Surrogate and reduced-order modeling: a comparison of approaches for large-scale statistical inverse problems [Chapter 7]The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters.
CitationFrangos, M., Y. Marzouk, K. Willcox, and B. van Bloemen Waanders (2010). Surrogate and reduced-order modeling: a comparison of approaches for large-scale statistical inverse problems.
The Arnoldi and Lanczos algorithms, which belong to the class of Krylov subspace methods, are increasingly used for model reduction of large scale systems.The standard versions of the algorithms tend to create reduced order models that poorly approximate low frequency dynamics. Rational Arnoldi and Lanczos algorithms produce reduced models that approximate dynamics at various frequencies. This paper tackles the issue of developing simple Arnoldi and Lanczos equations for the rational case. This allows a simple error analysis to be carried out for both algorithms and permits the development of computationally efficient model reduction algorithms, where the frequencies at which the dynamics are to be matched can be updated adaptively.
The development of efficient interior point methods has greatly enlarged the range of control problems with feasible numerical solution. These methods are nevertheless difficult to solve for large-scale problems. In this paper we suggest the use of a Krylov subspace technique for the efficient lowrank approximate solution to large-scale Sylvester equations. The suggested method is a novel restart scheme which improves further the computation efficiency and storage requirements of the standard Krylov subspace methods for the solution of largescale Sylvester equations.
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