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Item type:Publication, Access status: with files , A super-localized generalized finite element methodThis paper presents a novel multi-scale method for elliptic partial differential equations with arbitrarily rough coefficients. In the spirit of numerical homogenization, the method constructs problem-adapted ansatz spaces with uniform algebraic approximation rates. Localized basis functions with the same super-exponential localization properties as the recently proposed Super-Localized Orthogonal Decomposition enable an efficient implementation. The method’s basis stability is enforced using a partition of unity approach. A natural extension to higher order is presented, resulting in higher approximation rates and enhanced localization properties. We perform a rigorous a priori and a posteriori error analysis and confirm our theoretical findings in a series of numerical experiments. In particular, we demonstrate the method’s applicability for challenging high-contrast channeled coefficients.Publicationtype: Journal ArticleScopus© Citations 2 14 27 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , A note on Dekker’s FastTwoSum algorithmMore than 45 years ago, Dekker proved that it is possible to evaluate the exact error of a floating-point sum with only two additional floating-point operations, provided certain conditions are met. Today the respective algorithm for transforming a sum into its floating-point approximation and the corresponding error is widely referred to as FastTwoSum. Besides some assumptions on the floating-point system itself—all of which are satisfied by any binary IEEE 754 standard conform arithmetic, the main practical limitation of FastTwoSum is that the summands have to be ordered according to their exponents. In most preceding applications of FastTwoSum, however, a more stringent condition is used, namely that the summands have to be sorted according to their absolute value. In remembrance of Dekker’s work, this note reminds the original assumptions for an error-free transformation viaFastTwoSum. Moreover, we generalize the conditions for arbitrary bases and discuss a possible modification of the FastTwoSum algorithm to extend its applicability even further. Subsequently, a range of programs exploiting the wider applicability is presented. This comprises the OnlineExactSum algorithm by Zhu and Hayes, an error-free transformation from a product of three floating-point numbers to a sum of the same number of addends, and an algorithm for accurate summation proposed by Demmel and Hida.Publicationtype: Journal ArticleScopus© Citations 3 139 183 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , Restarting iterative projection methods for Hermitian nonlinear eigenvalue problems with minmax propertyIn this work we present a new restart technique for iterative projection methods for nonlinear eigenvalue problems admitting minmax characterization of their eigenvalues. Our technique makes use of the minmax induced local enumeration of the eigenvalues in the inner iteration. In contrast to global numbering which requires including all the previously computed eigenvectors in the search subspace, the proposed local numbering only requires a presence of one eigenvector in the search subspace. This effectively eliminates the search subspace growth and therewith the super-linear increase of the computational costs if a large number of eigenvalues or eigenvalues in the interior of the spectrum are to be computed. The new restart technique is integrated into nonlinear iterative projection methods like the Nonlinear Arnoldi and Jacobi-Davidson methods. The efficiency of our new restart framework is demonstrated on a range of nonlinear eigenvalue problems: quadratic, rational and exponential including an industrial real-life conservative gyroscopic eigenvalue problem modeling free vibrations of a rolling tire. We also present an extension of the method to problems without minmax property but with eigenvalues which have a dominant either real or imaginary part and test it on two quadratic eigenvalue problems.Publicationtype: Journal ArticleScopus© Citations 5 348 354 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Domain decomposition based H-LU preconditioning(2009-03-03); ; Hierarchical matrices provide a data-sparse way to approximate fully populated matrices. The two basic steps in the construction of an H-matrix are (a) the hierarchical construction of a matrix block partition, and (b) the blockwise approximation of matrix data by low rank matrices. In this paper, we develop a new approach to construct the necessary partition based on domain decomposition. Compared to standard geometric bisection based H-matrices, this new approach yields H-LU factorizations of finite element stiffness matrices with significantly improved storage and computational complexity requirements. These rigorously proven and numerically verified improvements result from an H-matrix block structure which is naturally suited for parallelization and in which large subblocks of the stiffness matrix remain zero in an LU factorization. We provide numerical results in which a domain decomposition based H-LU factorization is used as a preconditioner in the iterative solution of the discrete (three-dimensional) convection-diffusion equation.Publicationtype: Journal Article40 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Fast verification of solutions of matrix equationsIn this paper, we are concerned with a matrix equation Ax = b where A is an n × n real matrix and x and b are n-vectors. Assume that an approximate solution x is given together with an approximate LU decomposition. We will present fast algorithms for proving nonsingularity of A and for calculating rigorous error bounds for ∥A b - x̃∥ . The emphasis is on rigour of the bounds. The purpose of this paper is to propose different algorithms, the fastest with 2/3n flops computational cost for the verification step, the same as for the LU decomposition. The presented algorithms exclusively use library routines for LU decomposition and for all other matrix and vector operations. -1 3 ∞Publicationtype: Journal Article43