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This manual describes the use of PETSc for the numerical solution of partial differential equations and related problems on high-performance computers. The Portable, Extensible Toolkit for Scientific Computation (PETSc) is a suite of data structures and routines that provide the building blocks for the implementation of large-scale application codes on… (More)
We have divided this book into five main chapters. Chapter 1 gives the motivation for this book and the use of templates. Chapter 2 describes stationary and nonstationary iterative methods. In this chapter we present both historical development and state-of-the-art methods for solving some of the most challenging computational problems facing researchers.… (More)
— One of the main obstacles to the efficient solution of scientific problems is the problem of tuning software, both to the available architecture and to the user problem at hand. We describe approaches for obtaining tuned high-performance kernels, and for automatically choosing suitable algorithms. Specifically, we describe the generation of dense and… (More)
We survey the basic theory of the strengthened Cauchy-Buniakowskii-Schwarz inequality and its applications in multilevel methods for the solution of linear systems arising from nite element or nite diierence discretisation of elliptic partial diierential equations. Proofs are given both in a nite element context, and in purely algebraic form.
Nested recursivetwo-level factorizationmethodsfor nine-pointdiierencematricesare analyzed. Somewhat similar in construction to multilevel methods for nite element matrices, these methods use recursive red-black orderings of the meshes, approximating the nine-point stencils by ve-point ones in the red points and then forming the reduced system explicitly.… (More)
Description and comparison of several packages for the iterative solution of linear systems of equations.
We present computational aspects of mathematically equivalent implementations of the Conjugate Gradient method. Since the implementations have different performance characteristics, but compute the same quantities, an adaptive implementation can at run time pick the optimal choice.
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The challenge for the development of next generation software is the successful management of the complex grid environment while delivering to the scientist the full power of flexible compositions of the available al-gorithmic alternatives. Self-Adapting Numerical Software (SANS) systems are intended to meet this significant challenge. A SANS system… (More)
We present a method for automatically selecting optimal implementations of sparse matrix-vector operations. Our software 'AcCELS' (Accelerated Compress-storage Elements for Linear Solvers) involves a setup phase that probes machine characteristics, and a run-time phase where stored characteristics are combined with a measure of the actual sparse matrix to… (More)