Cankat Tilki

Cankat Tilki, Tobias Breiten, Serkan Gugercin

We develop the interpolatory $\mathcal{H}_2$ optimal model reduction framework for linear control systems posed on infinite dimensional state, input and output spaces. Specifically, we consider linear systems formulated as controlled abstract Cauchy problems on a Banach space and approximate them via Petrov-Galerkin projection onto finite dimensional trial and test subspaces. We show that the resulting reduced order transfer function interpolates the original at prescribed points, and we characterize precisely how the projection subspaces must be constructed to enforce this interpolation. Building on this, we develop a data-driven realization framework -- an infinite dimensional analogue of the Loewner approach -- that recovers the system behavior directly from input-output data without requiring access to the underlying operators. Finally, we derive $\mathcal{H}_2$ optimality conditions for the reduced model and show that the classical interpolatory characterization persists in this infinite dimensional setting: first-order optimality requires Hermite interpolation of the transfer function at the mirror images of the reduced model's poles. Taken together, these results establish that the interpolatory $\mathcal{H}_2$ optimal model reduction theory extends naturally and completely to infinite dimensional linear control systems with infinite dimensional input and output spaces.

Aurelian Gheondea, Cankat Tilki

We prove a few representer theorems for a localised version of the regularised and multiview support vector machine learning problem introduced by H.Q. Minh, L. Bazzani, and V. Murino, Journal of Machine Learning Research, 17(2016) 1-72, that involves operator valued positive semidefinite kernels and their reproducing kernel Hilbert spaces. The results concern general cases when convex or nonconvex loss functions and finite or infinite dimensional input spaces are considered. We show that the general framework allows infinite dimensional input spaces and nonconvex loss functions for some special cases, in particular in case the loss functions are Gateaux differentiable. Detailed calculations are provided for the exponential least squares loss functions that lead to systems of partially nonlinear equations for which a particular different types of Newton's approximation methods based on the interior point method can be used. Some numerical experiments are performed on a toy model that illustrate the tractability of the methods that we propose.

Cankat Tilki, Serkan Gugercin

We present an in-depth analysis of the Koopman semigroup via wavelet transform. Towards this goal, we start by introducing the wavelet-based observables and show that they are eigenfunctions of the Koopman semigroup when this semigroup is considered over the Banach space of continuous functions on a compact forward-invariant set endowed with the supremum norm. We then construct closed-form expressions of the action of the Koopman semigroup and its resolvent in terms of these observables. To approximate the action of Koopman semigroup numerically, we combine Extended Dynamic Mode Decomposition (EDMD) with the proposed wavelet-based observables leading to the Wavelet Dynamic Mode Decomposition via Continuous Wavelet Transform (cWDMD) algorithm. We validate our theoretical results on two numerical examples.