Adaptive Aggregation Based Domain Decomposition Multigrid for the Lattice Wilson Dirac Operator

In lattice QCD computations a substantial amount of work is spent in solving discretized versions of the Dirac equation. Conventional Krylov solvers show critical slowing down for large system sizes and physically interesting parameter regions. We present a domain decomposition adaptive algebraic multigrid method used as a precondtioner to solve the "clover improved" Wilson discretization of the Dirac equation. This approach combines and improves two approaches, namely domain decomposition and adaptive algebraic multigrid, that have been used seperately in lattice QCD before. We show in extensive numerical test conducted with a parallel production code implementation that considerable speed-up over conventional Krylov subspace methods, domain decomposition methods and other hierarchical approaches for realistic system sizes can be achieved.Comment: Additional comparison to method of arXiv:1011.2775 and to mixed-precision odd-even preconditioned BiCGStab. Results of numerical experiments changed slightly due to more systematic use of odd-even preconditionin

Some Preconditioning Techniques for Saddle Point Problems

Saddle point problems arise frequently in many applications in science and engineering, including constrained optimization, mixed finite element formulations of partial differential equations, circuit analysis, and so forth. Indeed the formulation of most problems with constraints gives rise to saddle point systems. This paper provides a concise overview of iterative approaches for the solution of such systems which are of particular importance in the context of large scale computation. In particular we describe some of the most useful preconditioning techniques for Krylov subspace solvers applied to saddle point problems, including block and constrained preconditioners.\ud \ud The work of Michele Benzi was supported in part by the National Science Foundation grant DMS-0511336

A semi-staggered dilation-free finite volume method for the numerical solution of viscoelastic fluid flows on all-hexahedral elements

The dilation-free semi-staggered finite volume method presented in Sabin [M. Sahin, A preconditioned semi-staggered dilation-free finite volume method for the incompressible Navier-Stokes equations on all-hexahedral elements, Int. J. Numer. Methods Fluids 49 (2005) 959-974] has been extended for the numerical solution of viscoelastic fluid flows on all-quadrilateral (2D) / hexahedral (3D) meshes. The velocity components are defined at element node points, while the pressure term and the extra stress tensor are defined at element centroids. The continuity equation is satisfied exactly within each element. An upwind least square method is employed for the calculation of the extra stresses at control volume faces in order to maintain stability for hyperbolic constitutive equations. The time stepping algorithm used decouples the calculation of the extra stresses from the evaluation of the velocity and pressure fields by solving a generalised Stokes problem. The resulting linear systems are solved using the GMRES method provided by the PETSc library with an ILU(k) preconditioner obtained from the HYPRE library. We apply the method to both two- and three-dimensional flow of an Oldroyd-B fluid past a confined circular cylinder in a channel with blockage ratio 0.5. Crown Copyright (C) 2007 Published by Elsevier B.V. All rights reserved

Exploiting spatial symmetries for solving Poisson's equation

This paper presents a strategy to accelerate virtually any Poisson solver by taking advantage of s spatial reflection symmetries. More precisely, we have proved the existence of an inexpensive block diagonalisation that transforms the original Poisson equation into a set of 2s fully decoupled subsystems then solved concurrently. This block diagonalisation is identical regardless of the mesh connectivity (structured or unstructured) and the geometric complexity of the problem, therefore applying to a wide range of academic and industrial configurations. In fact, it simplifies the task of discretising complex geometries since it only requires meshing a portion of the domain that is then mirrored implicitly by the symmetries’ hyperplanes. Thus, the resulting meshes naturally inherit the exploited symmetries, and their memory footprint becomes 2s times smaller. Thanks to the subsystems’ better spectral properties, iterative solvers converge significantly faster. Additionally, imposing an adequate grid points’ ordering allows reducing the operators’ footprint and replacing the standard sparse matrix-vector products with the sparse matrixmatrix product, a higher arithmetic intensity kernel. As a result, matrix multiplications are accelerated, and massive simulations become more affordable. Finally, we include numerical experiments based on a turbulent flow simulation and making state-of-theart solvers exploit a varying number of symmetries. On the one hand, algebraic multigrid and preconditioned Krylov subspace methods require up to 23% and 72% fewer iterations, resulting in up to 1.7x and 5.6x overall speedups, respectively. On the other, sparse direct solvers’ memory footprint, setup and solution costs are reduced by up to 48%, 58% and 46%, respectively.This work has been financially supported by two competitive R+D projects: RETOtwin (PDC2021-120970-I00), given by MCIN/AEI/10.13039/501100011033 and European Union Next GenerationEU/PRTR, and FusionCAT (001-P-001722), given by Generalitat de Catalunya RIS3CAT-FEDER. Àdel Alsalti-Baldellou has also been supported by the predoctoral grants DIN2018-010061 and 2019-DI-90, given by MCIN/AEI/10.13039/501100011033 and the Catalan Agency for Management of University and Research Grants (AGAUR), respectively.Peer ReviewedPostprint (published version