arXiv subjects

Marc Van Barel

Publications and source records attributed to Marc Van Barel.

At least 19 recordsLinked to original sources

arXiv2026

Updating and Downdating the Recurrences of Discrete Multiple Orthogonal Polynomials on the Real Line

We study multiple orthogonal polynomials with orthogonality defined by multiple positive discrete measures on the real line. Focusing on type~I and type~II multiple orthogonal polynomials and their step-line recurrence relations, we consider the reconstruction of the associated banded upper Hessenberg recurrence matrix from given nodes and weights of the measures. We propose efficient updating and downdating procedures that modify an existing recurrence matrix when nodes are added to or removed from the discrete measures. The updating strategy is based on solving an inverse eigenvalue problem, while the downdating procedure relies on an eigenvalue deflation technique inspired by a QR-type of algorithm. Numerical experiments confirm the stability and efficiency of the proposed approaches.

math.NA↗

arXiv2026

Krylov and core transformation algorithms for an inverse eigenvalue problem to compute recurrences of multiple orthogonal polynomials

In this paper, we develop algorithms for computing the recurrence coefficients corresponding to multiple orthogonal polynomials on the step-line. We reformulate the problem as an inverse eigenvalue problem, which can be solved using numerical linear algebra techniques. We consider two approaches: the first is based on the link with block Krylov subspaces and results in a biorthogonal Lanczos process with multiple starting vectors; the second consists of applying a sequence of Gaussian eliminations on a diagonal matrix to construct the banded Hessenberg matrix containing the recurrence coefficients. We analyze the accuracy and stability of the algorithms with numerical experiments on the ill-conditioned inverse eigenvalue problemshave related to Kravchuk and Hahn polynomials, as well as on other better conditioned examples.

math.NA↗

arXiv2026

Orthonormal polynomial wavelets associated with de la Vallée Poussin-type interpolation on $[-1,1]$

Starting from de la Vallée Poussin type (VP) interpolation, the authors have recently introduced a family of interpolating polynomial scaling and wavelet bases generating the approximation and detail spaces of a non-standard multiresolution analysis. Motivated by the fact that, in many applications, orthonormal rather than interpolating bases are preferable, the present study develops a new family of scaling and wavelet polynomials that provide well-localized and orthonormal bases for the same approximation and detail spaces. We show that the proposed new bases have a behavior very similar to the interpolating bases already introduced, presenting similar features although they are not interpolating but orthonormal. In particular, we study the Fourier projection corresponding to the proposed orthonormal scaling basis, and introduce a discrete version of it by approximating the Fourier--like coefficients. For both continuous and discrete orthogonal projections, we prove the uniform boundedness of the Lebesgue constants and the uniform convergence with an asymptotic rate comparable with the best uniform polynomial approximation. Numerical experiments confirm the theoretical results and compare the new orthonormal VP scaling and wavelet bases with the interpolating case previously treated by the authors.

math.NA↗

arXiv2025

A Parametric Family of Polynomial Wavelets for Signal and Image Processing

This paper investigates the potential applications of a parametric family of polynomial wavelets that has been recently introduced starting from de la Vallée Poussin (VP) interpolation at Chebyshev nodes. Unlike classical wavelets, which are constructed on the real line, these VP wavelets are defined on a bounded interval, offering the advantage of handling boundaries naturally while maintaining computational efficiency. In addition, the structure of these wavelets enables the use of fast algorithms for decomposition and reconstruction. Furthermore, the flexibility offered by a free parameter allows a better control of localized singularities, such as edges in images. On the basis of previous theoretical foundations, we show the effectiveness of the VP wavelets for basic signal denoising and image compression, emphasizing their potential for more advanced signal and image processing tasks.

math.NA↗

arXiv2025

On computing the zeros of a class of Sobolev orthogonal polynomials

A fast and weakly stable method for computing the zeros of a particular class of hypergeometric polynomials is presented. The studied hypergeometric polynomials satisfy a higher order differential equation and generalize Laguerre polynomials. The theoretical study of the asymptotic distribution of the spectrum of these polynomials is an active research topic. In this article we do not contribute to the theory, but provide a practical method to contribute to further and better understanding of the asymptotic behavior. The polynomials under consideration fit into the class of Sobolev orthogonal polynomials, satisfying a four--term recurrence relation. This allows computing the roots via a generalized eigenvalue problem. After condition enhancing similarity transformations, the problem is transformed into the computation of the eigenvalues of a comrade matrix, which is a symmetric tridiagonal modified by a rank--one matrix. The eigenvalues are then retrieved by relying on an existing structured rank based fast algorithm. Numerical examples are reported studying the accuracy, stability and conforming the efficiency for various parameter settings of the proposed approach.

math.NA↗

arXiv2025

Generalized cyclic symmetric decompositions for the matrix multiplication tensor

A new generalized cyclic symmetric structure in the factor matrices of polyadic decompositions of matrix multiplication tensors for non-square matrix multiplication is proposed to reduce the number of variables in the optimization problem and in this way improve the convergence. The structure is implemented in an existing numerical optimization algorithm. Extensive numerical experiments are given that the proposed structure indeed finds more (practical) decompositions.

math.NA↗

arXiv2025

Constructing Sobolev orthonormal rational functions via an updating procedure

In this paper, we generate the recursion coefficients for rational functions with prescribed poles that are orthonormal with respect to a continuous Sobolev inner product. Using a rational Gauss quadrature rule, the inner product can be discretized, thus allowing a linear algebraic approach. The presented approach involves reformulating the problem as an inverse eigenvalue problem involving a Hessenberg pencil, where the pencil will contain the recursion coefficients that generate the sequence of Sobolev orthogonal rational functions. This reformulation is based on the connection between Sobolev orthonormal rational functions and the orthonormal bases for rational Krylov subspaces generated by a Jordan-like matrix. An updating procedure, introducing the nodes of the inner product one after the other, is proposed and the performance is examined through some numerical examples.

math.NA↗

arXiv2025

A free parameter depending family of polynomial wavelets on a compact interval

On a compact interval, we introduce and study a whole family of wavelets depending on a free parameter that can be suitably modulated to improve performance. Such wavelets arise from de la Vallée Poussin (VP) interpolation at Chebyshev nodes, generalizing previous work by Capobianco and Themistoclakis who considered a special parameter setting. In our construction, both scaling and wavelet functions are interpolating polynomials at some Chebyshev zeros of 1st kind. Contrarily to the classical approach, they are not generated by dilations and translations of a single mother function and are naturally defined on the interval $[-1,1]$ to which any other compact interval can be reduced. In the paper, we provide a non-standard multiresolution analysis with fast (DCT-based) decomposition and reconstruction algorithms. Moreover, we state several theoretical results, particularly on convergence, laying the foundation for future applications.

math.NA↗

arXiv2024

An Arnoldi-based approach to polynomial and rational least squares problems

In this research, we solve polynomial, Sobolev polynomial, rational, and Sobolev rational least squares problems. Although the increase in the approximation degree allows us to fit the data better in attacking least squares problems, the ill-conditioning of the coefficient matrix fuels the dramatic decrease in the accuracy of the approximation at higher degrees. To overcome this drawback, we first show that the column space of the coefficient matrix is equivalent to a Krylov subspace. Then the connection between orthogonal polynomials or rational functions and orthogonal bases for Krylov subspaces in order to exploit Krylov subspace methods like Arnoldi orthogonalization is established. Furthermore, some examples are provided to illustrate the theory and the performance of the proposed approach.

math.NA↗

arXiv2024

A Riemannian rank-adaptive method for higher-order tensor completion in the tensor-train format

In this paper a new Riemannian rank adaptive method (RRAM) is proposed for the low-rank tensor completion problem (LRTCP) formulated as a least-squares optimization problem on the algebraic variety of tensors of bounded tensor-train (TT) rank. The method iteratively optimizes over fixed-rank smooth manifolds using a Riemannian conjugate gradient algorithm from Steinlechner (2016) and gradually increases the rank by computing a descent direction in the tangent cone to the variety. Additionally, a numerical method to estimate the amount of rank increase is proposed based on a theoretical result for the stationary points of the low-rank tensor approximation problem and a definition of an estimated TT-rank. Furthermore, when the iterate comes close to a lower-rank set, the RRAM decreases the rank based on the TT-rounding algorithm from Oseledets (2011) and a definition of a numerical rank. We prove that the TT-rounding algorithm can be considered as an approximate projection onto the lower-rank set which satisfies a certain angle condition to ensure that the image is sufficiently close to that of an exact projection. Several numerical experiments are given to illustrate the use of the RRAM and its subroutines in {\Matlab}. Furthermore, in all experiments the proposed RRAM outperforms the state-of-the-art RRAM for tensor completion in the TT format from Steinlechner (2016) in terms of computation time.

math.OC↗

arXiv2023

A generalization of Floater--Hormann interpolants

In this paper the interpolating rational functions introduced by Floater and Hormann are generalized leading to a whole new family of rational functions depending on $γ$, an additional positive integer parameter. For $γ= 1$, the original Floater--Hormann interpolants are obtained. When $γ>1$ we prove that the new rational functions share a lot of the nice properties of the original Floater--Hormann functions. Indeed, for any configuration of nodes in a compact interval, they have no real poles, interpolate the given data, preserve the polynomials up to a certain fixed degree, and have a barycentric-type representation. Moreover, we estimate the associated Lebesgue constants in terms of the minimum ($h^*$) and maximum ($h$) distance between two consecutive nodes. It turns out that, in contrast to the original Floater-Hormann interpolants, for all $γ> 1$ we get uniformly bounded Lebesgue constants in the case of equidistant and quasi-equidistant nodes configurations (i.e., when $h\sim h^*$). For such configurations, as the number of nodes tends to infinity, we prove that the new interpolants ($γ>1$) uniformly converge to the interpolated function $f$, for any continuous function $f$ and all $γ>1$. The same is not ensured by the original FH interpolants ($γ=1$). Moreover, we provide uniform and pointwise estimates of the approximation error for functions having different degrees of smoothness. Numerical experiments illustrate the theoretical results and show a better error profile for less smooth functions compared to the original Floater-Hormann interpolants.

math.NA↗

arXiv2023

Stability Improvements for Fast Matrix Multiplication

We implement an Augmented Lagrangian method to minimize a constrained least-squares cost function designed to find polyadic decompositions of the matrix multiplication tensor. We use this method to obtain new discrete decompositions and parameter families of decompositions. Using these parametrizations, faster and more stable matrix multiplication algorithms can be discovered.

math.NA↗

arXiv2023

Rank Estimation for Third-Order Tensor Completion in the Tensor-Train Format

We propose a numerical method to obtain an adequate value for the upper bound on the rank for the tensor completion problem on the variety of third-order tensors of bounded tensor-train rank. The method is inspired by the parametrization of the tangent cone derived by Kutschan (2018). A proof of the adequacy of the upper bound for a related low-rank tensor approximation problem is given and an estimated rank is defined to extend the result to the low-rank tensor completion problem. Some experiments on synthetic data illustrate the approach and show that the method is very robust, e.g., to noise on the data.

math.OC↗

arXiv2023

An Approximate Projection onto the Tangent Cone to the Variety of Third-Order Tensors of Bounded Tensor-Train Rank

An approximate projection onto the tangent cone to the variety of third-order tensors of bounded tensor-train rank is proposed and proven to satisfy a better angle condition than the one proposed by Kutschan (2019). Such an approximate projection enables, e.g., to compute gradient-related directions in the tangent cone, as required by algorithms aiming at minimizing a continuously differentiable function on the variety, a problem appearing notably in tensor completion. A numerical experiment is presented which indicates that, in practice, the angle condition satisfied by the proposed approximate projection is better than both the one satisfied by the approximate projection introduced by Kutschan and the proven theoretical bound.

math.OC↗

arXiv2023

Algorithms for Modifying Recurrence Relations of Orthogonal Polynomial and Rational Functions when Changing the Discrete Inner Product

Often, polynomials or rational functions, orthogonal for a particular inner product are desired. In practical numerical algorithms these polynomials are not constructed, but instead the associated recurrence relations are computed. Moreover, also typically the inner product is changed to a discrete inner product, which is the finite sum of weighted functions evaluated in specific nodes. For particular applications it is beneficial to have an efficient procedure to update the recurrence relations when adding or removing nodes from the inner product. The construction of the recurrence relations is equivalent to computing a structured matrix (polynomial) or pencil (rational) having prescribed spectral properties. Hence the solution of this problem is often referred to as solving an Inverse Eigenvalue Problem. In Van Buggenhout et al. (2022) we proposed updating techniques to add nodes to the inner product while efficiently updating the recurrences. To complete this study we present in this article manners to efficiently downdate the recurrences when removing nodes from the inner product. The link between removing nodes and the QR algorithm to deflate eigenvalues is exploited to develop efficient algorithms. We will base ourselves on the perfect shift strategy and develop algorithms, both for the polynomial case and the rational function setting. Numerical experiments validate our approach.

math.NA↗

arXiv2022

Equivalent Polyadic Decompositions of Matrix Multiplication Tensors

Invariance transformations of polyadic decompositions of matrix multiplication tensors define an equivalence relation on the set of such decompositions. In this paper, we present an algorithm to efficiently decide whether two polyadic decompositions of a given matrix multiplication tensor are equivalent. With this algorithm, we analyze the equivalence classes of decompositions of several matrix multiplication tensors. This analysis is relevant for the study of fast matrix multiplication as it relates to the question of how many essentially different fast matrix multiplication algorithms there exist. This question has been first studied by de~Groote, who showed that for the multiplication of $2\times2$ matrices with $7$ active multiplications, all algorithms are essentially equivalent to Strassen's algorithm. In contrast, the results of our analysis show that for the multiplication of larger matrices, (e.g., $2\times3$ by $3\times2$ or $3\times3$ by $3\times3$ matrices), two decompositions are very likely to be essentially different. We further provide a necessary criterion for a polyadic decomposition to be equivalent to a polyadic decomposition with integer entries. Decompositions with specific integer entries, e.g., powers of two, provide fast matrix multiplication algorithms with better efficiency and stability properties. This condition can be tested algorithmically and we present the conclusions obtained for the decompositions of small/medium matrix multiplication tensors.

cs.CC↗

arXiv2022

Adaptive cross approximation for Tikhonov regularization in general form

Many problems in Science and Engineering give rise to linear integral equations of the first kind with a smooth kernel. Discretization of the integral operator yields a matrix, whose singular values cluster at the origin. We describe the approximation of such matrices by adaptive cross approximation, which avoids forming the entire matrix. The choice of the number of steps of adaptive cross approximation is discussed. The discretized right-hand side represents data that commonly are contaminated by measurement error. Solution of the linear system of equations so obtained is not meaningful because the matrix determined by adaptive cross approximation is rank-deficient. We remedy this difficulty by using Tikhonov regularization and discuss how a fairly general regularization matrix can be used. Computed examples illustrate that the use of a regularization matrix different from the identity can improve the quality of the computed approximate solutions significantly.

math.NA↗

arXiv2021

Generation of orthogonal rational functions by procedures for structured matrices

The problem of computing recurrence coefficients of sequences of rational functions orthogonal with respect to a discrete inner product is formulated as an inverse eigenvalue problem for a pencil of Hessenberg matrices. Two procedures are proposed to solve this inverse eigenvalue problem, via the rational Arnoldi iteration and via an updating procedure using unitary similarity transformations. The latter is shown to be numerically stable. This problem and both procedures are generalized by considering biorthogonal rational functions with respect to a bilinear form. This leads to an inverse eigenvalue problem for a pencil of tridiagonal matrices. A tridiagonal pencil implies short recurrence relations for the biorthogonal rational functions, which is more efficient than the orthogonal case. However the procedures solving this problem must rely on nonunitary operations and might not be numerically stable.

math.NA↗