Schedule for: 25w5376 - Challenges, Opportunities, and New Horizons in Rational Approximation

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

Panel discussions on three key themes for the workshop. Theme 1 is rational approximation theory, and the discussion will be led by Daan Huybrechs and Nick Trefethen. Theme 2 is rational approximation in the age of data-driven modeling, and the discussion will be led by Heike Fassbender and Tobias Breiten. Theme 3 is software for rational approximation and rational approximation in practice, and the discussion will be led by Toby Driscoll and Steffen Werner.

Many attendees have already seen me speak about AAA approximation, so after just a brief review of the algorithm and its applications, this talk will turn to the question, when and how can AAA fail? There is more going on here than the binary question one might imagine: "can we prove it converges and if not why not?" On the contrary, there are different variants to consider, including AAA for near-best approximation, AAA-Lawson for best approximation, continuum AAA for approximation on domains such as intervals and disks, and a "sign" alternative to the original SVD iteration that proves more reliable for problems with a sign(z) flavor. There are also different failure modes to consider, including failure to converge, convergence but with approximations that are far from optimal, and the appearance of poles in unwanted locations. By investigating various aspects of these questions, we hope that one or more versions of AAA can be developed that are fast and effective and provably so.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

The RationalFunctionApproximation.jl software package offers simple access to three complementary approaches to finding rational approximations: linear least-squares using prescribed poles, AAA interpolation, and greedy adaptive Thiele interpolation. This implementation uses Julia to enable high performance and arbitrary precision while keeping enough flexibility and transparency to allow comparisons between and research into the underlying methods. Both discrete and continuous approximation over general regions in the complex plane are straightforward, and the package can be used as a library or interactively from Julia or Python.

From the approximation of complex-valued functions to the reduction of dynamical systems, Hardy spaces provide a powerful framework for the analysis and construction of optimal rational approximants. One of the most widely used methods in linear model order reduction is the iterative rational Krylov algorithm (IRKA) which was designed to construct rational interpolants that satisfy the first order optimality conditions introduced by Meier and Luenberger. This talk briefly reviews the classical $H_2$ optimal model reduction framework and provides possible extensions to a broader class of domains in the complex plane. In particular, two different approaches will be discussed: in the first case, reduced order models are assumed to have a rational structure; in the second case, the rational structure is obtained only after composition with a conformal map which relates to the underlying domain. For both approaches, connections between $H_2$- like optimality conditions and the Schwarz function will be discussed. Finally, the theoretical findings are illustrated by means of numerical examples.

Piecewise polynomial approximations play a prominent role throughout scientific computing. It turns out that functions well approximated by piecewise polynomials are typically well approximated by rational functions too - perhaps even better? The comparison is most interesting in the context of graded meshes. We understand this setting well for univariate functions and offer a few perspectives. The case of several variables is more fascinating, but very unexplored: this area is in need of new developments. In that setting there is also a third contender, namely neural networks, in particular neural network representations of continuous and piecewise linear functions.

TBA

\( \textbf{Maria del Carmen Quintana Ponce:}\) Loewner linearizations of structured rational matrices, \( \textbf{Michael Ackermann:} \)Robust learning of rational transfer functions from time-domain data, \( \textbf{Cade Ballew:}\) Polynomials: Better than you think, \( \textbf{Levent Batakci:}\) TBA, \( \textbf{Keaton Burns:}\) Rational preconditioners for sparse spectral methods, \( \textbf{Michael Chiwere:}\) Linear Barycentric Interpolation with Applications on the Sphere, \( \textbf{Simon Dirckx:} \) PQR-AAA: Accelerating set-valued rational approximation, \( \textbf{Evan Gawlik:}\) Rational finite elements, \( \textbf{Ion Victor Gosea:}\) The AAA algorithm as a tool for reduced-order modeling of dynamical systems: recent developments.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Women participants are encouraged to meet in front of the dining hall, have breakfast together, and take the opportunity to get to know other women in the field.

Each participant will choose to join a working group associated with either Theme 1, Theme 2 or Theme 3. The groups will meet and work on open problems.

TBA

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

This talk will explore several topics revolving around AAA and topics in (randomized) linear algebra. I will first discuss the use of AAA for solving nonlinear eigenvalue problems. The central question is: should we approximate $f(\lambda)$, or its resolvent $f(\lambda)^{-1}$ ? I will discuss the role of randomized sketching in this context, together with the benefits brought by oversampling. I will then turn to model order reduction, which shares several key aspects. Time permitting, I hope to touch on other aspects of rational approximation, such as approximating noisy functions, speeding up the AAA computation, and approximation by composite rational functions.

Barycentric algorithms, such as the AAA algorithm, generate near-best rational approximations with minimal prior knowledge of the function to be approximated. However, these methods do not allow direct control over the poles of the rational function, which is often desirable in applications. Assuming the locations of the singularities of the function are known, we demonstrate that accurate rational approximations can be obtained through simple least squares fitting. These approximations are constructed using partial fractions with preassigned poles. This talk provides an overview of existing results in both the univariate and multivariate cases and explores the properties of the partial fraction representation by addressing the question: what constitutes a 'good' basis?

TBA

\( \textbf{Sam Hocking:}\) Extended precision analytic continuation, \( \textbf{Kaitlynn Lilly:}\) From Oscillations to Approximations: The Rational Way, \( \textbf{Stefano Massei:}\) Error formula for block rational Krylov approximations of matrix functions, \( \textbf{Kyle McKee:}\) AAA for Fluid Flow and Advection-Diffusion, \( \textbf{Petar Mlinarić :}\) Riemannian Optimization over Rational Functions, \( \textbf{Sean Rieter:}\) $\mathcal{H}_2$-optimal model reduction by multivariate rational interpolation, \(\textbf{Zoran Tomljanović:}\) $H_{\infty}$ Analysis of Cooperative Multi-Agent Systems by Adaptive Interpolation, \(\textbf{Nick Trefethen:}\) Computing complex resonances with AAA, \(\textbf{Annan Yu:}\) How to train a rational function?

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

TBA.

This panel consists of speakers at various stages, with experiences in both academia and industry. In addition to sharing some insight about what has made their careers successful, panelists will answer questions from the audience and give advice about how to achieve success in the field of computational mathematics. We highly encourage graduate students, postdocs, and early career academics to attend! Panelists include Mark Embree, Akil Narayan, Barbara Wohlmuth, and Lucas Monzón.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Intrusive methods for model order reduction of dynamical systems is well understood. Recently nonintrusive (data driven) methods were introduced relying on rational approximation of the system’s output. In this talk, I will present variations of the AAA method and how they can be used for model order reduction of nonlinear and parametric systems. In the first part of the talk, I discuss variations of AAA to build rational approximations of a set of functions that use the same support points and weights. This has some advantages in the solution of nonlinear eigenvalue problems and dynamical systems with multiple input and outputs. The discussed methods are Set Valued AAA, Weighted AAA, QR-AAA. In the second part, I discuss the use of these methods for data driven model order reduction. First, I show Time Domain AAA as an alternative to Time Domain Vector Fitting. Second, I show the use of AAA for model order reduction of nonlinear systems with parameters. To this end, a combination of p-AAA and SV-AAA is made to deal with the difference in degree of dependency on parameters.

TBA

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Each participant will choose to join a working group associated with either Theme 1, Theme 2 or Theme 3. The groups will meet and work on open problems.

Kriging is a geostatistical interpolatory approximation method (known in the context of machine learning as Gaussian Process Regression) that in common usage predicts values of a random scalar field varying across a spatial domain by forming a weighted average of samples obtained at observed locations in the spatial domain with weights determined both from the spatial distribution of the observed locations and the (presumed known) spatial correlation structure of the quantity being observed. Standard kriging approximations can be associated with optimal spline approximation within a reproducing kernel Hilbert space determined by the given covariance function. ``Rational kriging" extends traditional kriging approaches through the use of rational basis functions to model nonstationary spatial dependence and drift. I will introduce elements of the basic framework of kriging and a version of rational kriging recently introduced by V. R. Joseph. This extension may be recast into an intrinsic Gaussian process framework and connected to optimal approximation in reproducing kernel Hilbert spaces in a way that parallels traditional kriging approaches, but now with reproducing kernels given as rational functions instead of splines. Some favorable computational consequences of this formulation will be discussed as well.

Network realizations of rational approximations have been known as a key component of electrical filter synthesis since the 1920s. In his seminal 1952 work, Mark Krein considered such networks and their embedding into strings with variable mass distributions. This gave rise to several fruitful applications, starting with finite-difference Gaussian quadratures for efficient PDE discretizations and extending to the solution of inverse scattering problems. In this talk, we present an application of this powerful approach to computing MIMO transfer functions of large dynamical systems via Krylov subspace projection. A block Gauss quadrature computed via (block)-Lanczos recursion yields block Padé approximations, which are numerically efficient for medium-sized problems thanks to their adaptation to the nonuniform spectral distributions of the underlying problems. However, the adaptation can be significantly damped for problems with dense spectra, such as large-scale discretizations of PDEs on unbounded domains. An acceleration for such problems, based on averaging Gauss and Gauss-Radau quadratures, was proposed in Zimmerling, Dr., and Simoncini, J. Sci. Comput., 2025. Here, we further improve convergence by introducing an adaptive block Krein-Nudelman extension of the Lanczos recursion. This approach yields a Hermite-Padé approximation, which is known to be competitive for problems with branch cuts. This low-cost modification outperforms the conventional Lanczos approximation for a variety of large-scale problems.

For the accurate modeling of real-world phenomena, high-dimensional nonlinear dynamical systems are indispensable. Thereby, many physical properties are encoded in the internal differential structure of these systems. Some examples of differential structures are second-order time derivatives in mechanical systems or time-delay terms. When using such models in computational settings, the high-dimensional nature represented by large numbers of differential equations describing the dynamics becomes the main computational bottleneck. A remedy to this problem is model order reduction, which is concerned with the construction of cheap-to-evaluate surrogate models that are described by a significantly smaller number of differential equations while accurately approximating the input-to-output behavior of the original high-dimensional system. It has been shown that many nonlinear phenomena can be equivalently modeled using only bilinear and quadratic terms. Dynamical systems with such terms can be represented in the Laplace domain using multivariate rational functions. In this work, we present a structure-preserving model reduction framework for nonlinear dynamical systems via multivariate rational function interpolation. This new approach allows the simulation-free construction of cheap-to-evaluate surrogate models for nonlinear dynamical systems with internal structures.

This talk discusses analytical estimates for a straightforward greedy construction of rational approximations for the transfer function of stable linear dynamical systems. This analysis, which immediately reveals corresponding algorithms, provides explicit estimates of transfer function rational approximability through sectorial properties of the system matrices. The theoretical constructions correspond to explicit Galerkin projection-based reduced order models of the original system, and hence as a bonus this analysis provides concrete estimates of Hankel singular value decay. The framework we discuss is a paradigm that forges concrete relationships between rational approximation, Kolmogorov n-widths, and Galerkin projection-based model reduction.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The working groups will share the outcomes of their work sessions.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.