Conference Agenda

We consider the problem of model order reduction in reproducible kernel Hilbert spaces (RKHSs). We prove a generalization of well-known necessary interpolatory conditions for norm-optimal reduced order models, originally stated for rational functions in the Hardy-Hilbert space $H_2$. Previous optimality conditions stated for $H_2$ spaces defined over various domains (e.g., the disk and the half plane) are special cases of the condition obtained in this talk. We introduce generalized Malmquist-Takenaka (GMT) functions that can be used to define orthonormal bases in an arbitrary RKHS. Using GMT functions, we construct an equivalent alternative to the iterative rational Krylov algorithm for RKHSs with shift invariant kernels to obtain norm-optimal reduced order models. In order to deal with non-shift invariant kernels, we propose a family of unitary transformations using GMT functions. The effectiveness of the proposed methods is demonstrated in several numerical experiments such as linear and nonlinear dynamical systems and machine learning models such as support vector machines and deep ReLU networks.

In this talk we present a method for computing quasi resonances in acoustic scattering based on a discretization of a boundary integral operators of the second kind. Quasi resonance is a minimum point of the function mapping the real wave number to the minimal singular value of the Calderon operator. We adaptively approximate this function by a piecewise rational function satisfying Hermite interpolation conditions in a weak sense. Hermite information is readily available in this setting since the main computational effort consists in assembling the discretization of the Calderon operator and the cost of assembling its derivatives is only a postprocessing task. The main algebraic effort consists in solving a family of SVD problems using a subspace accelerated randomized algorithm. The randomized algorithm is a further development of the standard approach in that we solve for the smallest singular value and we need to balance this error with the error in the construction of the rational surrogate function. We use BICGSTAB algorithm with H-matrix based preconditioner to this effect. This is a joint wotk with Luka Marohnic, TVZ.

The AAA algorithm was proposed in [Nakatsukasa/Sete/Treftehen '18] as a rational approximation tool that computes approximants as rational functions represented in barycentric form. The latter represents a numerically stable format that also imposes interpolation implicitly. The recent review paper [Nakatsukasa/Trefethen '26] surveys various improvements, generalizations, and applications of the AAA algorithm in the last 8 years. By harmoniously blending interpolation (as for non-iterative Loewner matrix methods) and least-squares fitting (as in the vector fitting approach), AAA aims to find a rational approximant by iteratively adjusting the barycentric form of the fitted model based on greedily selected interpolation points and a least-squares fit. In summary, AAA is applied iteratively, obtaining rational approximations of increased order, until a desired tolerance on the l2 approximation error is achieved. The model enrichment is performed by adding support points at the locations where the approximation error is largest. In particular, one neither needs to fix the surrogate model order nor (even more importantly) the support points in advance, since both are automatically chosen by the algorithm. Originally, the AAA algorithm was developed for approximation of scalar-valued functions, although recent extensions deal with vector- or matrix-valued functions as well (such as set-valued AAA, tangential AAA, or Block-AAA). We aim to extend the Block-AAA algorithm in [Gosea/Guettel '21] to more than two dimensions (going beyond 2D matrices to, e.g., multi-dimensional tensors). The reason for choosing this approach over other vector-valued extensions is that the latter typically fix the poles of the approximant for various parameter values, while Block-AAA offers more flexibility by allowing the location of poles to vary for both space and parameter values. This is particularly relevant for the class of problems we target here; the application of interest stems from structural dynamics, in which simulations and data analysis are challenged by large multi-dimensional data. Apart from two or three spatial directions, the problem coordinates include the dimension in frequency actuation and, possibly, dimensions to model uncertainties. If stored as a multidimensional array, these data quickly exceed storage capacities. Hence, efficient approximative representations are needed for data handling and, respectively, for simulations as a surrogate model. The proposed method aims precisely at attenuating this burden by harnessing the efficiency of canonical/hierarchical tensor decompositions, encoded explicitly in the generalized barycentric form of the approximant. The application studied here concerns modeling vibrating plates or vibroacoustic-coupled phenomena. Additionally, we consider uncertainty in the material parameters by modeling Young’s modulus as a log-normal random field. Various numerical results are presented to support the theoretical claims.

The motivation for our work comes from damage identification using guided ultrasonic wave propagation. By analyzing the refelction patterns caused by defects, the aim is to localize and characterize a damage. Such wave propagation can be described by a second order mechanical system. Solving the corresponding inverse problem involves a huge number of forward simulations which can be prohibitively expensive. Therefore, we look for surrogate models that are sufficiently accurate and cheaper to solve. It is well-known that projection-based model reduction such as e.g. classical POD are challenged in this setting due to the Kolmogorov barrier and stability issues need to be considered. In this work, we present a comparative study of intrusive and non-intrusive reduction approaches and take first steps towards the application of rational interpolation based reduced models to our application problem.