Accelerating the Induced Dimension Reduction method using spectral information

The Induced Dimension Reduction method (IDR(s)) (Sonneveld and van Gijzen, 2008) is a short-recurrences Krylov method to solve systems of linear equations. In this work, we accelerate this method using spectral information. We construct a Hessenberg relation from the IDR(s) residual recurrences formulas, from which we approximate the...

Induced Dimension Reduction algorithms for solving non-symmetric sparse matrix problems

In several applications in science and engineering, different types of matrix problems emerge from the discretization of partial differential equations.<br/>This thesis is devoted to the development of new algorithms to solve this<br/>kind of problems. In particular, when the matrices involved are sparse and<br/>non-symmetric. The new algorithms...

An MSSS-preconditioned matrix equation approach for the time-harmonic elastic wave equation at multiple frequencies

In this work, we present a new numerical framework for the efficient solution of the time-harmonic elastic wave equation at multiple frequencies. We show that multiple frequencies (and multiple right-hand sides) can be incorporated when the discretized problem is written as a matrix equation. This matrix equation can be solved efficiently...

Induced Dimension Reduction method for solving linear matrix equations

This paper discusses the solution of large-scale linear matrix equations using the Induced Dimension reduction method (IDR(s)). IDR(s) was originally presented to solve system of linear equations, and is based on the IDR(s) theorem. We generalize the IDR(s) theorem to solve linear problems in any finite-dimensional space. This generalization...

A Restarted Induced Dimension Reduction method to approximate eigenpairs of large unsymmetric matrices

A new algorithm to compute eigenpairs of large unsymmetric matrices is presented. Using the Induced Dimension Reduction method (IDR), which was originally proposed for solving linear systems, we obtain a Hessenberg decomposition from which we approximate the eigen-values and eigenvectors of a matrix. This decomposition has two main advantages....

Nested Krylov methods for shifted linear systems

An induced dimension reduction algorithm to approximate eigenpairs of large nonsymmetric matrices

This work presents an algorithm to approximate eigenpairs of large, sparse and nonsymmetric matrices based on the Induced Dimension Reduction method (IDR(s)) introduced in [1]. We obtain a Hessenberg relation from IDR(s) computations and in conjunction with Implicitly Restarting and shift-and-invert techniques [2] we created a short recurrence...

Exploiting BiCGstab(?) strategies to induce dimension reduction

IDR(s) [P. Sonneveld and M. B. van Gijzen, SIAM J. Sci. Comput., 31 (2008), pp. 1035–1062] and BiCGstab(?) [G. L. G. Sleijpen and D. R. Fokkema, Electron. Trans. Numer. Anal., 1 (1993), pp. 11–32] are two of the most efficient short-recurrence iterative methods for solving large nonsymmetric linear systems of equations. Which of the two is best...