An FPGA based floating point Gauss-Seidel iterative solver

It is known that the main computational costs in the FEM implementation are computing the global system of linear equations (material response computation and stiffness matrix ensemble) and its solution. With the aim of speeding up the total solving boundary value problem time with the help of the developed solver, algorithms of the parallel stiffness matrix ensemble and the system of linear equations solution were implemented. In this paper the results of using GPU to accelerate computations related to the solution of direct boundary value problems of crystal plasticity are present. The existing algorithms for solving the sparse systems of linear equations and parallel ensemble of the global stiffness matrix were implemented on both the CPU and GPU. As a result, a total acceleration of up to 10 times was obtained compared to the software implementation on the CPU.

The efficiency of numerical simulation is of great importance for the development of tight oil reservoirs. In this paper, a nonlinear seepage model of tight oil reservoirs was presented, and the physical model of multi-fractured horizontal well is meshed by unstructured triangular grids, and the mathematical model is discretized by finite element method. Then, a fully implicit iterative solution scheme was derived. A novel algebraic multigrid method (AMG-CG) was proposed to solve the large sparse linear equation system during the iteration process. In order to demonstrate the computational efficiency of the novel algebraic multigrid method, a comparison was made between the novel method and others. The results reveal that the AMG-CG method has a faster calculation speed in solving linear equation systems. The calculation time of the AMG-CG method is 1029 s, the calculation time of the AMG method is 2029 s, the calculation time of the generalized minimum residual method (GMRES) method is 3709 s, the calculation time of the Gauss–Seidel (GS) method is 8115 s. The calculation speed of the novel method is almost 7.88 times faster than the Gauss–Seidel method, the novel method is more effective in the numerical simulation of tight oil.

Application of modified iterative method to simulate rainfall infiltration in unsaturated soils

The DEFLATED-GMRES(m, k) method is one of the major iterative solvers for the large sparse linear systems of equations, x = b. This algorithm assembles a preconditioner adaptively for the GMRES(m) method based on eigencomponents gathered from the Arnoldi process during iterations. It is usually known that if a restarted GMRES(m) method is used to solve linear systems of equations, the information of the smallest eigencomponents is lost at each restart and the super-linear convergence may also be lost. In this paper, we propose an adaptive procedure that combines the DEFLATED-GMRES(m, k) algorithm and the determination of a restart frequency automatically. It is shown that a new algorithm combining elements of both will reduce the negative effects of the restarted procedure. The numerical experiments are presented on three test problems by using the MIMD parallel machine AP3000. From these numerical results, we show that the proposed algorithm leads to faster convergence than the conventional DEFLATED-GMRES(m, k) method. Copyright © 2000 John Wiley & Sons, Ltd.

Efficient SAT encoding scheme for schedulability analysis of non-preemptive tasks on multiple computational resources

The polarizable continuum model (PCM) is a well established method for computing solvation effects. Its attractiveness comes from the fact that the solution is modelled as a continuum instead of modelling each atom individually. This allows for more complex simulations. Nevertheless, there are still a couple of challenges to face. One of them being the computational time required at the limit of large systems, or when high accuracy is needed. The wavelet boundary element method can be used to overcome these problems, provided a reliable cavity generator is used. The PCM calculates the molecular free energy in solution as the sum of electrostatic, dispersion-repulsion and the cavitation energy by solving the underlying partial differential equations. Those differential equations can be transformed into integral equations, solely defined on the boundary of the molecular cavity, by applying the integral equation formalism (IEFPCM). The integral equations can then be discretized using the wavelet boundary element method. The resulting sparse system of linear equations can be solved reliably by iterative solvers, leading to high accuracy solutions even for large systems. The challenge of this approach lies in the generation and refinement of the molecular cavities on which the interactions take place. Since wavelets are defined as a tensor product of one-dimensional functions, the wavelet boundary element method needs a discretization into quadrangular patches of the molecular cavity. This feature is missing in common commercial mesh generation tools, which focus on generating triangular meshes. Making use of characteristic functions for defining which points are inside the cavity, a model for generating the molecular surface independently of its exact description is achieved. We present here a way for generating triangular meshes which are subsequently merged into quadrangular ones. We apply geometrical quantities that measure the fitness of the shapes involved and improve the position of individual points iteratively. Thereby, we generate quadrangular meshes with an unprecedented quality, which ultimately can be refined in a hierarchical manner for the use of wavelets. Overall, this method can generate reliable parametrizations on a variety of smooth cavities. With the help of the wavelet boundary element method, the PCM equations can thus be solved with high accuracy also on large systems.

Element-by-element iterative solution methods have been shown to be effective for solving large, sparse linear systems of equations with a positive definite stiffness matrix. In this paper the iterative method GMRES will be described. This solution method is also suited for solving nonsymmetric systems of equations. As acceleration technique an element-by-element preconditioner has been employed. Attention is focused mainly on the convergence behaviour of the method. The calculations described in this paper have been performed with the finite element package DIANA.

This paper presents a novel hardware solution for solving sparse matrices using emulation technology. The proposed solution introduces a bio-inspired technique, which is Genetic Algorithm (GA) to solve the sparse system of linear equations (SLEs). The proposed solution can be used with any physical system that can be modeled as SLEs. Computations involved in Finite Element Method (FEM) consume too much time, which affects the final time-to-market value. Profiling shows that the most time-consuming part in the simulation process is the solver part which is responsible for solving the resultant SLEs generated from the FEM. The total number of equations may reach thousands or millions of linear equations. Hardware implementation of GA and making a good use of parallelism in implementing the architecture accelerate the time taken by the solver part. Solving the system by classical numerical methods, such as conjugate gradient (CG) implemented in software is performed sequentially, here the architecture allows the parallelism in performing GA operators, such as selection, crossover, and mutation. The parallelism in the proposed architecture is the superiority point which we rely on. The results show that our proposed method to solve the SLEs outperforms the classical CG software implementation on run time and number of equations.

Iterative solvers allow for a trade-off between speed and accuracy. We propose an iterative method for the estimation of the internal states of a given discrete-time linear state-space model from a series of noisy measurements. In particular we identify the MAP estimate of those states as being the solution of a sparse system of linear equations and derive an iterative solver based on the conjugate gradient method. We derive convergence results to quantify the trade-off between speed and accuracy and finally apply the method to channel estimation where it is shown to outperform Kalman smoothing complexity-wise.

We present a parallel domain decomposition method based on a hybrid direct-iterative solver for 3D frequency-domain modelling of visco-acoustic waves. Frequency-domain seismic modelling reduces to the solution of a large and sparse system of linear equations for each frequency. The hybrid directiterative approach aims to overcome the memory requirement and the limited scalability of direct-solver approaches, on the one hand, and to iteratively solve better-conditioned system than in global iterative approaches on the other hand. The domain decomposition is based upon the algebraic Schur complement method. The reduced Schur complement system is solved with the global minimum residual method (GMRES) and is preconditioned by an algebraic additive Schwarz preconditioner. The MUMPS direct solver is used to factorize local impedance matrices defined on each subdomain. Simulations in the salt models for frequencies up to 12.5 Hz show that the number of iterations increases linearly with frequencies when the grid interval is matched to the frequency and the size of the subdomains is kept constant over frequencies. This makes the time complexity of the hybrid approach similar to that of global iterative solvers. Possible improvements of the method for multi-source simulation involve the use of block iterative solver and two levels of parallelism.

Sparse systems of linear equations and eigen-equations arise at the heart of many large-scale, vital simulations in DOE. Examples include the Accelerator Science and Technology SciDAC (Omega3P code, electromagnetic problem), the Center for Extended Magnetohydrodynamic Modeling SciDAC (NIMROD and M3D-C1 codes, fusion plasma simulation). The Terascale Optimal PDE Simulations (TOPS) is providing high-performance sparse direct solvers, which have had significant impacts on these applications. Over the past several years, we have been working closely with the other SciDAC teams to solve their large, sparse matrix problems arising from discretization of the partial differential equations. Most of these systems are very ill-conditioned, resulting in extremely poor convergence (sometimes no convergence) for many iterative solvers. we have successfully deployed our direct methods techniques in these applications, which achieved significant scientific results as well as performance gains. These successes were made possible through the SciDAC model of computer scientists and application scientists working together to take full advantage of terascale computing systems and new algorithms research.

During the past few years, algorithmic improvements alone have reduced the time required for the direct solution of unsymmetric sparse systems of linear equations by almost an order of magnitude. This paper compares the performance of some well-known software packages for solving general sparse systems. In particular, it demonstrates the consistently high level of performance achieved by WSMP---the most recent of such solvers. It compares the various algorithmic components of these solvers and discusses their impact on solver performance. Our experiments show that the algorithmic choices made in WSMP enable it to run more than twice as fast as the best among similar solvers and that WSMP can factor some of the largest sparse matrices available from real applications in only a few seconds on a 4-CPU workstation. Thus, the combination of advances in hardware and algorithms makes it possible to solve those general sparse linear systems quickly and easily that might have been considered too large until recently.

In this research paper a new modification of Gauss-Seidel method has been presented for solving the system of linear algebraic equations. The systems of linear algebraic equations have an important role in the field of science and engineering. This modification has been developed by using the procedure of Gauss-Seidel method and the concept of substitution techniques. Developed modification of Gauss-Seidel method is a fast convergent as compared to Gauss Jacobi’s method, Gauss-Seidel method and successive over-relaxation (SOR) method. It works on the diagonally dominant as well as positive definite symmetric systems of linear algebraic equations. Its solution has been compared with the Gauss Jacobi’s method, Gauss-Seidel method and Successive over-Relaxation method by taking different systems of linear algebraic equations and found that, it was reducing to the number of iterations and errors in each problem.

Tasks such as classification, clustering, and regression require the identification and elimination of outliers as part of the preprocessing of data. Without an adequate processing of outliers the results of data analysis will be biased and inexact. This article proposes a distance-based method for the detection of outliers in multivariate datasets. The proposed method takes advantage of the principal components for avoiding problems of collinearity in datasets and the high concentration of variance is used to increase the separation between outliers and inliers. The proposed method was compared against four outlier detection methods from the literature, two deterministic and two stochastic. The datasets used in the comparison were generated according to different works in the literature. These datasets follows different distributions and contains different amount of outliers and inliers, and different number of variables and instances. Per each distribution, low and high dimensionality were considered. Unlike to other methods in the state-of-the-art, the proposed method does not require apriori the definition of any value for its operation. Also, the calculation of distances of elements in the dataset takes lower processing time. According to the experiments, the proposed method is suitable to deal with dataset contaminated with low and high proportions of outliers, low and high dimensions, and simmetric and assimetric distributions. Also supports colineality in data.

We present a new family of sums of squares (SOS) relaxations to cones of positive polynomials. The SOS relaxations employed in the literature are cones of polynomials which can be represented as ratios, with an SOS as numerator and a fixed positive polynomial as denominator. We employ nonlinear transformations of the arguments instead. A fixed cone of positive polynomials, considered as a subset in an abstract coefficient space, corresponds to an infinite, partially ordered set of concrete cones of positive polynomials of different degrees and in a different number of variables. To each such concrete cone corresponds its own SOS cone, leading to a hierarchy of increasingly tighter SOS relaxations for the abstract cone.