Section: New Results

High performance solvers for large linear algebra problems

Divide and conquer symmetric tridiagonal eigensolver for multicore architectures

Computing eigenpairs of a symmetric matrix is a problem arising in many industrial applications, including quantum physics and finite-elements computation for automobiles. A classical approach is to reduce the matrix to tridiagonal form before computing eigenpairs of the tridiagonal matrix. Then, a back-transformation allows one to obtain the final solution. Parallelism issues of the reduction stage have already been tackled in different shared-memory libraries. In this work, we focus on solving the tridiagonal eigenproblem, and we describe a novel implementation of the Divide and Conquer algorithm. The algorithm is expressed as a sequential task-flow, scheduled in an out-of-order fashion by a dynamic runtime which allows the programmer to play with tasks granularity. The resulting implementation is between two and five times faster than the equivalent routine from the INTEL MKL library, and outperforms the best MRRR implementation for many matrices.

These contributions have be presented at the international conference IPDPS 2015 [32] in Hyderabad.

Blocking strategy optimizations for sparse direct linear solver on heterogeneous architectures

Solving sparse linear systems is a problem that arises in many scientific applications, and sparse direct solvers are a time consuming and key kernel to those applications or more advanced solvers such as hybrid direct-iterative solvers. That is why optimizing their performance on modern architectures is a crucial problem. The preprocessing steps of sparse direct solvers: ordering and symbolic factorization, are two major steps that lead to a reduced amount of computation and memory, and to a better task granularity to reach a good level of performance when using BLAS kernels. With the advent of GPUs, the granularity of the symbolic factorization became more important than ever. In this work, we present a reordering strategy that increases the block granularity. This strategy relies on the symbolic factorization to refine the ordering produced by tools such as METIS or Scotch , and does not impact the number of operations required to solve the problem. We integrated this algorithm in the PaStiX solver and show a reduction of the number of off-diagonal blocks by two to three on a large spectrum of matrices. This improvement leads to an efficiency on GPUs raised by up to $40\%$. These contributions have be presented at the Sparse Days [51] in Saint-Girons.

On the use of $\mathscr{H}$-Matrix Arithmetic in PaStiX : a Preliminary Study

The objective is to investigate innovative low­rank approximations based on $\mathscr{H}$­matrix variants for direct solver and Schur complements. The intent is to improve scalability of those components involved in preconditioners and hybrid solvers by reducing the computational and memory costs of the dense calculation. The quality of hybrid ordering algorithms combining top­down (such as nested dissection) and bottom­up (such as minimum degree) ordering techniques in the context of sparse linear solvers will be investigated.

In this work, we describe a preliminary fast direct solver using HODLR library to compress large blocks appearing in the symbolic structure of the PaStiX sparse direct solver. We present our general strategy before analyzing the practical gains in terms of memory and floating point operations with respect to a theoretical study of the problem. Finally, we discuss ways to enhance the overall performance of the solver.

Some contributions have already been presented at the Workshop on Fast Solvers [52] in Toulouse. This work is a joint effort between Professor Darve’s group at Stanford and the Inria HiePACS team within FastLA .

Data sparse techniques for parallel hybrid solvers

In this work we describe how data sparse techniques exploiting $\mathscr{H}$-matrix calculations can be implemented in a parallel hybrid sparse linear solver based on an algebraic non overlapping domain decomposition approach.

Various graph-based clustering techniques to approximate the local Schur complements are investigated, with the aim of optimally complying with the interface structure of the local interfaces of the subdomains. We consider strong-hierarchical (sH) matrix arithmetic as efficient means for obtaining low rank approximations in terms of workload distribution as well as memory consumption. We also show how sH-arithmetic can be utilized to form an effective global preconditioner for the iterative phase of the hybrid solver. Numerical and parallel experiments are presented to evaluate the advantages and drawbacks of the different variants.

This work is a joint effort between Professor Darve’s group at Stanford and the Inria HiePACS team within FastLA . Some intermediate progresses have already been presented [38] , [37]

Analysis of the rounding error accumulation in Conjugate Gradient to improve the maximal attainable accuracy of pipelined CG

Pipelined Krylov solvers typically offer better scalability in the strong scaling limit compared to standard Krylov methods. The synchronization bottleneck is mitigated by overlapping time-consuming global communications with useful computations in the algorithm. However, to achieve this communication hiding strategy, pipelined methods feature multiple recurrence relations on additional auxiliary variables to update the guess for the solution. This paper aims to study the influence of rounding errors on the convergence of the pipelined Conjugate Gradient method. It is analyzed why rounding effects have a significantly larger impact on the maximal attainable accuracy of the pipelined CG algorithm compared to the traditional CG method. Based on a rounding error model, we then propose an automated residual replacement strategy to reduce the effect of rounding errors on the final iterative solution. The resulting pipelined CG method with residual replacement improves the maximal attainable accuracy of pipelined CG while maintaining its efficient parallel performance.

This research effort was conduced in collaboration with colleagues S. Cools and W. Vanroose from the Applied Mathematics Group of Antwerp university within the framework of the EXA2CT project.