17 citations · 95 across the 41 of their papers we have counts for
46 papers
How different is rational approximation from piecewise polynomial approximation?
Daan Huybrechs
The first aim of this paper is to show that there is merit to the question posed in the title. Indeed, for certain function classes, approximation by rational functions and by piec…
Compact Rational Krylov for Parametrized Systems with Application to BEM Frequency Sweeping
Kobe Bruyninckx, Daan Huybrechs, Karl Meerbergen
In parametrized linear systems the system matrix depends nonlinearly on a parameter and solutions are sought for many values o…
On the algorithmic construction of deep ReLU networks
Daan Huybrechs
It is difficult to describe in mathematical terms what a neural network trained on data represents. On the other hand, there is a growing mathematical understanding of what neural…
Function Approximation in Numerically Rank-Deficient Bases
Astrid Herremans, Daan Huybrechs
We study linear function approximation in a finite basis under finite-precision arithmetic. In a highly non-orthogonal basis, certain directions are only weakly represented, so tha…
Uniform H-matrix Compression with Applications to Boundary Integral Equations
Kobe Bruyninckx, Daan Huybrechs, Karl Meerbergen
Boundary integral equations lead to dense system matrices when discretized, yet they are data-sparse. Using the -matrix format, this sparsity is exploited to achieve $…
Multivariate rational approximation of functions with curves of singularities
Nicolas Boullé, Astrid Herremans, Daan Huybrechs
Functions with singularities are notoriously difficult to approximate with conventional approximation schemes. In computational applications, they are often resolved with low-order…
QR-based Parallel Set-Valued Approximation with Rational Functions
Simon Dirckx, Karl Meerbergen, Daan Huybrechs
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 f…
Efficient Function Approximation in Enriched Approximation Spaces
Astrid Herremans, Daan Huybrechs
An enriched approximation space is the span of a conventional basis with a few extra functions included, for example to capture known features of the solution to a computational pr…
Efficient least squares approximation and collocation methods using radial basis functions
Yiqing Zhou, Daan Huybrechs
We describe an efficient method for the approximation of functions using radial basis functions (RBFs), and extend this to a solver for boundary value problems on irregular domains…
Sigmoid functions and multiscale resolution of singularities
Daan Huybrechs, Lloyd N. Trefethen
In this short, conceptual paper we observe that essentially the same mathematics applies in three contexts with disparate literatures: (1) sigmoidal and RBF approximation of smooth…
Resolution of singularities by rational functions
Astrid Herremans, Daan Huybrechs, Lloyd N. Trefethen
Results on the rational approximation of functions containing singularities are presented. We build further on the ''lightning method'', recently proposed by Trefethen and collabor…
Stable and accurate least squares radial basis function approximations on bounded domains
Ben Adcock, Daan Huybrechs, Cécile Piret
The computation of global radial basis function (RBF) approximations requires the solution of a linear system which, depending on the choice of RBF parameters, may be ill-condition…
On the computation of the SVD of Fourier submatrices
Simon Dirckx, Daan Huybrechs, Robbe Ongenae
Contiguous submatrices of the Fourier matrix are known to be ill-conditioned. In a recent paper in SIAM Review A. Barnett has provided new bounds on the rate of ill-conditioning of…
Arbitrary-order asymptotic expansions of Gaussian quadrature rules with classical and generalised weight functions
Peter Opsomer, Daan Huybrechs
Gaussian quadrature rules are a classical tool for the numerical approximation of integrals with smooth integrands and positive weight functions. We derive and expicitly list asymp…
AAA interpolation of equispaced data
Daan Huybrechs, Lloyd N. Trefethen
We propose AAA rational approximation as a method for interpolating or approximating smooth functions from equispaced data samples. Although it is always better to approximate from…
Stable approximation of Helmholtz solutions in the disk by evanescent plane waves
Emile Parolin, Daan Huybrechs, Andrea Moiola
Superpositions of plane waves are known to approximate well the solutions of the Helmholtz equation. Their use in discretizations is typical of Trefftz methods for Helmholtz proble…
An analysis of least-squares oversampled collocation methods for compactly perturbed boundary integral equations in two dimensions
Georg Maierhofer, Daan Huybrechs
In recent work (Maierhofer & Huybrechs, 2022, Adv. Comput. Math.), the authors showed that least-squares oversampling can improve the convergence properties of collocation methods…
Convergence analysis of oversampled collocation boundary element methods in 2D
Georg Maierhofer, Daan Huybrechs
Collocation boundary element methods for integral equations are easier to implement than Galerkin methods because the elements of the discretization matrix are given by lower-dimen…
Frequency extraction for BEM-matrices arising from the 3D scalar Helmholtz equation
Simon Dirckx, Daan Huybrechs, Karl Meerbergen
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…
Quaternionic Step Derivative: Machine Precision Differentiation of Holomorphic Functions using Complex Quaternions
Martin Roelfs, David Dudal, Daan Huybrechs
The known Complex Step Derivative (CSD) method allows easy and accurate differentiation up to machine precision of real analytic functions by evaluating them a small imaginary step…
On the adaptive spectral approximation of functions using redundant sets and frames
Vincent Coppé, Daan Huybrechs
The approximation of smooth functions with a spectral basis typically leads to rapidly decaying coefficients where the rate of decay depends on the smoothness of the function and v…
Efficient function approximation on general bounded domains using wavelets on a cartesian grid
Vincent Coppé, Daan Huybrechs
Fourier extension is an approximation method that alleviates the periodicity requirements of Fourier series and avoids the Gibbs phenomenon when approximating functions. We describ…
The AZ algorithm for least squares systems with a known incomplete generalized inverse
Vincent Coppe, Daan Huybrechs, Roel Matthysen +1
We introduce an algorithm for the least squares solution of a rectangular linear system , in which may be arbitrarily ill-conditioned. We assume that a complementary matr…
Fast hybrid numerical-asymptotic boundary element methods for high frequency screen and aperture problems based on least-squares collocation
Andrew Gibbs, David Hewett, Daan Huybrechs +1
We present a hybrid numerical-asymptotic (HNA) boundary element method (BEM) for high frequency scattering by two-dimensional screens and apertures, whose computational cost to ach…