We propose a block Krylov subspace version of the GCRO-DR method proposed in [Parks et al. SISC 2005], which is an iterative method allowing for the efficient minimization of the the residual over an augmented block Krylov subspace. We offer a clean derivation of the method and discuss methods of selecting recycling subspaces at restart as well as implementation decisions in the context of high...

Preconditioning  Krylov Subspace Methods are commonly used for solving linear system  Standard implementations are communication-bound due to required SpMV and orthogonalization in every iteration  Solution: rearrange algorithms to perform s iterations at a time without communicating (s-step methods)  SpMV in each iteration is replaced with a call to the Matrix Powers Kernel, which performs...

By introducing the second order Krylov subspace, a method for the reduction of second order systems is proposed leading to a reduced system of the same structure. This generalization of Krylov subspace involves two matrices and some starting vectors and the reduced order model is found by applying a projection directly to the second order model without any conversion to state space. A numerical...

Consider a system of linear algebraic equations Ax = b where A is an n by n real matrix and b a real vector of length n. Unlike in the linear iterative methods based on the idea of splitting of A, the Krylov subspace methods, which are used in computational kernels of various optimization techniques, look for some optimal approximate solution xn in the subspaces Kn(A, b) = span{b, Ab, . . . , A...

It has been shown that approximate extended Krylov subspaces can be computed –under certain assumptions– without any explicit inversion or system solves. Instead the necessary products A−1v are obtained in an implicit way retrieved from an enlarged Krylov subspace. In this paper this approach is generalized to rational Krylov subspaces, which contain besides poles at infinite and zero also fini...

Numerical simulations of the behavior of machine tools are usually based on a finite element (FE) discretization of their mechanical structure. In order to capture all necessary details FE models are in general very large and sparse. Hence the computation of the simulation takes an unacceptable long time and requires much memory space. To calculate the results in reasonable time typically modal...

This paper presents the Cholesky factor–alternating direction implicit (CF–ADI) algorithm, which generates a low rank approximation to the solution X of the Lyapunov equation AX + XAT = −BBT . The coefficient matrix A is assumed to be large, and the rank of the righthand side −BBT is assumed to be much smaller than the size of A. The CF–ADI algorithm requires only matrix-vector products and mat...

In Erlangga and Nabben [SIAM J. Sci. Comput., 30 (2008), pp. 1572–1595], a multilevel Krylov method is proposed to solve linear systems with symmetric and nonsymmetric matrices of coefficients. This multilevel method is based on an operator which shifts some small eigenvalues to the largest eigenvalue, leading to a spectrum which is favorable for convergence acceleration of a Krylov subspace me...

Non-linear image reconstruction methods are desirable for applications in electrical impedance tomography (EIT) such as brain or breast imaging where the assumptions of linearity are violated. We present a novel non-linear Newton-Krylov method for solving large-scale EIT inverse problems, which has the potential advantages of improved robustness and computational efficiency over previous method...

We propose a time-exact Krylov-subspace-based method for solving linear ODE (ordinary differential equation) systems of the form y ′ = −Ay+g(t), where y(t) is the unknown function. The method consists of two stages. The first stage is an accurate polynomial approximation of the source term g(t), constructed with the help of the truncated SVD (singular value decomposition). The second stage is a...