Preconditioning of Krylov subspace methods using recycling in Lattice QCD computations

Many applications in computational science and engineering require the solution of sequences of slowly changing linear systems. We focus on problems arising in Lattice QCD simulations. In order to generate an ensemble of configurations from which the values of physical observables can be obtained, we have to solve a linear system with a Dirac operator in each time step of the hybrid Monte-Carlo simulations. This operator changes just slightly from time step to time step. While recycling subspace information from the previous system like described in [1] reduces the number of necessary matrix-vector multiplications, the systems are still expensive to solve. To overcome this limitation, we include preconditioning in our implementation. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On asynchronous iterations in partially ordered spaces

We extend the idea of asynchronous iterations to self-mappings of product spaces with infinitely many components. In addition to giving a rather general convergence theorem we study in some detail the case of isotone and isotonically decomposable mappings in partially ordered spaces. In particular, we obtain relationships between asynchronous iterations and the total step method and results on enclosures for fixed points. They appear to be new, even for mappings defined on a product space with only finitely many components.     

A framework for existence tests based on the topological degree and homotopy

The invariance of the topological degree under certain homotopies is used to derive a framework for tests to computationally prove the existence of zeros of nonlinear mappings in . These tests use interval arithmetic to enclose the range of a function over a box and are provably more general than many other tests like the Moore–Kioustelidis test, a test based on the Krawczyk operator, and another degree–based test published recently. A numerical example is included.     

Über das Auftreten von Wirbeln und Strudeln (geschlossener und spiraliger Integralkurven) in der Umgebung rationaler Unbestimmtheitsstellen

Ohne Zusammenfassung     

Weighted max norms, splittings, and overlapping additive Schwarz iterations

Summary. Weighted max-norm bounds are obtained for Algebraic Additive Schwarz Iterations with overlapping blocks for the solution of Ax = b, when the coefficient matrix A is an M-matrix. The case of inexact local solvers is also covered. These bounds are analogous to those that exist using A-norms when the matrix A is symmetric positive definite. A new theorem concerning P-regular splittings is presented which provides a useful tool for the A-norm bounds. Furthermore, a theory of splittings is developed to represent Algebraic Additive Schwarz Iterations. This representation makes a connection with multisplitting methods. With this representation, and using a comparison theorem, it is shown that a coarse grid correction improves the convergence of Additive Schwarz Iterations when measured in weighted max norm. Received March 13, 1998 / Revised version received January 26, 1999     

Quasi-Norm interpolation error estimates for the piecewise linear finite element approximation of p-Laplacian problems

Summary. In this work, new interpolation error estimates have been derived for some well-known interpolators in the quasi-norms. The estimates are found to be essential to obtain the optimal a priori error bounds under the weakened regularity conditions for the piecewise linear finite element approximation of a class of degenerate equations. In particular, by using these estimates, we can close the existing gap between the regularity required for deriving the optimal error bounds and the regularity achievable for the smooth data for the 2-d and 3-d p-Laplacian.Mathematics Subject Classification (1991): 65N30