Definite Systems Of Linear Equations Research Articles

A breakdown-free block conjugate gradient method for large-scale discriminant analysis

<abstract><p>Rayleigh-Ritz discriminant analysis (RRDA) is an effective algorithm for linear discriminant analysis (LDA), but there are some drawbacks in its implementation. In this paper, we first improved Rayleigh-Ritz discriminant analysis (IRRDA) to make its framework more concise, and established the equivalence theory of the solution space between our discriminant analysis and RRDA. Second, we proposed a new model based on positive definite systems of linear equations for linear discriminant analysis, and certificated the rationality of the new model. Compared with the traditional linear regression model for linear discriminant analysis, the coefficient matrix of our model avoided forming a centralized matrix or appending the original data matrix, but the original matrix itself, which greatly reduced the computational complexity. According to the size of data matrix, we designed two solution schemes for the new model based on the block conjugate gradient method. Experiments in real-world datasets demonstrated the effectiveness and efficiency of our algorithm and it showed that our method was more efficient and faster than RRDA.</p></abstract>

El-WaveHoltz: A time-domain iterative solver for time-harmonic elastic waves

We consider the application of the WaveHoltz iteration to time-harmonic elastic wave equations with energy conserving boundary conditions. The original WaveHoltz iteration for acoustic Helmholtz problems is a fixed-point iteration that filters the solution of the wave equation with time-harmonic forcing and boundary data. As in the original WaveHoltz method, we reformulate the fixed point iteration as a positive definite linear system of equations that is iteratively solved by a Krylov method. We present two time-stepping schemes, one explicit and one (novel) implicit, which completely remove time discretization error from the WaveHoltz solution by performing a simple modification of the initial data and time-stepping scheme. Numerical experiments indicate an iteration scaling similar to that of the original WaveHoltz method, and that the convergence rate is dictated by the shortest (shear) wave speed of the problem. We additionally show that the implicit scheme can be advantageous in practice for meshes with disparate element sizes.

Ritz Vectors-Based Deflation Preconditioner for Linear System with Multiple Right-Hand Sides

Computational mathematics have many tools to solve the large systems of equations which may be linear or nonlinear. Iterative methods are used to solve the nonsymmetric definite system of linear equations like Krylov methods. Linear systems with multiple right-hand sides find application in many areas of engineering and science. Considering generality and indefiniteness, Krylov subspace methods are frequently used for such problems. However, problem with system with multiple right-hand side vectors requires constructing subspace for every right-hand side. GMRES produces Ritz vectors, approximation to eigenvectors during iterations. These Ritz values, recycled vectors are used in Krylov solve while solving system with second and subsequent right-hand side vectors. This is applied as a deflation preconditioner to GMRES. The numerical results show that the computational time, residuals and number of iterations is reduced as compare to simple GMRES. Deflation technique with Ritz vectors is not expensive as compare to the GMRES and as well as exact eigenvectors deflations.

Lagrangian Method to Model Advection-Dispersion-Reaction Transport in Drinking Water Pipe Networks.

A Lagrangian method to simulate the advection, dispersion, and reaction of a single chemical, biological, or physical constituent within drinking water pipe networks is presented. This Lagrangian approach removes the need for fixed computational grids typically required in Eulerian and Eulerian-Lagrangian methods and allows for nonuniform computational segments. This makes the method fully compatible with the advection-reaction water quality engine currently used in EPANET. An operator splitting approach is used, in which the advection-reaction process is modeled before the dispersion process for each water quality step. The dispersion equation is discretized using a segment-centered finite-difference scheme, and flux continuity boundary conditions are applied at network junctions. A staged approach is implemented to solve the dispersion equation for interconnected pipe networks. First, a linear relationship between the boundary and internal concentrations is established for every pipe. Second, a symmetric and positive definite linear system of equations is constructed to calculate the concentrations at network junctions. Last, pipe internal concentrations are updated based on the junction concentrations. The solution generates exact results when the analytical solutions are available and leads to more accurate water quality simulations than advection-reaction-only water quality models, especially in the areas where dispersion dominates advection.

EM-WaveHoltz: How to find Time-Harmonic Solutions by Time-Domain Solvers and Positive Definite Systems

We present EM-WaveHoltz, a new method for computing frequency domain solutions to Maxwell's equations in linear complex media. EM-WaveHoltz is based on a fixed point iteration which is applied to the initial data of a time domain solution. The fixed point iteration converges to a single time-harmonic component and can be formulated as a symmetric positive definite linear system of equations. The method is agnostic to the material model and to the time-domain solver used.

A conjugate gradient-based algorithm for large-scale quadratic programming problem with one quadratic constraint

In this paper, we consider the nonconvex quadratically constrained quadratic programming (QCQP) with one quadratic constraint. By employing the conjugate gradient method, an efficient algorithm is proposed to solve QCQP that exploits the sparsity of the involved matrices and solves the problem via solving a sequence of positive definite system of linear equations after identifying suitable generalized eigenvalues. Specifically, we analyze how to recognize hard case (case 2) in a preprocessing step, fixing an error in Sect. 2.2.2 of Pong and Wolkowicz (Comput Optim Appl 58(2):273–322, 2014) which studies the same problem with the two-sided constraint. Some numerical experiments are given to show the effectiveness of the proposed method and to compare it with some recent algorithms in the literature.

The Analysis of Dynamic Buckling of an Impacted Column using Difference Methods

In this paper, the object of our study is two coupled partial differential equation which have a rich physical background. With a reasonable change to the initial value, we use central difference method and compact finite difference method to get the numerical results of the governing equations. In the analysis of the latter, we use the Hermitian and skew-Hermitian splitting (HSS) method to solve a large sparse non-Hermitian positive definite system of linear equations. In this way, the computational efficiency can be improved and the convergence is guaranteed which is proved by Bai [Bai et al. [2003] “Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems,” SIAM J. Matrix Anal. Appl. 24, 603–626] and Chen [Chen et al. [2014] “Convergence analysis of the modified Newton-HSS method under the Hölder continuous condition,” J. Comput. Appl. Math. 264, 115–130]. And we also pay attention to the effect of initial conditions, on which we add a small perturbation to observe its influence by comparing numerical results.

Scaled norm minimization method for computing the parameters of the HSS and the two‐parameter HSS preconditioners

SummaryThe performance of the Hermitian and skew‐Hermitian splitting (HSS) preconditioner for the non‐Hermitian positive definite system of linear equations is largely dependent on the choice of its parameter value. In this work, an efficient scaled norm minimization (SNM) method is proposed to compute the parameter value of the HSS preconditioner. In addition, by choosing different parameters for the Hermitian and the skew‐Hermitian matrices in the HSS preconditioner, a two‐parameter HSS preconditioner is proposed. Moreover, an efficient and practical formula for computing the parameter values of this new preconditioner is also derived by using the SNM method. Numerical examples are illustrated to verify the performances of the HSS and the two‐parameter HSS preconditioners when their parameters are computed by the SNM method.

A Distributed Algorithm for Solving Positive Definite Linear Equations Over Networks With Membership Dynamics

This paper considers the problem of solving a symmetric positive definite system of linear equations over a network of agents with arbitrary asynchronous interactions and membership dynamics. The latter implies that each agent is allowed to join and leave the network at any time, for infinitely many times, and lose all its memory upon leaving. We develop Subset Equalizing (SE), a distributed asynchronous algorithm for solving such a problem. To design and analyze SE, we introduce a novel time-varying Lyapunov-like function, defined on a state space with changing dimension, and a generalized concept of network connectivity, capable of handling such interactions and membership dynamics. Based on them, we establish the boundedness, asymptotic convergence, and exponential convergence of SE, along with a bound on its convergence rate. Finally, through extensive simulation, we show that SE is effective in a volatile agent network and that a special case of SE, termed Groupwise Equalizing , is significantly more bandwidth/energy efficient than two existing algorithms in multi-hop wireless networks.

Time efficient aeroelastic simulations based on radial basis functions

Aeroelasticity studies the interaction between aerodynamic forces and structural responses, and is one of the fundamental problems to be considered in the design of modern aircraft. The fluid–structure interpolation (FSI) and mesh deformation are two key issues in the CFD–CSD coupling approach (the partitioned approach), which is the mainstream numerical strategy in aeroelastic simulations. In this paper, a time efficient coupling scheme is developed based on the radial basis function interpolations. During the FSI process, the positive definite system of linear equations is constructed with the introduction of pseudo structural forces. The acting forces on the structural nodes can be calculated more efficiently via the solution of the linear system, avoiding the costly computations of the aerodynamic/structural coupling matrix. The multi-layer sequential mesh motion algorithm (MSM) is proposed to improve the efficiency of the volume mesh deformations, which is adequate for large-scale time dependent applications with frequent mesh updates. Two-dimensional mesh motion cases show that the MSM algorithm can reduce the computing cost significantly compared to the standard RBF-based method. The computations of the AGARD 445.6 wing flutter and the static deflections of the three-dimensional high-aspect-ratio aircraft demonstrate that the developed coupling scheme is applicable to both dynamic and static aeroelastic problems.

Multi-step modified Newton-HSS methods for systems of nonlinear equations with positive definite Jacobian matrices

The Hermitian and skew-Hermitian splitting (HSS) method is an unconditionally convergent iterative method for solving large sparse non-Hermitian positive definite system of linear equations. In this paper, we establish a class of multi-step modified Newton-HSS (MMN-HSS) methods for solving large sparse system of nonlinear equations with positive definite Jacobian matrices. The MMN-HSS methods use the multi-step modified Newton methods to solve the nonlinear equations, and the HSS method to approximately solve the Newton equation. Local and semilocal convergence theorems are proved under proper conditions. Also, we present the global multi-step modified Newton-HSS methods with a backtracking strategy and analyse its global convergence. Finally, numerical results are given to confirm the feasibility and effectiveness of our method.

Positive Definite and Semi-Definite Splitting Methods for Non-Hermitian Positive Definite Linear Systems

In this paper, we further generalize the technique for constructing the normal (or positive definite) and skew-Hermitian splitting iteration method for solving large sparse nonHermitian positive definite system of linear equations. By introducing a new splitting, we establish a class of efficient iteration methods, called positive definite and semi-definite splitting (PPS) methods, and prove that the sequence produced by the PPS method converges unconditionally to the unique solution of the system. Moreover, we propose two kinds of typical practical choices of the PPS method and study the upper bound of the spectral radius of the iteration matrix. In addition, we show the optimal parameters such that the spectral radius achieves the minimum under certain conditions. Finally, some numerical examples are given to demonstrate the effectiveness of the considered methods.

The generalized HSS method with a flexible shift-parameter for non-Hermitian positive definite linear systems

Based on the Hermitian and skew-Hermitian splitting (HSS), we come up with a generalized HSS iteration method with a flexible shift-parameter for solving the non-Hermitian positive definite system of linear equations. This iteration method utilizes the optimization technique to obtain the optimal value of the flexible shift-parameter at iteration process. Both theory and experiment have shown that the new strategy is efficient.

On SSOR‐like preconditioners for non‐Hermitian positive definite matrices

SUMMARYWe construct, analyze, and implement SSOR‐like preconditioners for non‐Hermitian positive definite system of linear equations when its coefficient matrix possesses either a dominant Hermitian part or a dominant skew‐Hermitian part. We derive tight bounds for eigenvalues of the preconditioned matrices and obtain convergence rates of the corresponding SSOR‐like iteration methods as well as the corresponding preconditioned GMRES iteration methods. Numerical implementations show that Krylov subspace iteration methods such as GMRES, when accelerated by the SSOR‐like preconditioners, are efficient solvers for these classes of non‐Hermitian positive definite linear systems. Copyright © 2015 John Wiley & Sons, Ltd.

An Efficient Semi-Analytical Method to Compute Displacements and Stresses in an Elastic Half-Space with a Hemispherical Pit

AbstractThis paper presents an efficient method to calculate the displacement and stress fields in an isotropic elastic half-space having a hemispherical pit and being subject to gravity. The method is semi-analytical and takes advantage of the axisymmetry of the problem. The Boussinesq potentials are used to obtain an analytical solution in series form, which satisfies the equilibrium equations of elastostatics, traction-free boundary conditions on the infinite plane surface and decaying conditions at infinity. The boundary conditions on the free surface of the pit are then imposed numerically, by minimising a quadratic functional of surface elastic energy. The minimisation yields a symmetric and positive definite linear system of equations for the coefficients of the series, whose particular block structure allows its solution in an efficient and robust way. The convergence of the series is verified and the obtained semi-analytical solution is then evaluated, providing numerical results. The method is validated by comparing the semi-analytical solution with the numerical results obtained using a commercial finite element software.

Improved seed methods for symmetric positive definite linear equations with multiple right‐hand sides

SUMMARYWe consider symmetric positive definite systems of linear equations with multiple right‐hand sides. The seed conjugate gradient (CG) method solves one right‐hand side with the CG method and simultaneously projects over the Krylov subspace thus developed for the other right‐hand sides. Then the next system is solved and used to seed the remaining ones. Rounding error in the CG method limits how much the seeding can improve convergence. We propose three changes to the seed CG method: only the first right‐hand side is used for seeding, this system is solved past convergence, and the roundoff error is controlled with some reorthogonalization. We will show that results are actually better with only one seeding, even in the case of related right‐hand sides. Controlling rounding error gives the potential for rapid convergence for the second and subsequent right‐hand sides. Polynomial preconditioning can help reduce storage needed for reorthogonalization. The new seed methods are applied to examples including matrices from quantum chromodynamics. Copyright © 2013 John Wiley & Sons, Ltd.

Convergence analysis of modified Newton-HSS method for solving systems of nonlinear equations

Hermitian and skew-Hermitian splitting(HSS) method has been proved quite successfully in solving large sparse non-Hermitian positive definite systems of linear equations. Recently, by making use of HSS method as inner iteration, Newton-HSS method for solving the systems of nonlinear equations with non-Hermitian positive definite Jacobian matrices has been proposed by Bai and Guo. It has shown that the Newton-HSS method outperforms the Newton-USOR and the Newton-GMRES iteration methods. In this paper, a class of modified Newton-HSS methods for solving large systems of nonlinear equations is discussed. In our method, the modified Newton method with R-order of convergence three at least is used to solve the nonlinear equations, and the HSS method is applied to approximately solve the Newton equations. For this class of inexact Newton methods, local and semilocal convergence theorems are proved under suitable conditions. Moreover, a globally convergent modified Newton-HSS method is introduced and a basic global convergence theorem is proved. Numerical results are given to confirm the effectiveness of our method.

Application of the meshless Galerkin boundary node method to potential problems with mixed boundary conditions

Abstract In this paper, the meshless Galerkin boundary node method is developed for boundary-only analysis of two- and three-dimensional potential problems with mixed boundary conditions of Dirichlet and Neumann type. This meshless algorithm leads to a symmetric and positive definite system of linear equations. Additionally, boundary conditions can be implemented directly and easily despite the fact that the employed meshless shape functions lack the delta function property. Theoretical error analysis and numerical results indicate that it is an efficient and accurate numerical method.

On Newton-HSS Methods for Systems of Nonliear Equations with Positive-Definite Jacobian Matrices

The Hermitian and skew-Hermitian splitting (HSS) method is an unconditionally convergent iteration method for solving large sparse non-Hermitian positive definite system of linear equations. By making use of the HSS iteration as the inner solver for the Newton method, we establish a class of Newton-HSS methods for solving large sparse systems of nonlinear equations with positive definite Jacobian matrices at the solution points. For this class of inexact Newton methods, two types of local convergence theorems are proved under proper conditions, and numerical results are given to examine their feasibility and effectiveness. In addition, the advantages of the Newton-HSS methods over the Newton-USOR, the Newton-GMRES and the Newton-GCG methods are shown through solving systems of nonlinear equations arising from the finite difference discretization of a two-dimensional convection-diffusion equation perturbed by a nonlinear term. The numerical implementations also show that as preconditioners for the Newton-GMRES and the Newton-GCG methods the HSS iteration outperforms the USOR iteration in both computing time and iteration step. Mathematics subject classification: 65F10, 65W05.

Distributed Asynchronous Algorithms for Solving Positive Definite Linear Equations over Networks—Part II: Wireless Networks

—In this Part II of the two-part paper, we study the interplay among wireless communications, distributed algorithms, and control in solving symmetric positive definite systems of linear equations over multi-hop wireless networks with fixed topologies. Building on the results from Part I, we develop and analyze Pairwise, Groupwise, Random Hopwise, and Controlled Hopwise Equalizing (PE, GE, RHE, and CHE), showing along the way how the broadcast nature of wireless transmissions may be fully utilized, how undesirable overlapping iterations may be avoided, and how iterations may be feedback controlled in a greedy, decentralized, Lyapunov-based fashion, leading to CHE. We show that CHE yields a networked dynamical system with state-dependent switching, provable exponential convergence, and quantifiable worst-case convergence rate. Finally, through extensive simulation on random geometric graphs, we show that GE, RHE, and CHE are dramatically more efficient and scalable than two existing, average-consensus-based schemes, with CHE having the best performance.