308 research outputs found

    Inexact Solves in Interpolatory Model Reduction

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    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 H2{\mathcal H}_2 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 L2\mathcal{L}_2-optimal Reduced-order Modeling

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    We develop a unifying framework for interpolatory L2\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 H2\mathcal{H}_2-optimal model order reduction and leads to the interpolatory conditions for H2⊗L2\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 L2\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

    Reduced-Order Modelling of Parametric Systems via Interpolation of Heterogeneous Surrogates

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    Stochastic collocation on unstructured multivariate meshes

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    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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