Computational Mathematics and Applications Seminar

Data-driven reduced-order modeling aims at constructing models describing the underlying dynamics of unknown systems from measurements. This has become an increasingly preeminent discipline in the last few years. It is an essential tool in situations when explicit models in the form of state space formulations are not available, yet abundant input/output data are, motivating the need for data-driven modeling. Depending on the underlying physics, dynamical systems can inherit differential structures leading to specific physical interpretations. In this work, we concentrate on systems that are described by differential equations and possess linear dynamics. Extensions to more complicated, nonlinear dynamics are also possible and will be briefly covered here if time permits.

The methods developed in our study use rational approximation based on Loewner matrices. Starting with the approach by Antoulas and Anderson in '86, and moving forward to the one by Mayo and Antoulas in '07, the Loewner framework (LF) has become an established methodology in the model reduction and reduced-order modeling community. It is a data-driven approach in the sense that what is needed to compute the reduced models is solely data, i.e., samples of the system's transfer function. As opposed to conventional intrusive methods that require an actual large-scale model to reduce (described by many differential equations), the LF only needs measurements in compressed format. In the former category of approaches, we mention balanced truncation (BT), arguably one of the most prevalent model reduction methods. Introduced in the early 80s, this method constructs reduced-order models (ROMs) by using balancing and truncating steps (with respect to classical system theory concepts such as controllability and observability). We show that BT can be reinterpreted as a data-driven approach, by using again the Loewner matrix as a central ingredient. By making use of quadrature approximations of certain system theoretical quantities (infinite Gramian matrices), a novel method called QuadBT (quadrature-based BT) is introduced by G., Gugercin, and Beattie in '22. We show parallels with the LF and, if time permits, certain recent extensions of QuadBT. Finally, all theoretical considerations are validated on various numerical test cases.

Tensor decomposition express a tensor as a linear combination of elementary tensors. They have applications in chemometrics, computer science, machine learning, psychometrics, and signal processing. Their uniqueness properties render them suitable for data analysis tasks in which the elementary tensors are the quantities of interest. However, in applications, the idealized mathematical model is corrupted by measurement errors. For a robust interpretation of the data, it is therefore imperative to quantify how sensitive these elementary tensors are to perturbations of the whole tensor. I will give an overview of recent results on the condition number of tensor decompositions, established with my collaborators C. Beltran, P. Breiding, and N. Dewaele.

I will reflect on my time as Professor of Numerical Analysis.

Geometric evolutions can arise as simple models or fundamental building blocks in various applications with moving boundaries and time-dependent domains, such as grain boundaries in materials or deforming cell boundaries. Mesh-based methods require adaptation and smoothing, particularly in the case of strong deformations. We consider finite element schemes based on classical approaches for geometric evolution equations but augmented with the gradient of the Dirichlet energy or a variant of it, which is known to produce a tangential mesh movement beneficial for the mesh quality. We focus on the one-dimensional case, where convergence of semi-discrete schemes can be proved, and discuss two cases. For networks forming triple junctions, it is desirable to keep the impact of any additional, mesh smoothing terms on the geometric evolution as small as possible, which can be achieved with a perturbation approach. Regarding the elastic flow of curves, the Dirichlet energy can serve as a replacement of the usual penalty in terms of the length functional in that, modulo rescaling, it yields the same minimisers in the long run.

Placeholder entry, date subject to change. Preferred dates: 2, 9 or 16 November