Explicit preconditioned conjugate gradient schemes for solving biharmonic equations

A new class of approximate inverse matrix techniques based on the concept of sparse LU-type factorization procedures is introduced for computing explicitly inverses of arrowhead matrices without inverting the decomposition factors. Explicit preconditioned iterative schemes in conjunction with AIM techniques are presented for the efficient solution of linear systems. Applications of the method on a linear system are discussed and numerical results are given.

A new class of normalized explicit approximate inverse matrix techniques, based on normalized approximate factorization procedures, for solving sparse linear systems resulting from the finite difference discretization of partial differential equations in three space variables are introduced. A new parallel normalized explicit preconditioned conjugate gradient square method in conjunction with normalized approximate inverse matrix techniques for solving efficiently sparse linear systems on distributed memory systems, using Message Passing Interface (MPI) communication library, is also presented along with theoretical estimates on speedups and efficiency. The implementation and performance on a distributed memory MIMD machine, using Message Passing Interface (MPI) is also investigated. Applications on characteristic initial/boundary value problems in three dimensions are discussed and numerical results are given.

A class of Generalized Approximate Inverse Matrix (GAIM) techniques, based on the concept of LU-sparse factorization procedures, is introduced for computing explicitly approximate inverses of large sparse unsymmetric matrices of irregular structure, without inverting the decomposition factors. Explicit preconditioned iterative methods, in conjunction with modified forms of the GAIM techniques, are presented for solving numerically initial/boundary value problems on multiprocessor systems. Application of the new methods on linear boundary-value problems is discussed and numerical results are given.

PurposeTo propose novel parallel/distributed normalized explicit finite element (FE) approximate inverse preconditioning for solving sparse FE linear systems.Design/methodology/approachThe design of suitable methods was the main objective for which several families of the normalized approximate inverse, based on sparse normalized approximate factorization, are produced. The main motive for the derivation of the new normalized approximate inverse FE matrix algorithmic techniques is that they can be efficiently used in conjunction with normalized explicit preconditioned conjugate gradient (NEPCG) – type schemes on parallel and distributed systems. Theoretical estimates on the rate of convergence and computational complexity of the NEPCG method are also derived.FindingsApplication of the proposed method on a three‐dimensional boundary value problem is discussed and numerical results for uniprocessor systems along with speed‐ups and efficiency for multicomputer systems are given. These results tend to become optimum, which are in qualitative agreement with the theoretical results presented for uniprocessor and distributed memory systems, using message passing interface (MPI) communication library.Research limitations/implicationsFurther parallel algorithmic techniques will be investigated in order to improve the speed‐ups and the computational complexity of the parallel normalized explicit approximate inverse preconditioning.Originality/valueThe proposed parallel/distributed normalized explicit approximate inverse preconditioning, using approximate factorization and approximate inverse algorithms, is an efficient computational method that is valuable for computer scientists and for scientists and engineers in engineering computations.

During the last decades, explicit approximate inverse preconditioning methods have been used for efficiently solving sparse linear systems on multiprocessor systems. The effectiveness of explicit approximate inverse preconditioning schemes relies on the use of efficient preconditioners that are close approximants to the coefficient matrix and are fast to compute in parallel. A new parallel computational technique is proposed for the parallelization of the explicit preconditioned conjugate gradient type method on a Graphics Processing Unit (GPU). The proposed parallel methods have been implemented using Compute Unified Device Architecture (CUDA) developed by NVIDIA. The inherently parallel operations between vectors and matrices involved in the explicit preconditioned biconjugate conjugate gradient type schemes exhibit significant amounts of loop-level parallelism because of the matrix–vector and the vector–vector products that can lead to high performance gain on the GPU systems, specifically designed for such computations. Finally, numerical results for the performance of the explicit preconditioned biconjugate conjugate gradient type method for solving characteristic two dimensional boundary value problems, using the finite difference method, on a massive multiprocessor interface on a GPU are presented. The CUDA implementation issues of the proposed method are also discussed.

An experimental study of γ-ray attenuation in polyethylene—Lead shields

PurposeThe purpose of this paper is to introduce efficient methods for solving the 2D biharmonic equation with Dirichlet boundary conditions of second kind. This equation describes the deflection of loaded plate with boundary conditions of simply supported plate kind. Also it can be derived from the calculus of variations combined with the variational principle of minimum potential energy. Because of existing fourth derivatives in this equation, introducing high‐order accurate methods need to use artificial points. Also solving the resulted linear system of equations suffers from slow convergence when iterative methods are used. This paper aims to introduce efficient methods to overcome these problems.Design/methodology/approachThe paper considers several compact finite difference approximations that are derived on a nine‐point compact stencil using the values of the solution and its second derivatives as the unknowns. In these approximations there is no need to define special formulas near the boundaries and boundary conditions can be incorporated with these techniques. Several iterative linear systems solvers such as Krylov subspace and multigrid methods and their combination (with suitable preconditioner) have been developed to compare the efficiency of each method and to design powerful solvers.FindingsThe paper finds that the combination of compact finite difference schemes with multigrid method and Krylov iteration methods preconditioned by multigrid have excellent results for the second biharmonic equation, and that Krylov iteration methods preconditioned by multigrid are the most efficient methods.Originality/valueThe paper is of value in presenting, via some tables and figures, some numerical experiments which resulted from applying new methods on several test problems, and making comparison with conventional methods.

An improved iterative finite element solution for pin joints

A novel sparse matrix technique for the numerical analysis of semiconductor devices and its algorithms are presented. Storage scheme and calculation procedure, of the sparse matrix are described in detail. The sparse matrix technique in the device simulation can decrease, storage greatly with less CPU time and its implementation is very easy. Some algorithms and calculation examples to show the time and space characteristics of the sparse matrix are given.

A novel approximate technique is proposed for the estimation of call blocking probabilities in cellular mobile telephone networks where call blocking triggers customer retrials. The approximate analysis technique is based on Markovian models with state spaces whose cardinalities are proportional to the maximum number of calls that can be simultaneously in progress within cells. The accuracy of the approximate technique is assessed by comparison against results of detailed simulation experiments. Numerical results show that the proposed approximate technique is very accurate, in spite of the remarkably small state spaces of the Markovian models.

A sparse stiff chemistry solver based on dynamic adaptive integration for efficient combustion simulations

The problem of solving linear systems is one of the most fundamental problems in computer science, where given a satisfiable linear system (A,b), for A ∈ ℝ^{n×n} and b ∈ ℝⁿ, we wish to find a vector x ∈ ℝⁿ such that Ax = b. The current best algorithms for solving dense linear systems reduce the problem to matrix multiplication, and run in time O(n^ω). We consider the problem of finding ε-approximate solutions to linear systems with respect to the L₂-norm, that is, given a satisfiable linear system (A ∈ ℝ^{n×n}, b ∈ ℝⁿ), find an x ∈ ℝⁿ such that ||Ax - b||₂ ≤ ε||b||₂. Our main result is a fine-grained reduction from computing the rank of a matrix to finding ε-approximate solutions to linear systems. In particular, if the best known Õ(n^ω) time algorithm for computing the rank of n × O(n) matrices is optimal (which we conjecture is true), then finding an ε-approximate solution to a dense linear system also requires Ω̃(n^ω) time, even for ε as large as (1 - 1/poly(n)). We also prove (under some modified conjectures for the rank-finding problem) optimal hardness of approximation for sparse linear systems, linear systems over positive semidefinite matrices and well-conditioned linear systems. At the heart of our results is a novel reduction from the rank problem to a decision version of the approximate linear systems problem. This reduction preserves properties such as matrix sparsity and bit complexity.

A new compact 2-D full-wave algorithm based on the Yee's grid, is developed for propagation constant computation. Unlike other 2-D Yee's grid based techniques (which require an initial input parameter of phase constant value in order to extract the resonant domain information), this computes the propagation constant directly from a given frequency. The technique has no stability problems and offers potential fast computation speed and better accuracy over the existing 2D time-domain approaches. Sparse matrix technique is used to minimize the memory requirement and CPU time. Numerical results are presented and are compared to other literatures.

In this paper, we describe scalable parallel algorithms for symmetric sparse matrix factorization, analyze their performance and scalability, and present experimental results for up to 1,024 processors on a Gray T3D parallel computer. Through our analysis and experimental results, we demonstrate that our algorithms substantially improve the state of the art in parallel direct solution of sparse linear systems-both in terms of scalability and overall performance. It is a well known fact that dense matrix factorization scales well and can be implemented efficiently on parallel computers. In this paper, we present the first algorithms to factor a wide class of sparse matrices (including those arising from two- and three-dimensional finite element problems) that are asymptotically as scalable as dense matrix factorization algorithms on a variety of parallel architectures. Our algorithms incur less communication overhead and are more scalable than any previously known parallel formulation of sparse matrix factorization. Although, in this paper, we discuss Cholesky factorization of symmetric positive definite matrices, the algorithms can be adapted for solving sparse linear least squares problems and for Gaussian elimination of diagonally dominant matrices that are almost symmetric in structure. An implementation of one of our sparse Cholesky factorization algorithms delivers up to 20 GFlops on a Gray T3D for medium-size structural engineering and linear programming problems. To the best of our knowledge, this is the highest performance ever obtained for sparse Cholesky factorization on any supercomputer.

Influence of experimental design on hyperparameter estimates precision when performing a population pharmacokinetic-pharmacodynamic (PK-PD) analysis has been shown by several studies and various approaches have been proposed for optimizing or evaluating such designs. Some of these methods rely on the optimization of a suitable scalar function of the population information matrix. Unfortunately for the nonlinear models encountered in pharmacokinetics or pharmacodynamics the latter is particularly difficult to evaluate. Under some assumptions and after a linearization of the PK-PD model a closed form of this matrix can be obtained which considerably simplifies its calculation but leads to an approximation. The aim of this paper is to evaluate the quality of the latter and its potential impact, when comparing or optimizing population designs and to relate it to Bates and Watts curvature measures. Two models commonly used in PK-PD were considered and nominal hyperparameter values when chosen for each one. Several population designs were studied and the associated population information matrix was computed for each using the approximate procedure and also using a reference method. Design optimizations were calculated under constraints for each model from the reference and approximate population information matrix. Nonlinearity curvatures were also computed for every model and design. The impact of model linearization when calculating the population information matrix was then examined in terms of lower bound accuracies on the hyperparameter estimates, design criterion variation, as well as D-optimal population designs, these results being related to nonlinearity curvature measures. Our results emphasize the influence of the parameter effects curvature when deriving the lower bounds of the hyperparameter estimates precision for a given design from the approximate population information matrix especially for hyperparameters quantifying the PK-PD interindividual variability. No discrepancies were detected between the population D-optimal designs obtained from the approximate and reference matrix despite some minor differences in criterion variation with respect to the design. More pronounced differences were, however, observed when comparing the amplitudes of criterion variation which can lead to errors when calculating design efficiencies. From a practical point of view, a strategy easily applicable by the pharmacokineticist for avoiding such problems in the context of population design optimization or comparison is then proposed.