Preconditioning for nonsymmetry and time-dependence

In this short paper, we decribe at least one simple and frequently arising situation |that of nonsymmetric real Toeplitz (constant diagonal) matrices| where we can guarantee rapid convergence of the appropriate iterative method by manipulating the problem into a symmetric form without recourse to the normal equations. This trick can be applied regardless of the nonnormality of the Toeplitz matrix. We also propose a symmetric and positive definite preconditioner for this situation which is proved to cluster eigenvalues and is by consequence guaranteed to ensure convergence in a number of iterations independent of the matrix dimension

The Convergence Of Iterative Solution Methods For Symmetric And Indefinite Linear Systems

this paper we concentrate on convergence estimates for miminum residual iteration applied to a linear system Ax = b, (so that A represents the preconditioned coefficient matrix if preconditioning is employed). In particular we generalise the results of [25] to establish rigorous convergence estimates for families of matrices which depend on an asymptotically small parameter ff (in applications ff is typically a positive power of the mesh size parameter h). These results prove the superiority of the minimum residual approach over the solution of normal equations for all except one very special type of symmetric and indefinite matrix. More background and an easy introduction to this problem can be found in [8], pp. 310-31

A major role for matrix metalloproteinases in T cell injury in the gut

Activated lamina propria T cells responding to luminal Ags are thought to be important in celiac disease and Crohn's disease, and T cells responding to foreign MHC products are also important in intestinal graft-vs-host disease and intestinal transplant rejection. However, the mechanism(s) by which T cells mediate damage in the gut is not known. We have previously shown that activation of lamina propria T cells by PWM in explant cultures of second trimester human small intestine produces severe tissue injury, with epithelial cell shedding and loss of villi. In this study, we have investigated the role of matrix metalloproteinases in this system. Organ culture supernatants of explants stimulated with PWM showed a 3-fold increase in the concentration of interstitial collagenase and a 10-fold increase in stromelysin-1 compared with control explant culture supernatants. Tissue inhibitors of metalloproteinase-1 and -2 concentrations were unchanged. Increased metalloproteinase enzymatic activity was detected by gelatin and casein zymography. Western blotting revealed the active forms of interstitial collagenase and stromelysin-1 in PWM-stimulated culture supernatants. Up-regulation of mRNA for interstitial collagenase, stromelysin-1, and gelatinase-B was also seen. Nanomolar amounts of recombinant stromelysin-1 added directly to explants produced rapid severe tissue injury. PWM-induced mucosal injury was inhibited by a synthetic peptidomimetic inhibitor of matrix metalloproteinases. Mesenchymal cells isolated from the mucosa of human fetal small intestine produced increased amounts of interstitial collagenase, gelatinase A, and stromelysin-1 when stimulated with IL-1beta or TNF-alpha. These results suggest that T cell activation in the lamina propria results in increased production of matrix metalloproteinases, which by degrading the lamina propria matrix represent a major pathway by which T cells cause injury in the gut