Krylov eigensolvers are used in many scientific fields, such as nuclear physics, page ranking, oil and gas exploration, etc... In this paper, we focus on the ERAM Krylov eigensolver whose convergence is strongly correlated to the Krylov subspace size and the restarting vector v0, a unit norm vector. We focus on computing the restarting vector v0 to accelerate the ERAM convergence. First, we stu...

The Arnoldi method computes eigenvalues of large nonsymmetric matrices. Restarting is generally needed to reduce storage requirements and orthogonalization costs. However, restarting slows down the convergence and makes the choice of the new starting vector difficult if several eigenvalues are desired. We analyze several approaches to restarting and show why Sorensen’s implicit QR approach is g...

lanczos-type algorithms are well known for their inherent instability. they typically breakdown occurs when relevant orthogonal polynomials do not exist. current approaches to curing breakdown rely on jumping over the non-existent polynomials to resume computation. this may have to be used many times during the solution process. we suggest an alternative to jumping, which consists of restarting...

When using the Arnoldi method for approximating f(A)b, the action of a matrix function on a vector, the maximum number of iterations that can be performed is often limited by the storage requirements of the full Arnoldi basis. As a remedy, different restarting algorithms have been proposed in the literature, none of which was universally applicable, efficient, and stable at the same time. We ut...

In the present thesis we introduce several ways of extending restarting automata to devices for realizing binary (word) relations, which is mainly motivated by linguistics. In particular, we adapt the notion of input/output-relations and proper-relations to restarting automata and to parallel communicating systems of two restarting automata (PC-systems). Further, we introduce a new model, calle...

The generalized minimum residual method (GMRES) is well known for solving large nonsymmetric systems of linear equations. It generally uses restarting, which slows the convergence. However, some information can be retained at the time of the restart and used in the next cycle. We present algorithms that use implicit restarting in order to retain this information. Approximate eigenvectors determ...

The performance of Krylov subspace eigenvalue algorithms for large matrices can be measured by the angle between a desired invariant subspace and the Krylov subspace. We develop general bounds for this convergence that include the effects of polynomial restarting and impose no restrictions concerning the diagonalizability of the matrix or its degree of nonnormality. Associated with a desired se...

This paper discusses the numerical simulations of heat transfer problem, particularly during welding materials processing. process highly depends on temperature to determine metallurgical properties, strength, and sureness joint welding; hence it is desirable study its flow. Finite difference method (FDM) used discretize partial differential equations (PDEs) obtain systems linear (SLEs), which ...

The Lanczos method is one of the standard approaches for computing a few eigenpairs of a large, sparse, symmetric matrix. It is typically used with restarting to avoid unbounded growth of memory and computational requirements. Thick-restart Lanczos is a popular restarted variant because of its simplicity and numerically robustness. However, convergence can be slow for highly clustered eigenvalu...