12 research outputs found
Interpolatory methods for model reduction of multi-input/multi-output systems
We develop here a computationally effective approach for producing
high-quality -approximations to large scale linear
dynamical systems having multiple inputs and multiple outputs (MIMO). We extend
an approach for model reduction introduced by Flagg,
Beattie, and Gugercin for the single-input/single-output (SISO) setting, which
combined ideas originating in interpolatory -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
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
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 ..