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Hessian‐based model reduction for large‐scale systems with initial‐condition inputs
Author(s) -
Bashir O.,
Willcox K.,
Ghattas O.,
van Bloemen Waanders B.,
Hill J.
Publication year - 2007
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.2100
Subject(s) - hessian matrix , reduction (mathematics) , mathematical optimization , dimension (graph theory) , range (aeronautics) , scale (ratio) , optimization problem , sequence (biology) , eigenvalues and eigenvectors , inverse problem , fidelity , computer science , mathematics , engineering , mathematical analysis , telecommunications , physics , geometry , quantum mechanics , biology , pure mathematics , genetics , aerospace engineering
Reduced‐order models that are able to approximate output quantities of interest of high‐fidelity computational models over a wide range of input parameters play an important role in making tractable large‐scale optimal design, optimal control, and inverse problem applications. We consider the problem of determining a reduced model of an initial value problem that spans all important initial conditions, and pose the task of determining appropriate training sets for reduced‐basis construction as a sequence of optimization problems. We show that, under certain assumptions, these optimization problems have an explicit solution in the form of an eigenvalue problem, yielding an efficient model reduction algorithm that scales well to systems with states of high dimension. Furthermore, tight upper bounds are given for the error in the outputs of the reduced models. The reduction methodology is demonstrated for a large‐scale contaminant transport problem. Copyright © 2007 John Wiley & Sons, Ltd.