Null space correction and adaptive model order reduction in multi-frequency Maxwell's problem

نویسندگان

  • Michal Kordy
  • Elena Cherkaev
  • Philip E. Wannamaker
چکیده

A model order reduction method is developed for an operator with a non7 empty null-space and applied to numerical solution of a forward multi-frequency 8 eddy current problem using a rational interpolation of the transfer function in the 9 complex plane. The equation is decomposed into the part in the null space of the 10 operator, calculated exactly, and the part orthogonal to it which is approximated on 11 a low-dimensional rational Krylov subspace. For the Maxwell’s equations the null 12 space is related to the null space of the curl. The proposed null space correction is 13 related to divergence correction and uses the Helmholtz decomposition. In the case of 14 the finite element discretization with the edge elements, it is accomplished by solving 15 the Poisson equation on the nodal elements of the same grid. To construct the low16 dimenensional approximation we adaptively choose the interpolating frequencies, 17 defining the rational Krylov subspace, to reduce the maximal approximation error. 18 Communicated by: Carlos Garcia-Cervera We acknowledge the support of this work from the U.S. Dept. of Energy under contract DE-EE0002750 to PW. EC acknowledges the partial support of the U.S. National Science Foundation through grant DMS-1413454. * Michal Kordy [email protected] Elena Cherkaev [email protected] Philip Wannamaker [email protected] 1 Department of Mathematics, University of Utah, 155 S 1400 E JWB 233, Salt Lake City, UT 84112-0090, USA 2 Energy, Geoscience Institute, University of Utah, 423 Wakara Way, Suite 300, Salt Lake City, UT 84108, USA AUTHOR'S PROOF JrnlID 10444 ArtID 9482 Proof#1 07/09/2016 U N C O R R E C TE D P R O O F M. Kordy et al. We prove that in the case of an adaptive choice of shifts, the matrix spanning the 19 approximation subspace can never become rank deficient. The efficiency of the 20 developed approach is demonstrated by applying it to the magnetotelluric problem, 21 which is a geophysical electromagnetic remote sensing method used in mineral, 22 geothermal, and groundwater exploration. Numerical tests show an excellent per23 formance of the proposed methods characterized by a significant reduction of the 24 computational time without a loss of accuracy. The null space correction regularizes 25 the otherwise ill-posed interpolation problem. 26

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عنوان ژورنال:
  • Adv. Comput. Math.

دوره 43  شماره 

صفحات  -

تاریخ انتشار 2017