Large Systems Of Linear Equations Research Articles

COMPOSED COEFFICIENT TECHNIQUE FOR MODELLING BARRETTE GROUPS

Most of soil structure interaction methods for analyzing large-section supports such as barrette foundation modeling and the surrounding soil are using 3D finite element (FE) models. In which, the model leads to a large finite element mesh, and consequently a large system of linear equations to be solved. In this paper, Composed Coefficient Technique (CCT) is adapted for analyzing barrette group. The technique considers the 3D full interactions between barrettes and the surrounding soil. Due to the high rigidity of the barrettes relative to the surrounding soil, a uniform settlement for the barrettes can be considered. This is done to compose the stiffness coefficients of the soil matrix into composed coefficients, which consequently leads to a significant reduction in the soil stiffness matrix. An application for analyzing barrette group by CCT technique is carried out on a real subsoil. The application presents guidelines and diagrams for barrette group that may be used in real practice.

Arnoldi decomposition, GMRES, and preconditioning for linear discrete ill-posed problems

GMRES is one of the most popular iterative methods for the solution of large linear systems of equations that arise from the discretization of linear well-posed problems, such as boundary value problems for elliptic partial differential equations. The method is also applied to the iterative solution of linear systems of equations that are obtained by discretizing linear ill-posed problems, such as many inverse problems. However, GMRES does not always perform well when applied to the latter kind of problems. This paper seeks to shed some light on reasons for the poor performance of GMRES in certain situations, and discusses some remedies based on specific kinds of preconditioning. The standard implementation of GMRES is based on the Arnoldi process, which also can be used to define a solution subspace for Tikhonov or TSVD regularization, giving rise to the Arnoldi–Tikhonov and Arnoldi-TSVD methods, respectively. The performance of the GMRES, the Arnoldi–Tikhonov, and the Arnoldi-TSVD methods is discussed. Numerical examples illustrate properties of these methods.

A new greedy Kaczmarz algorithm for the solution of very large linear systems

We propose a new greedy Kaczmarz algorithm for the solution of very large systems of linear equations. In our proposed algorithm, control sequence is determined by a greedy rule, and relaxation parameters are determined adaptively. Convergence of our proposed algorithm is proved. Numerical results show that the proposed algorithm is feasible and has faster convergence rate than the greedy randomized Kaczmarz algorithm.

Superfast Iterative Solvers for Linear Matrix Equations

Superfast algorithms for solving large systems of linear equations are developed on the basis of an original method for multistep decomposition of a linear multidimensional dynamical system. Examples of analytical synthesis of iterative solvers for matrices of the general form and for large numerical systems of linear algebraic equations are given. For the analytical case, it is shown that convergence occurs at the second iteration.

QR-Factorization Algorithm for Computed Tomography (CT): Comparison With FDK and Conjugate Gradient (CG) Algorithms

Even though QR-factorization of the system matrix for tomographic devices has been already used for medical imaging, to date, no satisfactory solution has been found for solving large linear systems, such as those used in computed tomography (CT) (in the order of $10^{6}$ equations). In CT, the Feldkamp, Davis, and Kress back projection algorithm (FDK) and iterative methods like conjugate gradient (CG) are the standard methods used for image reconstruction. As the image reconstruction problem can be modeled by a large linear system of equations, QR-factorization of the system matrix could be used to solve this system. Current advances in computer science enable the use of direct methods for solving such a large linear system. The QR-factorization is a numerically stable direct method for solving linear systems of equations, which is beginning to emerge as an alternative to traditional methods, bringing together the best from traditional methods. QR-factorization was chosen because the core of the algorithm, from the computational cost point of view, is precalculated and stored only once for a given CT system, and from then on, each image reconstruction only involves a backward substitution process and the product of a vector by a matrix. Image quality assessment was performed comparing contrast to noise ratio and noise power spectrum; performances regarding sharpness were evaluated by the reconstruction of small structures using data measured from a small animal 3-D CT. Comparisons of QR-factorization with FDK and CG methods show that QR-factorization is able to reconstruct more detailed images for a fixed voxel size.

Accelerating compressed sensing in parallel imaging reconstructions using an efficient circulant preconditioner for cartesian trajectories.

PurposeDesign of a preconditioner for fast and efficient parallel imaging (PI) and compressed sensing (CS) reconstructions for Cartesian trajectories.TheoryPI and CS reconstructions become time consuming when the problem size or the number of coils is large, due to the large linear system of equations that has to be solved in ℓ1 and ℓ2‐norm based reconstruction algorithms. Such linear systems can be solved efficiently using effective preconditioning techniques.MethodsIn this article we construct such a preconditioner by approximating the system matrix of the linear system, which comprises the data fidelity and includes total variation and wavelet regularization, by a matrix that is block circulant with circulant blocks. Due to this structure, the preconditioner can be constructed quickly and its inverse can be evaluated fast using only two fast Fourier transformations. We test the performance of the preconditioner for the conjugate gradient method as the linear solver, integrated into the well‐established Split Bregman algorithm.ResultsThe designed circulant preconditioner reduces the number of iterations required in the conjugate gradient method by almost a factor of 5. The speed up results in a total acceleration factor of approximately 2.5 for the entire reconstruction algorithm when implemented in MATLAB, while the initialization time of the preconditioner is negligible.ConclusionThe proposed preconditioner reduces the reconstruction time for PI and CS in a Split Bregman implementation without compromising reconstruction stability and can easily handle large systems since it is Fourier‐based, allowing for efficient computations.

MODELING SINGLE BARRETTES AS ELASTIC SUPPORT BY CCT

Classical analyses of barrette foundation taking into account full interactions between barrette and the surrounding soil leads to a huge barrette stiffness matrix. Consequently, a large system of linear equations must be solved, especially for analyzing barrette group and barrette raft. To overcome this problem, a Composed Coefficient Technique (CCT) is developed for analyzing barrette. In the analysis, the elasticity of the barrette body is considered using the finite element method, while that of the soil elements is considered using flexibility coefficients. The compatibility between the vertical displacements of the barrette and the soil settlements at the soil-barrette interface is taken in the vertical direction only. This assumption is that the external load on the barrette head, which is expected to be heavy load, is applied in the vertical direction. For comparative examinations, the barrette elasticity is determined using either 1D or 3D finite elements. A series of examinations is carried out to verify the application for analyzing barrette by CCT. It was found that, treating the barrette as an elastic body and representing the barrette by either 1D or 3D finite elements, gives nearly the same results.

MODELING SINGLE BARRETTES AS ELASTIC SUPPORT BY CCT

Classical analyses of barrette foundation taking into account full interactions between barrette and the surrounding soil leads to a huge barrette stiffness matrix. Consequently, a large system of linear equations must be solved, especially for analyzing barrette group and barrette raft. To overcome this problem, a Composed Coefficient Technique (CCT) is developed for analyzing barrette. In the analysis, the elasticity of the barrette body is considered using the finite element method, while that of the soil elements is considered using flexibility coefficients. The compatibility between the vertical displacements of the barrette and the soil settlements at the soil-barrette interface is taken in the vertical direction only. This assumption is that the external load on the barrette head, which is expected to be heavy load, is applied in the vertical direction. For comparative examinations, the barrette elasticity is determined using either 1D or 3D finite elements. A series of examinations is carried out to verify the application for analyzing barrette by CCT. It was found that, treating the barrette as an elastic body and representing the barrette by either 1D or 3D finite elements, gives nearly the same results.

Modelling the wave phenomena in acoustic and elastic media with sharp variations of physical properties using the grid‐characteristic method

ABSTRACTThis paper introduces a novel method of modelling acoustic and elastic wave propagation in inhomogeneous media with sharp variations of physical properties based on the recently developed grid‐characteristic method which considers different types of waves generated in inhomogeneous linear‐elastic media (e.g., longitudinal, transverse, Stoneley, Rayleigh, scattered PP‐, SS‐waves, and converted PS‐ and SP‐waves). In the framework of this method, the problem of solving acoustic or elastic wave equations is reduced to the interpolation of the solutions, determined at earlier time, thus avoiding a direct solution of the large systems of linear equations required by the FD or FE methods. We apply the grid‐characteristic method to compare wave phenomena computed using the acoustic and elastic wave equations in geological medium containing a hydrocarbon reservoir or a fracture zone. The results of this study demonstrate that the developed algorithm can be used as an effective technique for modelling wave phenomena in the models containing hydrocarbon reservoir and/or the fracture zones, which are important targets of seismic exploration.

English

The finite element (FE) solution of geotechnical elasticity problems leads to the solution of a large system of linear equations. For solving the system, we use the preconditioned conjugate gradient (PCG) method with two-level additive Schwarz preconditioner. The preconditioning is realised in parallel. A coarse space is usually constructed using an aggregation technique. If the finite element spaces for coarse and fine problems on structural grids are fully compatible, relations between elements of matrices of the coarse and fine problems can be derived. By generalization of these formulae, we obtain an overlapping aggregation technique for the construction of a coarse space with smoothed basis functions. The numerical tests are presented at the end of the paper.

Development and convergence analysis of an effective and robust implicit Euler solver for 3D unstructured grids

This paper reports the development and convergence analysis in steady-state of an effective and robust implicit finite-volume solver for compressible Euler equations on three-dimensional unstructured grids. A second-order upwind scheme (MUSCL) was employed based on Roe's approximate Flux-Difference Scheme (FDS) by using Venkatrakishnan flux limiters. The construction of the linear system for the implicit scheme was performed by applying the backward Euler on the left-hand side of the conservation equation and Newton-type linearization on the right-hand side. The Jacobian matrix that resulted from the linearization process was computed analytically using Roe flux terms. In this phase, the defect-correction technique was employed allowing effective time-dependent computations by an implicit time-integration scheme. In this approach, the flux integral on the right-hand side is computed based on a high-order of accuracy whilst the left-hand side the Jacobian is performed based on the low-order. The resulting sparse and large system of linear equations is solved by a sequential Gauss–Seidel iterative method. Simulations were performed and the developed implicit defect-correction solver was validated and verified. In addition, convergence analysis comparing the implicit solver and the explicit Runge–Kutta of 5-steps using Implicit Residual Smoothing (IRS) were performed showing the significant speed-up of the implicit solver over the explicit one. Simulations were performed for case studies to demonstrate the robustness of the developed implicit defect-correction solver in solving typical problems of aerodynamic involving transonic condition and shock wave captures for internal and external flows. Finally, the main particularities of the implicit scheme were investigated and discussed considering the simulation results, showing also its capacity to serve as an effective preconditioner (start-up method) to other implicit techniques.

Indexed fast network proximity querying

Node proximity queries are among the most common operations on network databases. A common measure of node proximity is random walk based proximity, which has been shown to be less susceptible to noise and missing data. Real-time processing of random-walk based proximity queries poses significant computational challenges for larger graphs with over billions of nodes and edges, since it involves solution of large linear systems of equations. Due to the importance of this operation, significant effort has been devoted to developing efficient methods for random-walk based node proximity computations. These methods either aim to speed up iterative computations by exploiting numerical properties of random walks, or rely on computation and storage of matrix inverses to avoid computation during query processing. Although both approaches have been well studied, the speedup achieved by iterative approaches does not translate to real-time query processing, and the storage requirements of inversion-based approaches prohibit their use on very large graph databases. We present a novel approach to significantly reducing the computational cost of random walk based node proximity queries with scalable indexing. Our approach combines domain graph-partitioning based indexing with fast iterative computations during query processing using Chebyshev polynomials over the complex elliptic plane. This approach combines the query processing benefits of inversion techniques with the memory and storage benefits of iterative approache. Using real-world networks with billions of nodes and edges, and top- k proximity queries as the benchmark problem, we show that our algorithm, I-C hopper , significantly outperforms existing methods. Specifically, it drastically reduces convergence time of the iterative procedure, while also reducing storage requirements for indexing.

ANALYZING SINGLE BARRETTES AS RIGID SUPPORT BY COMPOSED COEFFICIENT TECHNIQUE

Most of soil structure interaction methods for analyzing large-section supports such as barrette foundation modeling the barrette and surrounding soil using 3D FE model. In which, the model leads to a large finite element mesh of a large system of linear equations to be solved. In this paper, a Composed Coefficient Technique (CCT) is adapted for analyzing barrette. The technique takes into account the 3D full interactions between barrette and the surrounding soil. Due to the high rigidity of the barrette relative to the surrounding soil, a uniform settlement along the barrette height can be considered. This enables to compose the stiffness coefficients of the soil matrix into composed coefficients, which consequently leads to a significant reduction in the soil stiffness matrix. The validity and the examinations presented in this research work were carried out with an application for analyzing barrette by CCT. This analysis can be extended for modeling the nonlinear behavior of single barrette and barrette-raft. In which, the barrette can be treated as single unit having a uniform settlement.

A Low-Rank Multigrid Method for the Stochastic Steady-State Diffusion Problem

We study a multigrid method for solving large linear systems of equations with tensor product structure. Such systems are obtained from stochastic finite element discretization of stochastic partial differential equations such as the steady-state diffusion problem with random coefficients. When the variance in the problem is not too large, the solution can be well approximated by a low-rank object. In the proposed multigrid algorithm, the matrix iterates are truncated to low rank to reduce memory requirements and computational effort. The method is proved convergent with an analytic error bound. Numerical experiments show its effectiveness in solving the Galerkin systems compared to the original multigrid solver, especially when the number of degrees of freedom associated with the spatial discretization is large.

A Multigrid Approach to SDP Relaxations of Sparse Polynomial Optimization Problems

We propose a multigrid approach for the global optimization of polynomial optimization problems with sparse support. The problems we consider arise from the discretization of infinite dimensional optimization problems, such as PDE optimization problems, boundary value problems, and some global optimization applications. In many of these applications, the level of discretization can be used to obtain a hierarchy of optimization models that capture the underlying infinite dimensional problem at different degrees of fidelity. This approach, inspired by multigrid methods, has been successfully used for decades to solve large systems of linear equations. However, multigrid methods are difficult to apply to semidefinite programming (SDP) relaxations of polynomial optimization problems. The main difficulty is that the information between grids is lost when the original problem is approximated via an SDP relaxation. Despite the loss of information, we develop a multigrid approach and propose prolongation operator...

Direct Image Reconstruction of Lissajous-Type Magnetic Particle Imaging Data Using Chebyshev-Based Matrix Compression

Image reconstruction in magnetic particle imaging (MPI) is done using an algebraic approach for Lissajous-type measurement sequences. By solving a large linear system of equations, the spatial distribution of magnetic nanoparticles can be determined. Despite the use of iterative solvers that converge rapidly, the size of the MPI system matrix leads to reconstruction times that are typically much longer than the actual data acquisition time. For this reason, matrix compression techniques have been introduced that transform the MPI system matrix into a sparse domain and then utilize this sparsity for accelerated reconstruction. Within this work, we investigate the Chebyshev transformation for matrix compression and show that it can provide better reconstruction results for high compression rates than the commonly applied Cosine transformation. By reducing the number of coefficients per matrix row to one, it is even possible to derive a direct reconstruction method that obviates the usage of iterative solvers.

Improvement of computational efficiency of circular function-based gas kinetic scheme by using Jacobian-free Newton–Krylov method

The gas-kinetic BGK scheme is a promising method for simulation of inviscid and viscous flows. Different from conventional Navier–Stokes (N-S) solvers, it evaluates the inviscid and viscous fluxes simultaneously by reconstructing the local solution of BGK Boltzmann equation at the cell interface. Due to its inherently superior dissipation property, it usually gives accurate and robust numerical results. However, a notable drawback of the BGK-type scheme is the low computational efficiency. Recently, aimed at reducing the computational effort, a circular function-based BGK (CBGK) scheme was developed. Nevertheless, it is still time consuming because the original scheme used an explicit way in the time integration as in most existing BGK schemes and the convergence speed can be slow. To improve the computational efficiency, an implicit CBGK scheme is developed in this paper by incorporating the Jacobian-free Newton–Krylov (JFNK) method into the scheme. Particularly, the generalized minimal residual (GMRES) approach is employed to iteratively solve the large linear equation system. With the help of the Jacobian-free approach, a faster convergence speed can be achieved without explicitly computing and storing the flux Jacobian, which is usually a large and sparse matrix. In order to reduce the number of GMRES iterations, the preconditioning is also adopted and the Lower-Upper symmetric Gauss-Seidel (LUSGS) scheme is employed as a preconditioner. For validation of the present CBGK-JFNK method, several two-dimensional inviscid and viscous test cases are investigated. The numerical results show that the needed computational time is significantly reduced as compared with the original explicit CBGK scheme and the CBGK-LUSGS method.

Operating Quantum States in Single Magnetic Molecules: Implementation of Grover's Quantum Algorithm.

Quantum algorithms use the principles of quantum mechanics, such as, for example, quantum superposition, in order to solve particular problems outperforming standard computation. They are developed for cryptography, searching, optimization, simulation, and solving large systems of linear equations. Here, we implement Grover's quantum algorithm, proposed to find an element in an unsorted list, using a single nuclear 3/2 spin carried by a Tb ion sitting in a single molecular magnet transistor. The coherent manipulation of this multilevel quantum system (qudit) is achieved by means of electric fields only. Grover's search algorithm is implemented by constructing a quantum database via a multilevel Hadamard gate. The Grover sequence then allows us to select each state. The presented method is of universal character and can be implemented in any multilevel quantum system with nonequal spaced energy levels, opening the way to novel quantum search algorithms.

Stability analysis of Bilinear Iterative Rational Krylov Algorithm

Models coming from different physical applications are very large in size. Simulation with such systems is expensive so one usually obtains a reduced model (by model reduction) that replicates the input-output behavior of the original full model. A recently proposed algorithm for model reduction of bilinear dynamical systems, Bilinear Iterative Rational Krylov Algorithm (BIRKA), does so in a locally optimal way. This algorithm requires solving very large linear systems of equations. Usually these systems are solved by direct methods (e.g., LU), which are very expensive. A better choice is iterative methods (e.g., Krylov). However, iterative methods introduce errors in linear solves because they are not exact. They solve the given linear system up to a certain tolerance. We prove that under some mild assumptions BIRKA is stable with respect to the error introduced by the inexact linear solves. We also analyze the accuracy of the reduced system obtained from using these inexact solves and support all our results by numerical experiments.

A new modified conjugate gradient coefficient for solving system of linear equations

Conjugate gradient (CG) method is an evolution of computational method in solving unconstrained optimization problems. This approach is easy to implement due to its simplicity and has been proven to be effective in solving real-life application. Although this field has received copious amount of attentions in recent years, some of the new approaches of CG algorithm cannot surpass the efficiency of the previous versions. Therefore, in this paper, a new CG coefficient which retains the sufficient descent and global convergence properties of the original CG methods is proposed. This new CG is tested on a set of test functions under exact line search. Its performance is then compared to that of some of the well-known previous CG methods based on number of iterations and CPU time. The results show that the new CG algorithm has the best efficiency amongst all the methods tested. This paper also includes an application of the new CG algorithm for solving large system of linear equations