The convergence of Jacobi-Davidson for Hermitian eigenproblems

Rayleigh Quotient iteration is an iterative method with some attractive convergence properties for nding (interior) eigenvalues of large sparse Hermitian matrices. However, the method requires the accurate (and, hence, often expensive) solution of a linear system in every iteration step. Unfortunately, replacing the exact solution with a cheaper approximation may destroy the convergence. The (Jacobi-)Davidson correction equation can be seen as a solution for this problem. In this paper we deduce quantitative results to support this viewpoint and we relate it to other methods. This should make some of the experimental observations in practice more quantitative in the Hermitian case. Asymptotic convergence bounds are given for xed preconditioners and for the special case if the correction equation is solved with some xed relative residual precision. A new dynamic tolerance is proposed and some numerical illustration is presented

Accurate approximations to eigenpairs using the harmonic Rayleigh Ritz Method

The problem in this paper is to construct accurate approximations from a subspace to eigenpairs for symmetric matrices using the harmonic Rayleigh-Ritz method. Morgan introduced this concept in [14] as an alternative forRayleigh-Ritz in large scale iterative methods for computing interior eigenpairs. The focus rests on the choice and in uence of the shift and error estimation. We also give a discussion of the dierences and similarities with the rened Ritz approach for symmetric matrices. Using some numerical experiments we compare dierent conditions for selecting appropriate harmonic Ritz vectors

Improving Inversions of the Overlap Operator

We present relaxation and preconditioning techniques which accelerate the inversion of the overlap operator by a factor of four on small lattices, with larger gains as the lattice size increases. These improvements can be used in both propagator calculations and dynamical simulations.Comment: lattice2004(machines

Optimal a priori error bounds for the Rayleigh-Ritz method

We derive error bounds for the Rayleigh-Ritz method for the approximation to extremal eigenpairs of a symmetric matrix. The bounds are expressed in terms of the eigenvalues of the matrix and the angle between the subspace and the eigenvector. We also present a sharp bound

A comparative study of numerical methods for the overlap Dirac operator--a status report

Improvements of various methods to compute the sign function of the hermitian Wilson-Dirac matrix within the overlap operator are presented. An optimal partial fraction expansion (PFE) based on a theorem of Zolotarev is given. Benchmarks show that this PFE together with removal of converged systems within a multi-shift CG appears to approximate the sign function times a vector most efficiently. A posteriori error bounds are given.Comment: 3 pages, poster contribution to Lattice2001(algorithms

Paradigm of biased PAR1 (protease-activated receptor-1) activation and inhibition in endothelial cells dissected by phosphoproteomics

Thrombin is the key serine protease of the coagulation cascade and mediates cellular responses by activation of PARs (protease-activated receptors). The predominant thrombin receptor is PAR1, and in endothelial cells (ECs), thrombin dynamically regulates a plethora of phosphorylation events. However, it has remained unclear whether thrombin signaling is exclusively mediated through PAR1. Furthermore, mechanistic insight into activation and inhibition of PAR1-mediated EC signaling is lacking. In addition, signaling networks of biased PAR1 activation after differential cleavage of the PAR1 N terminus have remained an unresolved issue. Here, we used a quantitative phosphoproteomics approach to show that classical and peptide activation of PAR1 induce highly similar signaling, that low thrombin concentrations initiate only limited phosphoregulation, and that the PAR1 inhibitors vorapaxar and parmodulin-2 demonstrate distinct antagonistic properties. Subsequent analysis of the thrombin-regulated phosphosites in the presence of PAR1 inhibitors revealed that biased activation of PAR1 is not solely linked to a specific G-protein downstream of PAR1. In addition, we showed that only the canonical thrombin PAR1 tethered ligand induces extensive early phosphoregulation in ECs. Our study provides detailed insight in the signaling mechanisms downstream of PAR1. Our data demonstrate that thrombin-induced EC phosphoregulation is mediated exclusively through PAR1, that thrombin and thrombin-tethered ligand peptide induce similar phosphoregulation, and that only canonical PAR1 cleavage by thrombin generates a tethered ligand that potently induces early signaling. Furthermore, platelet PAR1 inhibitors directly affect EC signaling, indicating that it will be a challenge to design a PAR1 antagonist that will target only those pathways responsible for tissue pathology

A Linux PC cluster for lattice QCD with exact chiral symmetry

A computational system for lattice QCD with exact chiral symmetry is described. The platform is a home-made Linux PC cluster, built with off-the-shelf components. At present this system constitutes of 64 nodes, with each node consisting of one Pentium 4 processor (1.6/2.0/2.5 GHz), one Gbyte of PC800/PC1066 RDRAM, one 40/80/120 Gbyte hard disk, and a network card. The computationally intensive parts of our program are written in SSE2 codes. The speed of this system is estimated to be 70 Gflops, and its price/performance is better than $1.0/Mflops for 64-bit (double precision) computations in quenched QCD. We discuss how to optimize its hardware and software for computing quark propagators via the overlap Dirac operator.Comment: 24 pages, LaTeX, 2 eps figures, v2:a note and references added, the version published in Int. J. Mod. Phys.

A note on Neuberger's double pass algorithm

We analyze Neuberger's double pass algorithm for the matrix-vector multiplication R(H).Y (where R(H) is (n-1,n)-th degree rational polynomial of positive definite operator H), and show that the number of floating point operations is independent of the degree n, provided that the number of sites is much larger than the number of iterations in the conjugate gradient. This implies that the matrix-vector product $(H)^{-1/2} Y \simeq R^{(n-1,n)}(H) \cdot Y$ can be approximated to very high precision with sufficiently large n, without noticeably extra costs. Further, we show that there exists a threshold $n_T$ such that the double pass is faster than the single pass for $n > n_T$, where $n_T \simeq 12 - 25$ for most platforms.Comment: 18 pages, v3: CPU time formulas are obtained, to appear in Physical Review