Approximating a Multi-Grid Solver

Multi-grid methods are numerical algorithms used in parallel and distributed processing. The main idea of multigrid solvers is to speedup the convergence of an iterative method by reducing the problem to a coarser grid a number of times. Multi-grid methods are widely exploited in many application domains, thus it is important to improve their performance and energy efficiency. This paper aims to reach this objective based on the following observation: Given that the intermediary steps do not require full accuracy, it is possible to save time and energy by reducing precision during some steps while keeping the final result within the targeted accuracy. To achieve this goal, we first introduce a cycle shape different from the classic V-cycle used in multi-grid solvers. Then, we propose to dynamically change the floating-point precision used during runtime according to the accuracy needed for each intermediary step. Our evaluation considering a state-of-the-art multi-grid solver implementation demonstrates that it is possible to trade temporary precision for time to completion without hurting the quality of the final result. In particular, we are able to reach the same accuracy results as with full double-precision while gaining between 15% and 30% execution time improvement.This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 708566 (DURO). The European Commission is not liable for any use that might be made of the information contained therein. This work has been supported by the Spanish Government (Severo Ochoa grant SEV2015-0493)Peer ReviewedPostprint (author's final draft

Preconditioned fully implicit PDE solvers for monument conservation

Mathematical models for the description, in a quantitative way, of the damages induced on the monuments by the action of specific pollutants are often systems of nonlinear, possibly degenerate, parabolic equations. Although some the asymptotic properties of the solutions are known, for a short window of time, one needs a numerical approximation scheme in order to have a quantitative forecast at any time of interest. In this paper a fully implicit numerical method is proposed, analyzed and numerically tested for parabolic equations of porous media type and on a systems of two PDEs that models the sulfation of marble in monuments. Due to the nonlinear nature of the underlying mathematical model, the use of a fixed point scheme is required and every step implies the solution of large, locally structured, linear systems. A special effort is devoted to the spectral analysis of the relevant matrices and to the design of appropriate iterative or multi-iterative solvers, with special attention to preconditioned Krylov methods and to multigrid procedures. Numerical experiments for the validation of the analysis complement this contribution.Comment: 26 pages, 13 figure

Tensor Numerical Methods in Quantum Chemistry: from Hartree-Fock Energy to Excited States

We resume the recent successes of the grid-based tensor numerical methods and discuss their prospects in real-space electronic structure calculations. These methods, based on the low-rank representation of the multidimensional functions and integral operators, led to entirely grid-based tensor-structured 3D Hartree-Fock eigenvalue solver. It benefits from tensor calculation of the core Hamiltonian and two-electron integrals (TEI) in $O(n\log n)$ complexity using the rank-structured approximation of basis functions, electron densities and convolution integral operators all represented on 3D $n\times n\times n$ Cartesian grids. The algorithm for calculating TEI tensor in a form of the Cholesky decomposition is based on multiple factorizations using algebraic 1D ``density fitting`` scheme. The basis functions are not restricted to separable Gaussians, since the analytical integration is substituted by high-precision tensor-structured numerical quadratures. The tensor approaches to post-Hartree-Fock calculations for the MP2 energy correction and for the Bethe-Salpeter excited states, based on using low-rank factorizations and the reduced basis method, were recently introduced. Another direction is related to the recent attempts to develop a tensor-based Hartree-Fock numerical scheme for finite lattice-structured systems, where one of the numerical challenges is the summation of electrostatic potentials of a large number of nuclei. The 3D grid-based tensor method for calculation of a potential sum on a $L\times L\times L$ lattice manifests the linear in $L$ computational work, $O(L)$, instead of the usual $O(L^3 \log L)$ scaling by the Ewald-type approaches

Bootstrap Multigrid for the Laplace-Beltrami Eigenvalue Problem

This paper introduces bootstrap two-grid and multigrid finite element approximations to the Laplace-Beltrami (surface Laplacian) eigen-problem on a closed surface. The proposed multigrid method is suitable for recovering eigenvalues having large multiplicity, computing interior eigenvalues, and approximating the shifted indefinite eigen-problem. Convergence analysis is carried out for a simplified two-grid algorithm and numerical experiments are presented to illustrate the basic components and ideas behind the overall bootstrap multigrid approach