Abstract: Joint work with Dr. Petr Tichy.
Reduced order modeling plays a central role in approximation of large-scale dynamical systems, and techniques based on Krylov subspaces are used in that area for decades, see, e.g. [2] and the recent monograph [1]. Krylov subspace methods can not only be viewed as tools for model reduction. Many of them by their nature represent model reductions based on matching moments. In order to explain the deep link between matching moments and Krylov subspace methods, we start with a modiﬁcation of the classical Stieltjes moment problem, and show that its solution is given:
in the language of orthogonal polynomials by the Gauss-Christoﬀel quadrature;
in the matrix form by the conjugate gradient method (see [4]).
In order to allow straightforward generalizations, we use the Vorobyev vector moment problem [3], and present some basic Krylov subspace methods from the moment matching model reduction point of view [5]. The second part of the talk will be devoted to application in numerical modeling of the diﬀraction of light on periodic media [6].
References
[1] A. C. Antoulas, Approximation of large-scale dynamical systems, vol. 6 of Advances in Design and Control, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2005.
[2] K. Gallivan, E. Grimme, and P. Van Dooren, Asymptotic wave-form evaluation via a Lanczos method, Appl. Math. Lett., 7 (1994), pp. 7580.
[3] Y. V. Vorobyev, Methods of moments in applied mathematics, Translated from the Russian by Bernard Seckler, Gordon and Breach Science Publishers, New York, 1965.
[4] Meurant, G. and Strakos, Z., The Lanczos and conjugate gradient algorithms in ﬁnite precision arithmetic, Acta Numerica, 15 (2006), pp. 471542.
[5] Strakos, Z., Model reduction using the Vorobyev moment problem, Numerical Algorithms, 51, (2008), pp. 363376.
[6] Hench, J. and Strakos, Z., The RCWA method - a Case study with open questions and perspectives of algebraic computations, Electronic Transactions on Numerical Analysis, 31 (2009), pp. 331357. |