Subspace Recycling for Sequences of Shifted Systems with Applications in Image Recovery

For many applications involving a sequence of linear systems with slowly changing system matrices, subspace recycling, which exploits relationships among systems and reuses search space information, can achieve huge gains in iterations across the total number of linear system solves in the sequence. However, for general (i.e., non-identity) shifted systems with the shift value varying over a wide range, the properties of the linear systems vary widely as well, which makes recycling less effective. If such a sequence of systems is embedded in a nonlinear iteration, the problem is compounded, and special approaches are needed to use recycling effectively. In this paper, we develop new, more efficient, Krylov subspace recycling approaches for large-scale image reconstruction and restoration techniques that employ a nonlinear iteration to compute a suitable regularization matrix. For each new regularization matrix, we need to solve regularized linear systems, ${\bf A} + \gamma_\ell {\bf E}_k$, for a sequence of regularization parameters, $\gamma_\ell$, to find the optimally regularized solution that, in turn, will be used to update the regularization matrix. In this paper, we analyze system and solution characteristics to choose appropriate techniques to solve each system rapidly. Specifically, we use an inner-outer recycling approach with a larger, principal recycle space for each nonlinear step and smaller recycle spaces for each shift. We propose an efficient way to obtain good initial guesses from the principle recycle space and smaller shift-specific recycle spaces that lead to fast convergence. Our method is substantially reduces the total number of matrix-vector products that would arise in a naive approach. Our approach is more generally applicable to sequences of shifted systems where the matrices in the sum are positive semi-definite

Optimal Interface Conditions for Domain Decomposition Methods

We define optimal interface conditions for the additive Schwarz method (ASM) in the sense that convergence is achieved in a number of steps equals to the number of subdomains. Since these boundary conditions are difficult to use, we approximate them by partial differential operators that are easier to use. We present numerical results using these approximate interface conditions for the ASM and Schur type methods (substructuring). We also give a new result of convergence for BiCG which is then used for BiCGSTAB

Islamic influences on urban form in Sumatra in the seventeenth to nineteenth centuries CE

This article focuses on the evolution of three urban centres: Palembang, Padang and Tanjung Pinang. Where appropriate, brief information about other towns is added which shows that the three towns are typical for towns on the east coast, the west coast and the islands in the Straits of Malacca respectively. Unfortunately, there is no place in the Minangkabau highlands for which historical sources exist that can help to reconstruct the townscape in a comparably detailed way. The descriptions of Palembang, Padang and Tanjung Pinang give details of Islamic buildings and provide information about the development of the settlements as a whole. These morphological histories have a value in their own right. They form a baseline to assess fully the specific Islamic influence on urban form in the disruption of some Islamic transformations. The Dutch changes bring out the previous Islamic influences more sharply. In the last section the emic (indigenous) conceptions of 'urban' will be analysed, by exploring the contrast between town and village and the role of Islamic buildings to accentuate the difference. The conclusion will list the most important empirical generalisations drawn from the descriptions. © 2004 Editors, Indonesia and the Malay World

Nested Krylov methods based on GCR

AbstractRecently the GMRESR method for the solution of linear systems of equations has been introduced by Vuik and Van der Vorst (1991). Similar methods have been proposed by Axelsson and Vassilevski (1991) and Saad (1993) (FGMRES11Since FGMRES and GMRESR are very similar, the ideas presented will be relevant for FGMRES as well.). GMRESR involves an outer and an inner method. The outer method is GCR, which is used to compute the optimal approximation over a given set of search vectors in the sense that the residual is minimized. The inner method is GMRES, which computes a new search vector by approximately solving the residual equation. This search vector is then used by the outer algorithm to compute a new approximation. However, the optimality of the approximation over the space of search vectors is ignored in the inner GMRES algorithm. This leads to suboptimal corrections to the solution in the outer algorithm. Therefore, we propose to preserve the orthogonality relations of GCR in the inner GMRES algorithm. This gives optimal corrections to the solution and also leads to solving the residual equation in a smaller subspace and with an “improved” operator, which should also lead to faster convergence. However, this involves using Krylov methods with a singular, nonsymmetric operator. We will discuss some important properties of this. We will show by experiments that in terms of matrix-vector products, this modification (almost) always leads to better convergence. Because we do more orthogonalizations, it does not always give an improved performance in time. This depends on the costs of the matrix-vector products relative to the costs of the orthogonalizations. Of course, we can also use methods other than GMRES as the inner method. Methods with short recurrences like BiCGSTAB seem especially interesting. The experimental results indicate that, especially for such methods, it is advantageous to preserve the orthogonality in the inner method