Tridiagonal System Solver Research Articles

Parallel prefix operations on GPU: tridiagonal system solvers and scan operators

Modern GPUs can achieve high computing power at low cost, but still requires much time and effort. Tridiagonal system and scan solvers are one example of widely used algorithms which can take advantage of these devices. In this article, one tridiagonal system solver and two scan primitive operators are implemented on CUDA GPUs. To do so, a tuning strategy based on three phases is developed. Additionally, a performance analysis is performed for two different CUDA GPU architectures, resulting in a huge improvement with respect to the state of the art.

Solving Large Problem Sizes of Index-Digit Algorithms on GPU: FFT and Tridiagonal System Solvers

Current Graphics Processing Units (GPUs) are capable of obtaining high computational performance in scientific applications. Nevertheless, programmers have to use suitable parallel algorithms for these architectures and usually have to consider optimization techniques in the implementation in order to achieve said performance. There are many efficient proposals for limited-size problems which fit directly in the shared memory of CUDA GPUs, however, there are few GPU proposals that tackle the design of efficient algorithms for large problem sizes that exceed shared memory storage capacity. In this work, we present a tuning strategy that addresses this problem for some parallel prefix algorithms that can be represented according to a set of common permutations of the digits of each of its element indices [1], denoted as Index-Digit (ID) algorithms. Specifically, our strategy has been applied to develop flexible Multi-Stage (MS) algorithms for the Fast Fourier Transform (FFT) algorithm (MS-ID-FFT) and a tridiagonal system solver (MS-ID-TS) on the GPU. The resulting implementation is compact and outperforms other well-known and commonly used state-of-the-art libraries, with an improvement of up to 1.47x with respect to NVIDIA's complex CUFFT, and up to 33.2x in comparison with NVIDIA's CUSPARSE for real data tridiagonal systems.

A cost-optimal parallel implementation of a tridiagonal system solver using skeletons

We address the task of systematically designing efficient programs for parallel machines. Our approach starts with a sequential algorithm and proceeds by expressing it in terms of standard, pre-implemented parallel components called skeletons. We demonstrate the skeleton-based design process using a tridiagonal system solver as our example application. We develop a cost-optimal parallel version of our application and implement it in message passing interface (MPI). The performance of our solution is demonstrated experimentally on a Cray T3E machine.

COST OPTIMALITY AND PREDICTABILITY OF PARALLEL PROGRAMMING WITH SKELETONS

Skeletons are reusable, parameterized program components with well-defined semantics and pre-packaged efficient parallel implementation. This paper develops a new, provably cost-optimal implementation of the DS (double-scan) skeleton for programming divide-and-conquer algorithms. Our implementation is based on a novel data structure called plist (pointed list); implementation's performance is estimated using an analytical model. We demonstrate the use of the DS skeleton for parallelizing a tridiagonal system solver and report experimental results for its MPI implementation on a Cray T3E and a Linux cluster: they confirm the performance improvement achieved by the cost-optimal implementation and demonstrate its good predictability by our performance model.

A parallel version of some block preconditionings

The block preconditioned conjugate gradient (BPCG) methods, even if very effective for solving the linear systems deriving from the discretization of partial differential equations, are not efficient for parallel computers. The bottleneck for their parallel implementation is represented by the solution of the linear system to obtain the preconditioned residual. Here, we present a parallel version of some BPCG algorithms. They are based on an approximate solver for tridiagonal systems, which utilizes an incomplete 2 X 2 block odd-even reduction. The stability properties of the approximate solver are investigated. Some numerical tests, on a net of transputers, are also included.

A cost-optimal parallel tridiagonal system solver

We first show how to transform the solution of an n × n tridiagonal system into suffix computations of continued fractions. Then a parallel substitution scheme is introduced to compute the suffix values. The derived parallel algorithm allows the tridiagonal system to be solved in O(log n) time on an unshuffle network with Θ( n /log n) processors. It is cost-optimal in the sense that processor number times execution time is minimized. Our solver is conceptually simple and easy for implementation.

Benchmark calculations for higher-order parabolic equations

Benchmark solutions generated with parabolic equation (PE) models are presented for range-dependent underwater acoustic propagation problems involving both penetrable and perfectly reflecting ocean bottoms. The solution of the wide-angle PE of Claerbout [J. F. Claerbout, Fundamentals of Geophysical Data Processing (McGraw-Hill, New York, 1976), pp. 206–207] agrees with the outgoing coupled-mode solution for the problems involving penetrable bottoms. The solution of the higher-order PE of Bamberger et al. [Bamberger et al., ‘‘Higher Order Paraxial Wave Equation Approximations in Heterogeneous Media,’’ SIAM J. Appl. Math. 48, 129–154 (1988)], which is a generalization of Claerbout’s PE, agrees with the outgoing coupled-mode solution for problems involving large variations in sound speed and propagation nearly orthogonal to the preferred direction. The computer code FEPE was used to generate the benchmark solutions and was found to run several times faster than the IFDPE computer code due to a tridiagonal system solver in FEPE that is optimized for range-dependent problems.