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

Tensor Computation: A New Framework for High-Dimensional Problems in EDA

Many critical EDA problems suffer from the curse of dimensionality, i.e. the very fast-scaling computational burden produced by large number of parameters and/or unknown variables. This phenomenon may be caused by multiple spatial or temporal factors (e.g. 3-D field solvers discretizations and multi-rate circuit simulation), nonlinearity of devices and circuits, large number of design or optimization parameters (e.g. full-chip routing/placement and circuit sizing), or extensive process variations (e.g. variability/reliability analysis and design for manufacturability). The computational challenges generated by such high dimensional problems are generally hard to handle efficiently with traditional EDA core algorithms that are based on matrix and vector computation. This paper presents "tensor computation" as an alternative general framework for the development of efficient EDA algorithms and tools. A tensor is a high-dimensional generalization of a matrix and a vector, and is a natural choice for both storing and solving efficiently high-dimensional EDA problems. This paper gives a basic tutorial on tensors, demonstrates some recent examples of EDA applications (e.g., nonlinear circuit modeling and high-dimensional uncertainty quantification), and suggests further open EDA problems where the use of tensor computation could be of advantage.Comment: 14 figures. Accepted by IEEE Trans. CAD of Integrated Circuits and System

Paraiso : An Automated Tuning Framework for Explicit Solvers of Partial Differential Equations

We propose Paraiso, a domain specific language embedded in functional programming language Haskell, for automated tuning of explicit solvers of partial differential equations (PDEs) on GPUs as well as multicore CPUs. In Paraiso, one can describe PDE solving algorithms succinctly using tensor equations notation. Hydrodynamic properties, interpolation methods and other building blocks are described in abstract, modular, re-usable and combinable forms, which lets us generate versatile solvers from little set of Paraiso source codes. We demonstrate Paraiso by implementing a compressive hydrodynamics solver. A single source code less than 500 lines can be used to generate solvers of arbitrary dimensions, for both multicore CPUs and GPUs. We demonstrate both manual annotation based tuning and evolutionary computing based automated tuning of the program.Comment: 52 pages, 14 figures, accepted for publications in Computational Science and Discover

Random walk on fixed spheres for Laplace and Lamé equations

The Random Walk on Fixed Spheres (RWFS) introduced in our previous paper is presented in details for Laplace and Lam'e equations governing static elasticity problems. The approach is based on the Poisson type integral formulae written for each disc of a domain consisting of a family of overlapping discs. The original differential boundary value problem is equivalently reformulated in the form of a system of integral equations defined on the intersection surfaces (arches, in 2D, and caps, if generalized to 3D spheres). To solve the obtained system of integral equations, a Random Walk procedure is constructed where the random walks are living on the intersecting surfaces. Since the spheres are fixed, it is convenient to construct also discrete random walk methods for solving the system of linear equations approximating the system of integral equations. We develop here two classes of special Monte Carlo iterative methods for solving these systems of linear algebraic equations which are constructed as a kind of randomized versions of the Chebyshev iteration method and Successive Over Relaxation (SOR) method. It is found that in this class of randomized SOR methods, the Gauss-Seidel method has a minimal variance. In our prevoius paper we have concluded that in the case of classical potential theory, the Random Walk on Fixed Spheres considerably improves the convergence rate of the standard Random Walk on Spheres method. More interesting, we succeeded there to extend the algorithm to the system of Lam'e equations which cannot be solved by the conventional Random Walk on Spheres method. We present here a series of numerical experiments for 2D domains consisting of 5, 10, and 17 discs, and analyze the dependence of the variance on the number of discs and elastic constants. Further generalizations to Neumann and Dirichlet-Neumann boundary conditions are also possible

Kinetics and scaling in ballistic annihilation

We study the simplest irreversible ballistically-controlled reaction, whereby particles having an initial continuous velocity distribution annihilate upon colliding. In the framework of the Boltzmann equation, expressions for the exponents characterizing the density and typical velocity decay are explicitly worked out in arbitrary dimension. These predictions are in excellent agreement with the complementary results of extensive Monte Carlo and Molecular Dynamics simulations. We finally discuss the definition of universality classes indexed by a continuous parameter for this far from equilibrium dynamics with no conservation laws

Random walk on fixed spheres method for electro- and elastostatics problems

Stochastic algorithms for solving Dirichlet boundary value problems for the Laplace and Lame equations governing 2D elasticity problems are developed. The approach presented is based on the Poisson integral formula written for each disc of a domain consisting of a family of overlapping discs. The original differential boundary value problem is reformulated in the form of equivalent system of integral equations defined on the intersection surfaces, i.e., arcs in 2D. A Random Walk algorithm can be applied then directly to the obtained system of integral equations where the random walks are living on the intersecting surfaces. We develop also a discrete random walk technique for solving the system of linear equations approximating the system of integral equations. We construct a randomized version of the successive over relaxation (SOR) method. In [6] we have demonstrated that in the case of classical potential theory our method considerably improves the convergence rate of the standard Random Walk on Spheres method. In this paper we extend the algorithm to the system of Lame equations which cannot be solved by the conventional Random Walk on Spheres method. Illustrating computations for 2D Laplace and Lame equations, and comparative analysis of different stochastic algorithms are presented

Automated Code Generation for Lattice Quantum Chromodynamics and beyond

We present here our ongoing work on a Domain Specific Language which aims to simplify Monte-Carlo simulations and measurements in the domain of Lattice Quantum Chromodynamics. The tool-chain, called Qiral, is used to produce high-performance OpenMP C code from LaTeX sources. We discuss conceptual issues and details of implementation and optimization. The comparison of the performance of the generated code to the well-established simulation software is also made

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