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This paper deals with adaptively preconditioned iterative methods for solving large and sparse systems of linear equations. In particular, the paper discusses preconditioning where adaptive dropping reflects the quality of preserving the relation U Z = I between the direct factor U and the inverse factor Z that satisfy A = U T U and A − 1 = Z Z T . The proposed strategy significantly extends and refines the approach from [1], see also [2], by using a specific multilevel framework. Numerical experiments with two levels demonstrate that the new preconditioning strategy is very promising. Namely, we show a surprising fact that in our approach the Schur complement is better to form in a more sophisticated way than by a standard sparse matrix-matrix multiplication. [ABSTRACT FROM AUTHOR]

It is well-known that analysis of incomplete Cholesky and LU decompositions with a general dropping is very difficult and of limited applicability, see, for example, the results on modified decompositions (Dupont et al., 1968; Gustafsson, 1978; Bern et al., 2006) and later results based on similar concepts. This is true not only for the dropping based on magnitude of entries but it also applies to algorithms that use a prescribed sparsity pattern. This paper deals with dropping strategies for a class of AINV-type incomplete decompositions (Benzi et al., 1996) that are based on the generalized Gram–Schmidt process. Its behavior in finite precision arithmetic has been discussed in Rozložník et al. (2012). This analysis enables better understanding of the incomplete process, and the main goal of the paper is to propose a new adaptive dropping strategy and to illustrate its efficiency for problems in structural mechanics. In addition, we add a brief comparison with another approximate inverse preconditioning strategy that is based on different principles and used in engineering applications. [ABSTRACT FROM AUTHOR]

Abstract: We present a roundoff error analysis of the method for solving the linear least squares problem with full column rank matrix A, using only factors Σ and V from the SVD decomposition of . This method (called here) is an analogue of the method of seminormal equations ( ), where the solution is computed from using only the factor R from the QR factorization of A. Such methods have practical applications when A is large and sparse and if one needs to solve least squares problems with the same matrix A and multiple right-hand sides. However, in general both and are not forward stable. We analyze one step of fixed precision iterative refinement to improve the accuracy of the method. We show that, under the condition , this method (called ) produces a forward stable solution, where denotes the condition number of the matrix A and u is the unit roundoff. However, for problems with only it is generally not forward stable, and has similar numerical properties to the corresponding method. Our forward error bounds for the are slightly better than for the since the terms are not present. We illustrate our analysis by numerical experiments. [Copyright &y& Elsevier]

• Novel relation between the modal and the static stiffness of a structure. • Identifiability of smallest structural stiffness based on modal properties. • Alternative method for localising the smallest structural stiffness based on structural compliance maximisation. Structural stiffness plays an important role in engineering design. The analysis of stiffness requires precise experiments and computational models that can be difficult or time–consuming to procure. A novel relation between modal and static stiffness based on modal decomposition is introduced in this study. This relation allows analysing the smallest structural stiffness and its direction. Further, it is shown that the smallest stiffness can be found using an optimisation algorithm that is based on the maximisation of structural compliance. Both approaches are compared on several computational examples leading to similar results in terms of smallest stiffness and its direction. The proposed approaches serve as quantitative/qualitative tools for the analyses of structural stiffness, particularly in structural health monitoring. [ABSTRACT FROM AUTHOR]