arXiv subjects

Eric Darve

Publications and source records attributed to Eric Darve.

82 records · Page 5Linked to original sources

arXiv2015

A Fast and Memory Efficient Sparse Solver with Applications to Finite-Element Matrices

In this article, we introduce a fast and memory efficient solver for sparse matrices arising from the finite element discretization of elliptic partial differential equations (PDEs). We use a fast direct (but approximate) multifrontal solver as a preconditioner, and use an iterative solver to achieve a desired accuracy. This approach combines the advantages of direct and iterative schemes to arrive at a fast, robust and accurate solver. We will show that this solver is faster ($\sim$ 2x) and more memory efficient ($\sim$ 2--3x) than a conventional direct multifrontal solver. Furthermore, we will demonstrate that the solver is both a faster and more effective preconditioner than other preconditioners such as the incomplete LU preconditioner. Specific speed-ups depend on the matrix size and improve as the size of the matrix increases. The solver can be applied to both structured and unstructured meshes in a similar manner. We build on our previous work and utilize the fact that dense frontal and update matrices, in the multifrontal algorithm, can be represented as hierarchically off-diagonal low-rank (HODLR) matrices. Using this idea, we replace all large dense matrix operations in the multifrontal elimination process with $O(N)$ HODLR operations to arrive at a faster and more memory efficient solver.

math.NA↗

arXiv2015

A Fast Block Low-Rank Dense Solver with Applications to Finite-Element Matrices

This article presents a fast solver for the dense "frontal" matrices that arise from the multifrontal sparse elimination process of 3D elliptic PDEs. The solver relies on the fact that these matrices can be efficiently represented as a hierarchically off-diagonal low-rank (HODLR) matrix. To construct the low-rank approximation of the off-diagonal blocks, we propose a new pseudo-skeleton scheme, the boundary distance low-rank approximation, that picks rows and columns based on the location of their corresponding vertices in the sparse matrix graph. We compare this new low-rank approximation method to the adaptive cross approximation (ACA) algorithm and show that it achieves betters speedup specially for unstructured meshes. Using the HODLR direct solver as a preconditioner (with a low tolerance) to the GMRES iterative scheme, we can reach machine accuracy much faster than a conventional LU solver. Numerical benchmarks are provided for frontal matrices arising from 3D finite element problems corresponding to a wide range of applications.

math.NA↗

arXiv2014

The Inverse Fast Multipole Method

This article introduces a new fast direct solver for linear systems arising out of wide range of applications, integral equations, multivariate statistics, radial basis interpolation, etc., to name a few. \emph{The highlight of this new fast direct solver is that the solver scales linearly in the number of unknowns in all dimensions.} The solver, termed as Inverse Fast Multipole Method (abbreviated as IFMM), works on the same data-structure as the Fast Multipole Method (abbreviated as FMM). More generally, the solver can be immediately extended to the class of hierarchical matrices, denoted as $\mathcal{H}^2$ matrices with strong admissibility criteria (weak low-rank structure), i.e., \emph{the interaction between neighboring cluster of particles is full-rank whereas the interaction between particles corresponding to well-separated clusters can be efficiently represented as a low-rank matrix}. The algorithm departs from existing approaches in the fact that throughout the algorithm the interaction corresponding to neighboring clusters are always treated as full-rank interactions. Our approach relies on two major ideas: (i) The $N \times N$ matrix arising out of FMM (from now on termed as FMM matrix) can be represented as an extended sparser matrix of size $M \times M$, where $M \approx 3N$. (ii) While solving the larger extended sparser matrix, \emph{the fill-in's that arise in the matrix blocks corresponding to well-separated clusters are hierarchically compressed}. The ordering of the equations and the unknowns in the extended sparser matrix is strongly related to the local and multipole coefficients in the FMM~\cite{greengard1987fast} and \emph{the order of elimination is different from the usual nested dissection approach}. Numerical benchmarks on $2$D manifold confirm the linear scaling of the algorithm.

math.NA↗

arXiv2013

Computing reaction rates in bio-molecular systems using discrete macro-states

Computing reaction rates in biomolecular systems is a common goal of molecular dynamics simulations. The reactions considered often involve conformational changes in the molecule, either changes in the structure of a protein or the relative position of two molecules, for example when modeling the binding of a protein and ligand. Here we will consider the general problem of computing the rate of transfer from a subset A of the conformational space Omega to a subset B of Omega. It is assumed that A and B are associated with minimum energy basins and are long-lived states. Rates can be obtained using many different methods. We review some of the most popular approaches. We organize the different approaches roughly in chronological order and under four main categories: reactive flux, transition path sampling, conformation dynamics. The fourth class of methods, to which we do not give any specific name, in some sense attempts to combine features from transition path sampling and conformation dynamics. They include non-equilibrium umbrella sampling (Warmflash et al. [2007], Dickson et al. [2009b]), and weighted ensemble dynamics (Huber and Kim [1996]).

math.DS↗

arXiv2013

Method and Advantages of Genetic Algorithms in Parameterization of Interatomic Potentials: Metal-Oxides

The method and the advantages of an evolutionary computing based approach using a steady state genetic algorithm (GA) for the parameterization of interatomic potentials for metal oxides within the shell model framework are developed and described. We show that the GA based methodology for the parameterization of interatomic force field functions is capable of (a) simultaneous optimization of the multiple phases or properties of a material in a single run, (b) facilitates the incremental re-optimization of the whole system as more data is available for either additional phases or material properties not included in previous runs, and (c) successful global optimization in the presence of multiple local minima in the parameter space. As an example, we apply the method towards simultaneous optimization of four distinct crystalline phases of Barium Titanate (BaTiO3 or BTO) using an ab initio density functional theory (DFT) based reference dataset. We find that the optimized force field function is capable of the prediction of the two phases not used in the optimization procedure, and that many derived physical properties such as the equilibrium lattice constants, unit cell volume, elastic properties, coefficient of thermal expansion, and average electronic polarization are in good agreement with the experimental results available from the literature.

cond-mat.mtrl-sci↗

arXiv2012

Optimized M2L Kernels for the Chebyshev Interpolation based Fast Multipole Method

A fast multipole method (FMM) for asymptotically smooth kernel functions (1/r, 1/r^4, Gauss and Stokes kernels, radial basis functions, etc.) based on a Chebyshev interpolation scheme has been introduced in [Fong et al., 2009]. The method has been extended to oscillatory kernels (e.g., Helmholtz kernel) in [Messner et al., 2012]. Beside its generality this FMM turns out to be favorable due to its easy implementation and its high performance based on intensive use of highly optimized BLAS libraries. However, one of its bottlenecks is the precomputation of the multiple-to-local (M2L) operator, and its higher number of floating point operations (flops) compared to other FMM formulations. Here, we present several optimizations for that operator, which is known to be the costliest FMM operator. The most efficient ones do not only reduce the precomputation time by a factor up to 340 but they also speed up the matrix-vector product. We conclude with comparisons and numerical validations of all presented optimizations.

math.NA↗

arXiv2012

Pipelining the Fast Multipole Method over a Runtime System

Fast Multipole Methods (FMM) are a fundamental operation for the simulation of many physical problems. The high performance design of such methods usually requires to carefully tune the algorithm for both the targeted physics and the hardware. In this paper, we propose a new approach that achieves high performance across architectures. Our method consists of expressing the FMM algorithm as a task flow and employing a state-of-the-art runtime system, StarPU, in order to process the tasks on the different processing units. We carefully design the task flow, the mathematical operators, their Central Processing Unit (CPU) and Graphics Processing Unit (GPU) implementations, as well as scheduling schemes. We compute potentials and forces of 200 million particles in 48.7 seconds on a homogeneous 160 cores SGI Altix UV 100 and of 38 million particles in 13.34 seconds on a heterogeneous 12 cores Intel Nehalem processor enhanced with 3 Nvidia M2090 Fermi GPUs.

cs.DC↗

arXiv2011

Fourier Based Fast Multipole Method for the Helmholtz Equation

The fast multipole method (FMM) has had great success in reducing the computational complexity of solving the boundary integral form of the Helmholtz equation. We present a formulation of the Helmholtz FMM that uses Fourier basis functions rather than spherical harmonics. By modifying the transfer function in the precomputation stage of the FMM, time-critical stages of the algorithm are accelerated by causing the interpolation operators to become straightforward applications of fast Fourier transforms, retaining the diagonality of the transfer function, and providing a simplified error analysis. Using Fourier analysis, constructive algorithms are derived to a priori determine an integration quadrature for a given error tolerance. Sharp error bounds are derived and verified numerically. Various optimizations are considered to reduce the number of quadrature points and reduce the cost of computing the transfer function.

math.NA↗

arXiv2011

Extension and optimization of the FIND algorithm: computing Green's and less-than Green's functions (with technical appendix)

The FIND algorithm is a fast algorithm designed to calculate certain entries of the inverse of a sparse matrix. Such calculation is critical in many applications, e.g., quantum transport in nano-devices. We extended the algorithm to other matrix inverse related calculations. Those are required for example to calculate the less-than Green's function and the current density through the device. For a 2D device discretized as an N_x x N_y mesh, the best known algorithms have a running time of O(N_x^3 N_y), whereas FIND only requires O(N_x^2 N_y). Even though this complexity has been reduced by an order of magnitude, the matrix inverse calculation is still the most time consuming part in the simulation of transport problems. We could not reduce the order of complexity, but we were able to significantly reduce the constant factor involved in the computation cost. By exploiting the sparsity and symmetry, the size of the problem beyond which FIND is faster than other methods typically decreases from a 130x130 2D mesh down to a 40x40 mesh. These improvements make the optimized FIND algorithm even more competitive for real-life applications.

math.NA↗

arXiv2007

N-Body Simulations on GPUs

Commercial graphics processors (GPUs) have high compute capacity at very low cost, which makes them attractive for general purpose scientific computing. In this paper we show how graphics processors can be used for N-body simulations to obtain improvements in performance over current generation CPUs. We have developed a highly optimized algorithm for performing the O(N^2) force calculations that constitute the major part of stellar and molecular dynamics simulations. In some of the calculations, we achieve sustained performance of nearly 100 GFlops on an ATI X1900XTX. The performance on GPUs is comparable to specialized processors such as GRAPE-6A and MDGRAPE-3, but at a fraction of the cost. Furthermore, the wide availability of GPUs has significant implications for cluster computing and distributed computing efforts like Folding@Home.

cs.CE↗