Interpolatory methods for H∞\mathcal{H}_\infty model reduction of multi-input/multi-output systems

We develop here a computationally effective approach for producing high-quality $\mathcal{H}_\infty$-approximations to large scale linear dynamical systems having multiple inputs and multiple outputs (MIMO). We extend an approach for $\mathcal{H}_\infty$ model reduction introduced by Flagg, Beattie, and Gugercin for the single-input/single-output (SISO) setting, which combined ideas originating in interpolatory $\mathcal{H}_2$-optimal model reduction with complex Chebyshev approximation. Retaining this framework, our approach to the MIMO problem has its principal computational cost dominated by (sparse) linear solves, and so it can remain an effective strategy in many large-scale settings. We are able to avoid computationally demanding $\mathcal{H}_\infty$ norm calculations that are normally required to monitor progress within each optimization cycle through the use of "data-driven" rational approximations that are built upon previously computed function samples. Numerical examples are included that illustrate our approach. We produce high fidelity reduced models having consistently better $\mathcal{H}_\infty$ performance than models produced via balanced truncation; these models often are as good as (and occasionally better than) models produced using optimal Hankel norm approximation as well. In all cases considered, the method described here produces reduced models at far lower cost than is possible with either balanced truncation or optimal Hankel norm approximation

Data-adaptive harmonic spectra and multilayer Stuart-Landau models

Harmonic decompositions of multivariate time series are considered for which we adopt an integral operator approach with periodic semigroup kernels. Spectral decomposition theorems are derived that cover the important cases of two-time statistics drawn from a mixing invariant measure. The corresponding eigenvalues can be grouped per Fourier frequency, and are actually given, at each frequency, as the singular values of a cross-spectral matrix depending on the data. These eigenvalues obey furthermore a variational principle that allows us to define naturally a multidimensional power spectrum. The eigenmodes, as far as they are concerned, exhibit a data-adaptive character manifested in their phase which allows us in turn to define a multidimensional phase spectrum. The resulting data-adaptive harmonic (DAH) modes allow for reducing the data-driven modeling effort to elemental models stacked per frequency, only coupled at different frequencies by the same noise realization. In particular, the DAH decomposition extracts time-dependent coefficients stacked by Fourier frequency which can be efficiently modeled---provided the decay of temporal correlations is sufficiently well-resolved---within a class of multilayer stochastic models (MSMs) tailored here on stochastic Stuart-Landau oscillators. Applications to the Lorenz 96 model and to a stochastic heat equation driven by a space-time white noise, are considered. In both cases, the DAH decomposition allows for an extraction of spatio-temporal modes revealing key features of the dynamics in the embedded phase space. The multilayer Stuart-Landau models (MSLMs) are shown to successfully model the typical patterns of the corresponding time-evolving fields, as well as their statistics of occurrence.Comment: 26 pages, double columns; 15 figure

Applied Mathematics And Computation

. The work is giving estimations of the discrepancy between solutions of the initial and the homogenized problems for a one--dimensional second order elliptic operators with random coefficients satisfying strong or uniform mixing conditions. We obtain several sharp estimates in terms of the corresponding mixing coefficient. CONTRIBUTED TALKS TUESDAY, 10:45--11:10, SECTION B Computation of Power--Series Expansions in Homogenisation of Nonlinear Equations Nenad Antoni'c and Martin Lazar Abstract. In the theory of homogenisation it is of particular interest to determine the classes of problems which are stable on taking the homogenisation limits. A notable situation where the limit enlarges the class of original problems is known as memory (nonlocal) effects. A number of results in that direction has been obtained for linear problems. Tartar (1990) innitiated the study of the effective equation corresponding to nonlinear equation: @ t u n + a n u 2 n = f: Significant progress has ..