623 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
Consistent Dynamic Mode Decomposition
We propose a new method for computing Dynamic Mode Decomposition (DMD)
evolution matrices, which we use to analyze dynamical systems. Unlike the
majority of existing methods, our approach is based on a variational
formulation consisting of data alignment penalty terms and constitutive
orthogonality constraints. Our method does not make any assumptions on the
structure of the data or their size, and thus it is applicable to a wide range
of problems including non-linear scenarios or extremely small observation sets.
In addition, our technique is robust to noise that is independent of the
dynamics and it does not require input data to be sequential. Our key idea is
to introduce a regularization term for the forward and backward dynamics. The
obtained minimization problem is solved efficiently using the Alternating
Method of Multipliers (ADMM) which requires two Sylvester equation solves per
iteration. Our numerical scheme converges empirically and is similar to a
provably convergent ADMM scheme. We compare our approach to various
state-of-the-art methods on several benchmark dynamical systems
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