative linear system solver becomes cheaper when the poles lie further away from the eigenvalues. In the paper, we propose a selection of poles inspired by the shifted Laplace preconditioner for the Helmholtz equation. We show numerical experiments from finite element models of vibrations. We compare the shifted Laplace rational filter with rational filters based on quadrature rules for contour integration.

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arXiv2025

Algorithms for Parallel Shared-Memory Sparse Matrix-Vector Multiplication on Unstructured Matrices

The sparse matrix-vector (SpMV) multiplication is an important computational kernel, but it is notoriously difficult to execute efficiently. This paper investigates algorithm performance for unstructured sparse matrices, which are more common than ever because of the trend towards large-scale data collection. The development of an SpMV multiplication algorithm for this type of data is hard due to two factors. First, parallel load balancing issues arise because of the unpredictable nonzero structure. Secondly, SpMV multiplication algorithms are inevitably memory-bound because the sparsity causes a low arithmetic intensity. Three state-of-the-art algorithms for parallel SpMV multiplication on shared-memory systems are discussed. Six new hybrid algorithms are developed which combine optimization techniques of the current algorithms. These techniques include parallelization strategies, storage formats, and nonzero orderings. A modern and high-performance implementation of all discussed algorithms is provided as open-source software. Using this implementation the algorithms are compared. Furthermore, SpMV multiplication algorithms require the matrix to be stored in a specific storage format. Therefore, the conversion time between these storage formats is also analyzed. Both tests are performed for multiple unstructured sparse matrices on different machines: two multi-CPU and two single-CPU architectures. We show that one of the newly developed algorithms outperforms the current state-of-the-art by 19% on one of the multi-CPU architectures. When taking conversion time into consideration, we show that 472 SpMV multiplications are needed to cover the cost of converting to a new storage format for one of the hybrid algorithms on a multi-CPU machine.

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arXiv2024

Efficient parallel inversion of ParaOpt preconditioners

Recently, the ParaOpt algorithm was proposed as an extension of the time-parallel Parareal method to optimal control. ParaOpt uses quasi-Newton steps that each require solving a system of matching conditions iteratively. The state-of-the-art parallel preconditioner for linear problems leads to a set of independent smaller systems that are currently hard to solve. We generalize the preconditioner to the nonlinear case and propose a new, fast inversion method for these smaller systems, avoiding disadvantages of the current options with adjusted boundary conditions in the subproblems.

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arXiv2024

QR-based Parallel Set-Valued Approximation with Rational Functions

In this article a fast and parallelizable algorithm for rational approximation is presented. The method, called (P)QR-AAA, is a (parallel) set-valued variant of the AAA algorithm for scalar functions. It builds on the set-valued AAA framework introduced by Lietaert, Meerbergen, P{é}rez and Vandereycken, accelerating it by using an approximate orthogonal basis obtained from a truncated QR decomposition. We demonstrate both theoretically and numerically this method's accuracy and efficiency. We show how it can be parallelized while maintaining the desired accuracy, with minimal communication cost.

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arXiv2024

The shift-and-invert Arnoldi method for singular matrix pencils

A popular method for solving large sparse regular eigenvalue problem is the shift-and-invert Arnoldi method. This paper aims to use the method for large sparse singular pencils. In three recent papers, {\em Hochstenbach, Mehl, and Plestenjak, 2019, 2023, and 2024}, propose regularization of the singular pencil, using randomly chosen regularization matrices. We propose sparse regularization matrices obtained from the pivoting sequence of a sparse LU factorization. As a side effect, the LU factorization often is rank revealing, which facilitates finding a regularization. Numerical examples illustrate that the LU factorization mostly detects the normal rank and finds a suitable sparse regularization. A rank correction method is proposed for the cases where the normal rank is not determined correctly. For full rank rectangular eigenvalue problems, the pivoting sequence of existing sparse direct system solvers can be used. We compare with randomized regularization methods: preservation of sparsity is beneficial for performance, and often, the accuracy of the eigenvalue solver.

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arXiv2024

On generalized preconditioners for time-parallel parabolic optimal control

The ParaDiag family of algorithms solves differential equations by using preconditioners that can be inverted in parallel through diagonalization. In the context of optimal control of linear parabolic PDEs, the state-of-the-art ParaDiag method is limited to solving self-adjoint problems with a tracking objective. We propose three improvements to the ParaDiag method: the use of alpha-circulant matrices to construct an alternative preconditioner, a generalization of the algorithm for solving non-self-adjoint equations, and the formulation of an algorithm for terminal-cost objectives. We present novel analytic results about the eigenvalues of the preconditioned systems for all discussed ParaDiag algorithms in the case of self-adjoint equations, which proves the favorable properties the alpha-circulant preconditioner. We use these results to perform a theoretical parallel-scaling analysis of ParaDiag for self-adjoint problems. Numerical tests confirm our findings and suggest that the self-adjoint behavior, which is backed by theory, generalizes to the non-self-adjoint case. We provide a sequential, open-source reference solver in Matlab for all discussed algorithms.

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arXiv2023

Diagonalization-based preconditioners and generalized convergence bounds for ParaOpt

The ParaOpt algorithm was recently introduced as a time-parallel solver for optimal-control problems with a terminal-cost objective, and convergence results have been presented for the linear diffusive case with implicit-Euler time integrators. We reformulate ParaOpt for tracking problems and provide generalized convergence analyses for both objectives. We focus on linear diffusive equations and prove convergence bounds that are generic in the time integrators used. For large problem dimensions, ParaOpt's performance depends crucially on having a good preconditioner to solve the arising linear systems. For the case where ParaOpt's cheap, coarse-grained propagator is linear, we introduce diagonalization-based preconditioners inspired by recent advances in the ParaDiag family of methods. These preconditioners not only lead to a weakly-scalable ParaOpt version, but are themselves invertible in parallel, making maximal use of available concurrency. They have proven convergence properties in the linear diffusive case that are generic in the time discretization used, similarly to our ParaOpt results. Numerical results confirm that the iteration count of the iterative solvers used for ParaOpt's linear systems becomes constant in the limit of an increasing processor count. The paper is accompanied by a sequential MATLAB implementation.

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arXiv2022

Time integration of finite element models with nonlinear frequency dependencies

The analysis of sound and vibrations is often performed in the frequency domain, implying the assumption of steady-state behaviour and time-harmonic excitation. External excitations, however, may be transient rather than time-harmonic, requiring time-domain analysis. Some material properties, e.g.\ often used to represent for damping treatments, are still described in the frequency domain, which complicates simulation in time. In this paper, we present a method for the linearization of finite element models with nonlinear frequency dependencies. The linearization relies on the rational approximation of the finite element matrices by the AAA method. We introduce the Extended AAA method, which is classical AAA combined with a degree two polynomial term to capture the second order behaviour of the models. A filtering step is added for removing unstable poles.

math.NA↗

arXiv2022

Contour Integration for Eigenvector Nonlinearities

Solving polynomial eigenvalue problems with eigenvector nonlinearities (PEPv) is an interesting computational challenge, outside the reach of the well-developed methods for nonlinear eigenvalue problems. We present a natural generalization of these methods which leads to a contour integration approach for computing all eigenvalues of a PEPv in a compact region of the complex plane. Our methods can be used to solve any suitably generic system of polynomial or rational function equations.

math.NA↗

arXiv2022

Frequency extraction for BEM-matrices arising from the 3D scalar Helmholtz equation

The discretisation of boundary integral equations for the scalar Helmholtz equation leads to large dense linear systems. Efficient boundary element methods (BEM), such as the fast multipole method (FMM) and $\Hmat$ based methods, focus on structured low-rank approximations of subblocks in these systems. It is known that the ranks of these subblocks increase linearly with the wavenumber. We explore a data-sparse representation of BEM-matrices valid for a range of frequencies, based on extracting the known phase of the Green's function. Algebraically, this leads to a Hadamard product of a frequency matrix with an $\Hmat$. We show that the frequency dependency of this $\Hmat$ can be determined using a small number of frequency samples, even for geometrically complex three-dimensional scattering obstacles. We describe an efficient construction of the representation by combining adaptive cross approximation with adaptive rational approximation in the continuous frequency dimension. We show that our data-sparse representation allows to efficiently sample the full BEM-matrix at any given frequency, and as such it may be useful as part of an efficient sweeping routine.

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arXiv2021

Linearizability of eigenvector nonlinearities

We present a method to linearize, without approximation, a specific class of eigenvalue problems with eigenvector nonlinearities (NEPv), where the nonlinearities are expressed by scalar functions that are defined by a quotient of linear functions of the eigenvector. The exact linearization relies on an equivalent multiparameter problem (MEP) that contains the exact solutions of the NEPv. Due to the characterization of MEPs in terms of a generalized eigenvalue problem this provides a direct way to compute all NEPv solutions for small problems, and it opens up the possibility to develop locally convergent iterative methods for larger problems. Moreover, the linear formulation allows us to easily determine the number of solutions of the NEPv. We propose two numerical schemes that exploit the structure of the linearization: inverse iteration and residual inverse iteration. We show how symmetry in the MEP can be used to improve reliability and reduce computational cost of both methods. Two numerical examples verify the theoretical results and a third example shows the potential of a hybrid scheme that is based on a combination of the two proposed methods.

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arXiv2020

Tensor-Krylov method for computing eigenvalues of parameter-dependent matrices

In this paper we extend the Residual Arnoldi method for calculating an extreme eigenvalue (e.g. largest real part, dominant,...) to the case where the matrices depend on parameters. The difference between this Arnoldi method and the classical Arnoldi algorithm is that in the former the residual is added to the subspace. We develop a Tensor-Krylov method that applies the Residual Arnoldi method (RA) for a grid of parameter points at the same time. The subspace contains an approximate Krylov space for all these points. Instead of adding the residuals for all parameter values to the subspace we create a low-rank approximation of the matrix consisting of these residuals and add only the column space to the subspace. In order to keep the computations efficient, it is needed to limit the dimension of the subspace and to restart once the subspace has reached the prescribed maximal dimension. The novelty of this approach is twofold. Firstly, we observed that a large error in the low-rank approximations is allowed without slowing down the convergence, which implies that we can do more iterations before restarting. Secondly, we pay particular attention to the way the subspace is restarted, since classical restarting techniques give a too large subspace in our case. We motivate why it is good enough to just keep the approximation of the searched eigenvector. At the end of the paper we extend this algorithm to shift-and-invert Residual Arnoldi method to calculate the eigenvalue close to a shift $σ$ for a specific parameter dependency. We provide theoretical results and report numerical experiments. The Matlab code is publicly available.

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