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In multiple areas of image processing, such as Computed Tomography, in which data acquisition is based on counting particles that hit a detector surface, Poisson noise occurs. Using variance-stabilizing transformations, the Poisson noise can be approximated by a Gaussian one, for which classical denoising filters can be used. This paper presents an experimental performance study of a parallel implementation of the Poissonian image restoration algorithm, introduced in Harizanov etźal. (2013). Hybrid parallelization based on MPI and OpenMP standards is investigated. The convergence rate of the algorithm heavily depends on both the image size and the choice of input parameters ( ź , ź ) , thus maximizing its parallel efficiency is vital for real-life applications. The implementation is tested for high-resolution radiographic images, on Linux clusters with Intel processors and on an IBM supercomputer.

The numerical solution of 3D linear elasticity equations is considered. The problem is described by a coupled system of second-order elliptic partial differential equations. This system is discretized by trilinear brick finite elements. The PCG iterative method is used for solving the large-scale linear algebraic systems arising after the FEM discretization of the problem. Displacement decomposition technique is applied at the first step to construct a preconditioner using the decoupled block-diagonal part of the original matrix. Then circulant block-factorization is used for preconditioning of the obtained block-diagonal matrix. New construction of a parallel algorithm for the discussed preconditioning method is proposed. The theoretical part of this study includes analysis of the execution time on various parallel architectures and asymptotic estimates of the parallel speedup and the parallel efficiency. The parallel performance estimates indicate that the proposed algorithm will be especially efficient on coarse-grain parallel systems, which is also confirmed by the numerical experiments. A portable MPI parallel code is developed. Numerical tests on three symmetric multiprocessor systems: SUN Enterprise 3000, SUN SPARCstation 10 and Origin 2000 are presented. The reported speedup and parallel efficiency illustrate well the features of the proposed method and its implementation.

We consider the time dependent Stokes equation on a finite time interval and on a uniform rectangular mesh, written in terms of velocity and pressure. In a parallel algorithm, based on a new direction splitting approach, the pressure equation is derived from a perturbed form of the continuity equation, in which the incompressibility constraint is penalized in a negative norm induced by the direction splitting. The scheme used in the algorithm is composed of: pressure prediction, velocity update, penalty step, and pressure correction. In order to achieve good parallel performance, the solution of the Poison problem for the pressure correction is replaced by solving a sequence of one-dimensional second order elliptic boundary value problems in each spatial direction. The parallel code was developed using MPI and tested on modern computer systems. The performed numerical tests illustrate the parallel efficiency, and the scalability, of the direction-splitting based algorithm.

New circulant block-factorization preconditioners are introduced and studied. The general approach is first formulated for the case of block tridiagonal sparse matrices. Then estimates of the relative condition number for a model Dirichlet boundary value problem are derived. In the case ofy-periodic problems the circulant block-factorization preconditioner is shown to give an optimal convergence rate. Finally, using a proper imbedding of the original Dirichlet boundary value problem to ay-periodic one a preconditioner of optimal convergence rate for the general case is obtained. The total computational cost of the preconditioner isO (N logN) (based on FFT), whereN is the number of unknowns. That is, the algorithm is nearly optimal. Various numerical tests that demonstrate the features of the circulant block-factorization preconditioners are presented.

We consider the b-adic diaphony as a tool to measure the uniform distribution of sequences, as well as to investigate pseudo-random properties of sequences. The study of pseudo-random properties of uniformly distributed nets is extremely important for quasi-Monte Carlo integration. It is known that the error of the quasi-Monte Carlo integration depends on the distribution of the points of the net. On the other hand, the b-adic diaphony gives information about the points distribution of the net. Several particular constructions of sequences (xi) are considered. The b-adic diaphony of the two dimensional nets {yi = (xi, xi+1)} is calculated numerically. The numerical results show that if the two dimensional net {yi} is uniformly distributed and the sequence (xi) has good pseudorandom properties, then the value of the b-adic diaphony decreases with the increase of the number of the points. The analysis of the results shows a direct relation between pseudo-randomness of the points of the constructed sequences and nets and the b-adic diaphony as well as the discrepancy.

We consider the numerical solution of 3D linear elasticity equations. The investigated problem is described by a coupled system of second order elliptic partial differential equations. This system is then discretized by conforming or nonconforming finite elements. After applying the Finite Element Method (FEM) based discretization, a system of linear algebraic equations has to be solved. In this system the stiffness matrix is large, sparse and symmetric positive definite. In the solution process we utilize a well-known fact that the preconditioned conjugate gradient method is the best tool for efficient solution of large-scale symmetric systems with sparse positive definite matrices. In this context, the displacement decomposition (DD) technique is applied at the first step to construct a preconditioner that is based on a decoupled block diagonal part of the original matrix. Then two preconditioners, namely the Modified Incomplete Cholesky factorization MIC(0) and the Circulant Block-Factorization (CBF) preconditioning, are used to precondition thus obtained block diagonal matrix. As far as the parallel implementation of the proposed solution methods is concerned, we utilize the Message Passing Interface (MPI) communication libraries. The aim of our work is to compare the performance of the two proposed preconditioners: the DD MIC(0) and the DD CBF. The presented comparative analysis is based on the execution times of actual codes run on modern parallel computers. Performed numerical tests demonstrate the level of parallel efficiency and robustness of the proposed algorithms. Furthermore, we discuss the number of iterations resulting from utilization of both preconditioners.

In this paper we consider numerical solution of 3D linear elasticity equations described by a coupled system of second order elliptic partial differential equations. This system is discretized by trilinear parallelepipedal finite elements. Preconditioned Conjugate Gradient iterative method is used for solving large-scale linear algebraic systems arising after the Finite Element Method (FEM) discretization of the problem. The displacement decomposition technique is applied at the first step to construct a preconditioner using the decoupled block diagonal part of the original matrix. Then circulant block factorization is used to precondition thus obtained block diagonal matrix. Since both preconditioning techniques, displacement decomposition and circulant block factorization, are highly parallelizable, a portable parallel FEM code utilizing MPI for communication is implemented. Results of numerical tests performed on a number of modern parallel computers using real life engineering problems from the geosciences (geomechanics in particular) are reported and discussed.

The recently introduced circulant block-factorization preconditioners are studied in this paper. The general approach is first formulated for the case of block-tridiagonal sparse matrices. Then an estimate of the condition number of the preconditioned matrix for a model anisotropic Dirichlet boundary value problem is derived in the formκ

Numerical solution of 3D linear elasticity equations is considered. Problem is described by a coupled system of second order elliptic partial differential equations. This system is discretized by trilinear parallelepipedal finite elements. Preconditioned Conjugate Gradient iterative method is used for solving large-scale linear algebraic systems arising after the Finite Element Method (FEM) discretization of the problem. The displacement decomposition technique is applied at the first step to construct a preconditioner using the decoupled block- diagonal part of the original matrix. Then circulant block- factorization is used to precondition thus obtained block- diagonal matrix. Since both preconditioning techniques, displacement decomposition and circulant block- factorization, are highly parallelizable, a portable parallel FEM code based on MPI is developed. Results of numerical tests performed on a number of modern parallel computers using real-life engineering problems from the geomechanics in geosciences are reported and discussed.

The numerical solution of 3D linear elasticity equations is considered. The problem is described by a coupled system of second-order elliptic partial differential equations. This system is discretized by trilinear parallelepipedal finite elements.The preconditioned conjugate gradient iterative method is used for solving of the large-scale linear algebraic systems arising after the finite element method (FEM) discretization of the problem. Displacement decomposition technique is applied at the first step to construct a preconditioner using the decoupled block-diagonal part of the original matrix. Then circulant block-factorization is used for preconditioning of the obtained block-diagonal matrix. Both techniques, displacement decomposition and circulant block-factorization, are highly parallelizable.A parallel algorithm is invented for the proposed preconditioner. The theoretical analysis of the execution time shows that the algorithm is highly efficient for coarse-grain parallel computer systems.A portable MPI parallel FEM code is developed. Numerical tests for real-life engineering problems of the geomechanics in geosciences on a number of modern parallel computers are presented. The reported speed-up and parallel efficiency well illustrate the parallel features of the proposed method and its implementation.

The robustness of the recently introduced circulant blockfactorization (CBF) preconditioners is studied in the case of finite element matrices arising from the discretization of the 2D Navier equations of elasticity. Conforming triangle finite elements are used for the numerical solution of the differential problem. The proposed preconditioner M C is constructed by CBF approximation of the block-diagonal part of the stiffness matrix. In other words, we implement in our algorithm the circulant block-factorization into the framework of the displacement decomposition technique. The estimate κ(M C −1 K)=O(√N/1−v) is proved asymptotically on N, where N is the size of the discrete problem. Note, that the corresponding known estimate for the widely used incomplete factorization displacement decomposition preconditioner M ILU is κ(M ILU −1 K)=O(√N/1−v)

The numerical solution of systems of convection-diffusion equations is considered. The problem is described by a system of second order partial differential equations (PDEs). This system is discretized by Courant-elements. The preconditioned conjugate gradient method is used for the iterative solution of the large-scale linear algebraic systems arising after the finite element discretization of the problem. Discrete Helmholtz preconditioners are applied to obtain a mesh independent superlinear convergence of the iterative method. A parallel algorithm is derived for the proposed preconditioner. A portable parallel code using Message Passing Interface (MPI) is developed. Numerical tests well illustrate the performance of the proposed method on a parallel computer architecture.

The numerical solution of 3D linear elasticity equations is considered. The problem is described by a coupled system of second order elliptic partial differential equations. This system is discretized by trilinear parallelepipedal finite elements.The Preconditioned Conjugate Gradient iterative method is used for solving of the large-scale linear algebraic systems arising after the Finite Element Method (FEM) discretization of the problem. Displacement decomposition technique is applied at the first step to construct a preconditioner using the decoupled block-diagonal part of the original matrix. Then circulant block-factorization is used for preconditioning of the obtained block-diagonal matrix. Both preconditioning techniques, displacement decomposition and circulant block-factorization, are highly parallelizable.A parallel algorithm is invented for the proposed preconditioner. The theoretical analysis of the execution time shows that the algorithm is highly efficient for coarse-grain parallel computer systems.A portable parallel FEM code based on MPI is developed. Numerical tests for real-life engineering problems in computational geomechanics are performed on a number of modern parallel computers: Cray T3E, Sunfire 6800, and Beowulf cluster. The reported speed-up and parallel efficiency well illustrate the parallel features of the proposed method and its implementation.