This paper presents a short survey of recent research on Krylov subspace methods with emphasis on implementation on vector and parallel computers. Conjugate gradient methods have proven very useful on traditional scalar computers, and their popularity is likely to increase as three dimensional

. For a bounded linear operator A on a Hilbert space H we study local spectral sets and their relation to the spectrum of the local operator. By the local operator we mean A restricted to a Krylov subspace spanfb; Ab; A 2 b; :::g for a generic b 2 H. Moreover we investigate the relation between

. Adaptive mesh refinement (AMR) is one efficient way to reduce the computational cost. We propose a new AMR scheme for topology optimization that results in more robust and efficient solutions. For large sparse symmetric linear systems arising in topology optimization, Krylov subspace methods are required

. Computing the linear least-squares estimate of a high-dimensional random quantity given noisy data requires solving a large system of linear equations. In many situations, one can solve this system efficiently using a Krylov subspace method, such as the conjugate gradient (CG) algorithm

We present a general framework for a number of techniques based on projection methods on `augmented Krylov subspaces'. These methods include the deflated GMRES algorithm, an inner-outer FGMRES iteration algorithm, and the class of block Krylov methods. Augmented Krylov subspace methods

Summary. Krylov subspace methods are well-known for their nice properties, but they have to be implemented with care. In this article the mathematical consequences encountered during implementation of Krylov subspace methods in an existing layout-simulator are discussed. Briefly

Abstract. In this paper we describe and analyze Krylov subspace techniques for accelerating the convergence of waveform relaxation for solving time-dependent problems. A new class of accelerated waveform methods, convolution Krylov subspace methods, is presented. In particular, we give convolution

Residual norm estimates are derived for a general class of methods based on projection techniques on subspaces of the form Km + W, where Km is the standard Krylov subspace associated with the original linear system, and W is some other subspace. These `augmented Krylov subspace methods

The aspects of implementing Krylov subspace methods on parallel computers are investigated. It is shown how to increase the parallel performance by restructuring standard sequential versions of the algorithms, with some trade-off in stability. Further, we discuss how the computational kernels

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