Null space correction and adaptive model order reduction in multi-frequency Maxwell’s problem
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- 作者：Michal Kordy ; Elena Cherkaev ; Philip Wannamaker
- 关键词：Rational Krylov subspace ; Model order reduction ; Helmholtz decomposition ; Frequency ; domain Maxwell’s system
- 刊名：Advances in Computational Mathematics
- 出版年：2017
- 出版时间：February 2017
- 年：2017
- 卷：43
- 期：1
- 页码：171-193
- 全文大小：
- 刊物类别：Computer Science
- 刊物主题：Computational Mathematics and Numerical Analysis; Mathematical Modeling and Industrial Mathematics; Mathematical and Computational Biology; Computational Science and Engineering; Visualization;
- 出版者：Springer US
- ISSN：1572-9044
- 卷排序：43
文摘
A model order reduction method is developed for an operator with a non-empty null-space and applied to numerical solution of a forward multi-frequency eddy current problem using a rational interpolation of the transfer function in the complex plane. The equation is decomposed into the part in the null space of the operator, calculated exactly, and the part orthogonal to it which is approximated on a low-dimensional rational Krylov subspace. For the Maxwell’s equations the null space is related to the null space of the curl. The proposed null space correction is related to divergence correction and uses the Helmholtz decomposition. In the case of the finite element discretization with the edge elements, it is accomplished by solving the Poisson equation on the nodal elements of the same grid. To construct the low-dimensional approximation we adaptively choose the interpolating frequencies, defining the rational Krylov subspace, to reduce the maximal approximation error. We prove that in the case of an adaptive choice of shifts, the matrix spanning the approximation subspace can never become rank deficient. The efficiency of the developed approach is demonstrated by applying it to the magnetotelluric problem, which is a geophysical electromagnetic remote sensing method used in mineral, geothermal, and groundwater exploration. Numerical tests show an excellent performance of the proposed methods characterized by a significant reduction of the computational time without a loss of accuracy. The null space correction regularizes the otherwise ill-posed interpolation problem.