Extended local fourier analysis for multigrid optimal smoothing, coarse grid correction, and preconditioning

Multigrid methods are fast iterative solvers for partial di erential equations. Especially for elliptic equations they have been proven to be highly e cient. For problems with nonelliptic and nonsymmetric features--as they often occur in typical real life applications--a rigorous mathematical theory is generally not available. For such situations Fourier smoothing and two-grid analysis can be considered as the main analysis tools to obtain quantitative convergence estimates and to optimize different multigrid components like smoothers or inter-grid transfer operators. In general, it is difficult to choose the correct multigrid components for large classes of problems. A popular alternative to construct a robust solver is the use of multigrid as a preconditioner for a Krylov subspace acceleration method like GMRES. Our contributions to the Fourier analysis for multigrid are two-fold. Firstly we extend the range of situations for which the Fourier analysis can be applied. More precisely, the Fourier analysis is generalized to k-grid cycles and to multigrid as a preconditioner. With a k-grid analysis it is possible to investigate real multigrid effects which cannot be captured by the classical two-grid analysis. Moreover, the k-grid analysis allows for a more detailed investigation of possible coarse grid correction difficulties. Additional valuable insight is obtained by evaluating multigrid as a preconditioner for GMRES. Secondly we extend the range of discretizations and multigrid components for which detailed Fourier analysis results exist. We consider four well-known singularly perturbed model problems to demonstrate the usefulness of the above generalizations: The anisotropic Poisson equation, the rotated anisotropic diffusion equation, the convection diffusion equation with dominant convection, and the driven cavity problem governed by the incompressible Navier Stokes equations. Each of these equations represents a larger class of problems with similar features and complications which are of practical relevance. With the help of the newly developed Fourier analysis methods, a comprehensive study of characteristic difficulties for singular perturbation problems can be performed. Based on the insights from this analysis it is possible to identify remedies resulting in an improved multigrid convergence. The theoretical considerations are validated by numerical test calculations

Local Fourier Analysis of the Complex Shifted Laplacian preconditioner for Helmholtz problems

In this paper we solve the Helmholtz equation with multigrid preconditioned Krylov subspace methods. The class of Shifted Laplacian preconditioners are known to significantly speed-up Krylov convergence. However, these preconditioners have a parameter beta, a measure of the complex shift. Due to contradictory requirements for the multigrid and Krylov convergence, the choice of this shift parameter can be a bottleneck in applying the method. In this paper, we propose a wavenumber-dependent minimal complex shift parameter which is predicted by a rigorous k-grid Local Fourier Analysis (LFA) of the multigrid scheme. We claim that, given any (regionally constant) wavenumber, this minimal complex shift parameter provides the reader with a parameter choice that leads to efficient Krylov convergence. Numerical experiments in one and two spatial dimensions validate the theoretical results. It appears that the proposed complex shift is both the minimal requirement for a multigrid V-cycle to converge, as well as being near-optimal in terms of Krylov iteration count.Comment: 20 page

Observation of deflection of a beam of multi-GeV electrons by a thin crystal

We report on an experiment performing channeling and volume reflection of a high-energy electron beam using a quasimosaic, bent silicon (111) crystal at the End Station A Test Beam at SLAC. The experiment uses beams of 3.35 and 6.3 GeV. In the channeling orientation, deflections of the beam of 400μrad for both energies with about 22% efficiency are observed, while in the volume-reflection orientation, deflection of the beam by 120μrad at 3.35 GeV and by 80μrad at 6.3 GeV is observed with 86%-95% efficiency. Quantitative measurements of the channeling efficiency, surface transmission, and dechanneling length are taken. These are the first quantitative measurements of channeling and volume reflection using a primary beam of multi-GeV electrons

Parameters for a Super-Flavor-Factory

A Super Flavor Factory, an asymmetric energy e+e- collider with a luminosity of order 10^36 cm-2s-1, can provide a sensitive probe of new physics in the flavor sector of the Standard Model. The success of the PEP-II and KEKB asymmetric colliders in producing unprecedented luminosity above 10^34 cm-2s-1 has taught us about the accelerator physics of asymmetric e+e- colliders in a new parameter regime. Furthermore, the success of the SLAC Linear Collider and the subsequent work on the International Linear Collider allow a new Super-Flavor collider to also incorporate linear collider techniques. This note describes the parameters of an asymmetric Flavor-Factory collider at a luminosity of order 10^36 cm-2s-1 at the Upsilon(4S) resonance and about 10^35 cm-2s-1 at the Tau production threshold. Such a collider would produce an integrated luminosity of about 10,000 fb-1 (10 ab-1) in a running year (10^7 sec) at the Upsilon(4S) resonance.Comment: Flavor Physics & CP Violation Conference, Vancouver, 200

Spin dynamic tool developments and study regarding the Super-B project

International audience

Tripartite phase separation of two signal effectors with vesicles priming B cell responsiveness.

Antibody-mediated immune responses rely on antigen recognition by the B cell antigen receptor (BCR) and the proper engagement of its intracellular signal effector proteins. Src homology (SH) 2 domain-containing leukocyte protein of 65 kDa (SLP65) is the key scaffold protein mediating BCR signaling. In resting B cells, SLP65 colocalizes with Cbl-interacting protein of 85 kDa (CIN85) in cytoplasmic granules whose formation is not fully understood. Here we show that effective B cell activation requires tripartite phase separation of SLP65, CIN85, and lipid vesicles into droplets via vesicle binding of SLP65 and promiscuous interactions between nine SH3 domains of the trimeric CIN85 and the proline-rich motifs (PRMs) of SLP65. Vesicles are clustered and the dynamical structure of SLP65 persists in the droplet phase in vitro. Our results demonstrate that phase separation driven by concerted transient interactions between scaffold proteins and vesicles is a cellular mechanism to concentrate and organize signal transducers