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

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

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

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

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

Numerical Methods for the QCD Overlap Operator:III. Nested Iterations

The numerical and computational aspects of chiral fermions in lattice quantum chromodynamics are extremely demanding. In the overlap framework, the computation of the fermion propagator leads to a nested iteration where the matrix vector multiplications in each step of an outer iteration have to be accomplished by an inner iteration; the latter approximates the product of the sign function of the hermitian Wilson fermion matrix with a vector. In this paper we investigate aspects of this nested paradigm. We examine several Krylov subspace methods to be used as an outer iteration for both propagator computations and the Hybrid Monte-Carlo scheme. We establish criteria on the accuracy of the inner iteration which allow to preserve an a priori given precision for the overall computation. It will turn out that the accuracy of the sign function can be relaxed as the outer iteration proceeds. Furthermore, we consider preconditioning strategies, where the preconditioner is built upon an inaccurate approximation to the sign function. Relaxation combined with preconditioning allows for considerable savings in computational efforts up to a factor of 4 as our numerical experiments illustrate. We also discuss the possibility of projecting the squared overlap operator into one chiral sector.Comment: 33 Pages; citations adde

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

Current status of Dynamical Overlap project

We discuss the adaptation of the Hybrid Monte Carlo algorithm to overlap fermions. We derive a method which can be used to account for the delta function in the fermionic force caused by the differential of the sign function. We discuss the algoritmic difficulties that have been overcome, and mention those that still need to be solved.Comment: Talk given at Workshop on Computational Hadron Physics, Nicosia, September 2005. 8 page

Nonperturbative renormalisation of composite operators with overlap quarks

We compute non-perturbatively the renormalisation constants of composite operators on a $16^3 \times 32$ lattice with lattice spacing $a$ = 0.093 fm for the overlap fermion action by using the regularisation independent (RI) scheme. The quenched gauge configurations are generated by tadpole improved plaquette plus rectangle action. We test the perturbative continuum relation $Z_A = Z_V$ and $Z_S=Z_P$ and find that they agree well above $\mu$ = 1.6 GeV. We also perform a Renormalisation Group analysis at the next-to-next-to-leading order and convert the renormalisation constants to the $\bar{MS}$ scheme.Comment: Talk given at LHP2003, Cairns, Australi