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This paper extends the weighted low rank approximation (WLRA) approach to linearly structured matrices. In the case of Hankel matrices with a special block structure, an equivalent unconstrained optimization problem is derived and an algorithm for solving it is proposed. Copyright © 2005 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY Saddle point systems arise widely in optimization problems with constraints. The utility of Schur complement approximation is now broadly appreciated in the context of solving such saddle point systems by iteration. In this short manuscript, we present a new Schur complement approximation for PDE-constrained optimization, an important class of these problems. Block diagonal and block triangular preconditioners have previously been designed to be used to solve such problems along with MINRES and non-standard Conjugate Gradients, respectively; with appropriate approximation blocks, these can be optimal in the sense that the time required for solution scales linearly with the problem size, however small the mesh size we use. In this paper, we extend this work to designing such preconditioners for which this optimality property holds independently of both the mesh size and the Tikhonov regularization parameter β that is used. This also leads to an effective symmetric indefinite preconditioner that exhibits mesh and β independence. We motivate the choice of these preconditioners based on observations about approximating the Schur complement obtained from the matrix system, derive eigenvalue bounds that verify the effectiveness of the approximation and present numerical results that show that these new preconditioners work well in practice. Copyright © 2011 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY In this paper, we consider the NP-hard problem of finding global minimum of quadratically constrained multivariate bi-quadratic optimization. We present some bounds of the considered problem via approximately solving the related bi-linear semidefinite programming (SDP) relaxation. Based on the bi-linear SDP relaxation, we also establish some approximation solution methods, which generalize the methods for the quadratic polynomial optimization in ( SIAM J. Optim. 2003; 14:268-283). Finally, we present a special form, whose bi-linear SDP relaxation can be approximately solved in polynomial time. Copyright © 2011 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]

This paper extends the weighted low rank approximation (WLRA) approach towards linearly structured matrices. In the case of Hankel matrices an equivalent unconstrained optimization problem is derived and an algorithm for solving it is proposed. The correctness of the latter algorithm is verified on a benchmark problem. Finally the statistical accuracy and numerical efficiency of the proposed algorithm is compared with that of STLNB, a previously proposed algorithm for solving Hankel WLRA problems. Copyright © 2004 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

SUMMARY The problem of symmetric rank-one approximation of symmetric tensors is important in independent components analysis, also known as blind source separation, as well as polynomial optimization. We derive several perturbative results that are relevant to the well-posedness of recovering rank-one structure from approximately-rank-one symmetric tensors. We also specialize the analysis of the shifted symmetric higher-order power method, an algorithm for computing symmetric tensor eigenvectors, to approximately-rank-one symmetric tensors. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We are interested in the numerical solution of large structured indefinite symmetric linear systems arising in mixed finite element approximations of the magnetostatic problem; in particular, we analyse definite block-diagonal and indefinite symmetric preconditioners. Relating the algebraic characteristics of the resulting preconditioned matrix to the properties of the continuous problem and of its finite element discretization, we show that the preconditioning strategies considered make the Krylov subspace solver used insensitive to the mesh refinement parameter, in terms of the number of iterations. In order to achieve computational efficiency, we also analyse algebraic approximations to the optimal preconditioners, and discuss their performance on real two- and three-dimensional application problems. Copyright © 2000 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]