308 research outputs found
Inexact Solves in Interpolatory Model Reduction
We investigate the use of inexact solves for interpolatory model reduction
and consider associated perturbation effects on the underlying model reduction
problem. We give bounds on system perturbations induced by inexact solves and
relate this to termination criteria for iterative solution methods. We show
that when a Petrov-Galerkin framework is employed for the inexact solves, the
associated reduced order model is an exact interpolatory model for a nearby
full-order system; thus demonstrating backward stability. We also give evidence
that for \h2-optimal interpolation points, interpolatory model reduction is
robust with respect to perturbations due to inexact solves. Finally, we
demonstrate the effecitveness of direct use of inexact solves in optimal
approximation. The result is an effective model reduction
strategy that is applicable in realistically large-scale settings.Comment: 42 pages, 5 figure
A Unifying Framework for Interpolatory -optimal Reduced-order Modeling
We develop a unifying framework for interpolatory -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
-optimal model order reduction and leads to the interpolatory
conditions for -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 -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
Stochastic collocation on unstructured multivariate meshes
Collocation has become a standard tool for approximation of parameterized
systems in the uncertainty quantification (UQ) community. Techniques for
least-squares regularization, compressive sampling recovery, and interpolatory
reconstruction are becoming standard tools used in a variety of applications.
Selection of a collocation mesh is frequently a challenge, but methods that
construct geometrically "unstructured" collocation meshes have shown great
potential due to attractive theoretical properties and direct, simple
generation and implementation. We investigate properties of these meshes,
presenting stability and accuracy results that can be used as guides for
generating stochastic collocation grids in multiple dimensions.Comment: 29 pages, 6 figure
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