A Unifying Framework for Interpolatory L2\mathcal{L}_2-optimal Reduced-order Modeling

We develop a unifying framework for interpolatory $\mathcal{L}_2$-optimal reduced-order modeling for a wide classes of problems ranging from stationary models to parametric dynamical systems. We first show that the framework naturally covers the well-known interpolatory necessary conditions for $\mathcal{H}_2$-optimal model order reduction and leads to the interpolatory conditions for $\mathcal{H}_2 \otimes \mathcal{L}_2$-optimal model order reduction of multi-input/multi-output parametric dynamical systems. Moreover, we derive novel interpolatory optimality conditions for rational discrete least-squares minimization and for $\mathcal{L}_2$-optimal model order reduction of a class of parametric stationary models. We show that bitangential Hermite interpolation appears as the main tool for optimality across different domains. The theoretical results are illustrated on two numerical examples.Comment: 20 pages, 2 figure

Aproksimacija linearnih dinamičkih sistema u prostoru H2p×m(C+)\mathcal{H}_2^{p \times m}(\mathbf{C}_+)

Dinamički sistemi su svuda oko nas, od raznih prirodnih do industrijskih procesa. Za te procese se razvijaju modeli, koji se onda koriste za simuliranje ponašanja procesa i/ili za upravljanje nad njima. U svrhu povećanja preciznosti, javljaju se sve kompliciraniji modeli s kojima nije moguće efektivno računati. Stoga je potrebno tražiti reducirane modele, modele koji su jednostavniji, a dovoljno bliski kompliciranim modelima. U radu se promatraju linearni, vremensko-invarijantni, neprekidno-vremenski, konačnodimenzionalni, realni, stabilni sistemi oblika $Ex'(t) = Ax(t) + Bu(t), y(t) = Cx(t)$, gdje su $E, A \in \mathbb{R}^{n \times n}, B \in \mathbb{R}^{n \times m}$ i $C \in \mathbb{R}^{p \times n}$ konstantne matrice, a $x(t) \in \mathbb{R}^n, u(t) \in \mathbb{R}^m i y(t) \in \mathbb{R}^p$ su stanje, ulaz i izlaz sistema. Za ovaj sistem se kaže da je $n$-dimenzionalan. Cilj redukcije modela je, za zadani $n$-dimenzionalni sistem, pronaći $r$-dimenzionalni sistem (gdje je $r \ll n$ takoder zadan), a da izlaz reduciranog modela bude što bliži izlazu punog modela. Smisao bliskosti modela koji se promatraju u radu je onaj dan normom Hardyjevog prostora $\mathcal{H}_2^{p \times m}(\mathbb{C}_+)$. U radu je obrađena metoda koja traži reducirani model koji zadovoljava neke nužne uvjete optimalnosti u $\mathcal{H}_2$ normi, dane u obliku tangencijalne Hermiteove interpolacije. Također je obrađena i metoda bazirana na Loewnerovim matricama koja ne ovisi realizaciji ($E, A, B, C$) i primjenjiva je za beskonačnodimenzionalne sisteme. Sve metode su testirane na primjerima iz Oberwolfach i NICONET baza te su rezultati uspoređeni s rezultatima dobivenim metodom balansiranog rezanja.Dynamical systems are all around us, from various natural to industrial processes. Models are developed for these processes, which are then used to simulate the behavior of processes and/or to control over them. In order to increase the precision, there are more complicated models which can’t be effectively used. It is therefore necessary to look for reduced models, models that are simpler, but close enough to complicated models. Linear, time-invariant, continuous-time, finite dimensional, real, stable systems of the form $Ex'(t) = Ax(t) + Bu(t), y(t) = Cx(t)$, are observed, where $E, A \in \mathbb{R}^{n \times n}, B \in \mathbb{R}^{n \times m}$ and $C \in \mathbb{R}^{p \times n}$ are constant matrices, and $x(t) \in \mathbb{R}^n, u(t) \in \mathbb{R}^m$ and $y(t) \in \mathbb{R}^p$ are the state, input and output of the system. This system is said to be $n$-dimensional. The goal of model reduction is, for given $n$-dimensional system, to find an $r$-dimensional system (where $r \ll n$ is also given), such that the output of the reduced model is as close as possible to the output of the full model. The meaning of closeness which is used in this work is that which is given by the norm in the Hardy’s space $\mathcal{H}_2^{p \times m}(\mathbb{C}_+)$. This work presents a method that finds a reduced model satisfying some necessary conditions of optimality in $\mathcal{H}_2$ norm, given in the form of tangential Hermite interpolation. It also presents a method based on Loewner’s matrices which does not depend on any realization ($E, A, B, C$) and is applicable to infinite dimensional systems. All methods have been tested on examples from Oberwolfach and NICONET databases and the results are compared with the results obtained using balanced truncation

Interpolatory H2\mathcal{H}_2-optimality Conditions for Structured Linear Time-invariant Systems

Interpolatory necessary optimality conditions for $\mathcal{H}_2$-optimal reduced-order modeling of unstructured linear time-invariant (LTI) systems are well-known. Based on previous work on $\mathcal{L}_2$-optimal reduced-order modeling of stationary parametric problems, in this paper we develop and investigate optimality conditions for $\mathcal{H}_2$-optimal reduced-order modeling of structured LTI systems, in particular, for second-order, port-Hamiltonian, and time-delay systems. We show that across all these different structured settings, bitangential Hermite interpolation is the common form for optimality, thus proving a unifying optimality framework for structured reduced-order modeling.Comment: 20 page

L2\mathcal{L}_2-optimal Reduced-order Modeling Using Parameter-separable Forms

We provide a unifying framework for $\mathcal{L}_2$-optimal reduced-order modeling for linear time-invariant dynamical systems and stationary parametric problems. Using parameter-separable forms of the reduced-model quantities, we derive the gradients of the $\mathcal{L}_2$ cost function with respect to the reduced matrices, which then allows a non-intrusive, data-driven, gradient-based descent algorithm to construct the optimal approximant using only output samples. By choosing an appropriate measure, the framework covers both continuous (Lebesgue) and discrete cost functions. We show the efficacy of the proposed algorithm via various numerical examples. Furthermore, we analyze under what conditions the data-driven approximant can be obtained via projection.Comment: 21 pages, 10 figure