Numerically Stable Recurrence Relations for the Communication Hiding Pipelined Conjugate Gradient Method

Pipelined Krylov subspace methods (also referred to as communication-hiding methods) have been proposed in the literature as a scalable alternative to classic Krylov subspace algorithms for iteratively computing the solution to a large linear system in parallel. For symmetric and positive definite system matrices the pipelined Conjugate Gradient method outperforms its classic Conjugate Gradient counterpart on large scale distributed memory hardware by overlapping global communication with essential computations like the matrix-vector product, thus hiding global communication. A well-known drawback of the pipelining technique is the (possibly significant) loss of numerical stability. In this work a numerically stable variant of the pipelined Conjugate Gradient algorithm is presented that avoids the propagation of local rounding errors in the finite precision recurrence relations that construct the Krylov subspace basis. The multi-term recurrence relation for the basis vector is replaced by two-term recurrences, improving stability without increasing the overall computational cost of the algorithm. The proposed modification ensures that the pipelined Conjugate Gradient method is able to attain a highly accurate solution independently of the pipeline length. Numerical experiments demonstrate a combination of excellent parallel performance and improved maximal attainable accuracy for the new pipelined Conjugate Gradient algorithm. This work thus resolves one of the major practical restrictions for the useability of pipelined Krylov subspace methods.Comment: 15 pages, 5 figures, 1 table, 2 algorithm

Minimizing inner product data dependencies in conjugate gradient iteration

The amount of concurrency available in conjugate gradient iteration is limited by the summations required in the inner product computations. The inner product of two vectors of length N requires time c log(N), if N or more processors are available. This paper describes an algebraic restructuring of the conjugate gradient algorithm which minimizes data dependencies due to inner product calculations. After an initial start up, the new algorithm can perform a conjugate gradient iteration in time c*log(log(N))

A bibliography on parallel and vector numerical algorithms

This is a bibliography of numerical methods. It also includes a number of other references on machine architecture, programming language, and other topics of interest to scientific computing. Certain conference proceedings and anthologies which have been published in book form are listed also

Solution of partial differential equations on vector and parallel computers

The present status of numerical methods for partial differential equations on vector and parallel computers was reviewed. The relevant aspects of these computers are discussed and a brief review of their development is included, with particular attention paid to those characteristics that influence algorithm selection. Both direct and iterative methods are given for elliptic equations as well as explicit and implicit methods for initial boundary value problems. The intent is to point out attractive methods as well as areas where this class of computer architecture cannot be fully utilized because of either hardware restrictions or the lack of adequate algorithms. Application areas utilizing these computers are briefly discussed

Analyzing the effect of local rounding error propagation on the maximal attainable accuracy of the pipelined Conjugate Gradient method

Pipelined Krylov subspace methods typically offer improved strong scaling on parallel HPC hardware compared to standard Krylov subspace methods for large and sparse linear systems. In pipelined methods the traditional synchronization bottleneck is mitigated by overlapping time-consuming global communications with useful computations. However, to achieve this communication hiding strategy, pipelined methods introduce additional recurrence relations for a number of auxiliary variables that are required to update the approximate solution. This paper aims at studying the influence of local rounding errors that are introduced by the additional recurrences in the pipelined Conjugate Gradient method. Specifically, we analyze the impact of local round-off effects on the attainable accuracy of the pipelined CG algorithm and compare to the traditional CG method. Furthermore, we estimate the gap between the true residual and the recursively computed residual used in the algorithm. Based on this estimate we suggest an automated residual replacement strategy to reduce the loss of attainable accuracy on the final iterative solution. The resulting pipelined CG method with residual replacement improves the maximal attainable accuracy of pipelined CG, while maintaining the efficient parallel performance of the pipelined method. This conclusion is substantiated by numerical results for a variety of benchmark problems.Comment: 26 pages, 6 figures, 2 tables, 4 algorithm

Preliminary Report on High-Performance Computational Structures for Robot Control

In this report we present some initial results of our work completed thus far on Computational Structures for Robot Control . A SIMD architecture with the crossbar interprocessor network which achieves the parallel processing execution time lower bound of o( [a1n ]), where a1 is a constant and n is the number of manipulator joints, for the computation of the inverse dynamics problem, is discussed. A novel SIMD task scheduling algorithm that optimizes the parallel processing performance on the indicated architecture is also delineated. Simulations performed on this architecture show speedup factor of 3.4 over previous related work completed for the evaluation of the specified problem, is achieved. Parallel processing of PUMA forward and inverse kinematics solutions is next investigated using a particular scheduling algorithm. In addition, a custom bit-serial array architecture is designed for the computation of the inverse dynamics problem within the bit-serial execution time lower bound of o(c1k + c2kn), where c1 and c2 are specified constants, k is the word length, and n is the number of manipulator joints. Finally, mapping of the Newton-Euler equations onto a fixed systolic array is investigated. A balanced architecture for the inverse dynamics problem which achieves the systolic execution time lower bound for the specified problem is depicted. Please note again that these results are only preliminary and improvements to our algorithms and architectures are currently still being made