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Helmholtz equations with variable coefficients are known to be hard to solve both analytically and numerically. In this paper, we introduce a numerical multigrid solver for one-dimensional Helmholtz eigenvalue problems with periodic potentials and solutions. The solvers employ wave–ray methodology suggested by Brandt, Livshits for Helmholtz equations with constant coefficients. The paper concludes with numerical experiments and the discussion of future efforts for solving two-dimensional problems. Copyright © 2004 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY Although there have been significant advances in robust algebraic multigrid (AMG) methods in recent years, numerical studies and emerging hardware architectures continue to favor structured-grid approaches. Specifically, implementations of logically structured robust variational multigrid algorithms, such as the black box multigrid (BoxMG) solver, have been shown to be 10 times faster than AMG for three-dimensional heterogeneous diffusion problems on structured grids. BoxMG offers important features such as operator-induced interpolation for robustness, while taking advantage of direct data access and bounded complexity in the Galerkin coarse-grid operator. Moreover, because BoxMG uses a variational framework, it can be used to explore advances of modern adaptive AMG approaches in a structured setting. In this paper, we show how to extend the adaptive multigrid methodology to the BoxMG setting. This extension not only retains the favorable properties of the adaptive framework but also sheds light on the relationship between BoxMG and AMG. In particular, we show how classical BoxMG can be viewed as a special case of classical AMG and how this viewpoint leads to a richer family of adaptive BoxMG approaches. We present numerical results that explore this family of adaptive methods and compare its robustness and efficiency to the classical BoxMG solver.Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we study the quadratic model updating problems by using symmetric low-rank correcting, which incorporates the measured model data into the analytical quadratic model to produce an adjusted model that matches the experimental model data, and minimizes the distance between the analytical and updated models. We give a necessary and sufficient condition on the existence of solutions to the symmetric low-rank correcting problems under some mild conditions, and propose two algorithms for finding approximate solutions to the corresponding optimization problems. The good performance of the two algorithms is illustrated by numerical examples. Copyright © 2008 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]