Solution Of Linear Systems Research Articles

Comparison theorems for splittings of M-matrices in (block) Hessenberg form

Some variants of the (block) Gauss–Seidel iteration for the solution of linear systems with M-matrices in (block) Hessenberg form are discussed. Comparison results for the asymptotic convergence rate of some regular splittings are derived: in particular, we prove that for a lower-Hessenberg M-matrix \(\rho (P_{GS})\ge \rho (P_S)\ge \rho (P_{AGS})\), where \(P_{GS}, P_S, P_{AGS}\) are the iteration matrices of the Gauss–Seidel, staircase, and anti-Gauss–Seidel method. This is a result that does not seem to follow from classical comparison results, as these splittings are not directly comparable. It is shown that the concept of stair partitioning provides a powerful tool for the design of new variants that are suited for parallel computation.

Integral Representation of the Solutions for Neutral Linear Fractional System with Distributed Delays

In the present paper, first we obtain sufficient conditions for the existence and uniqueness of the solution of the Cauchy problem for an inhomogeneous neutral linear fractional differential system with distributed delays (even in the neutral part) and Caputo type derivatives, in the case of initial functions with first kind discontinuities. This result allows to prove that the corresponding homogeneous system possesses a fundamental matrix C(t,s) continuous in t,t∈[a,∞),a∈R. As an application, integral representations of the solutions of the Cauchy problem for the considered inhomogeneous systems are obtained.

A new solver for incompressible non-isothermal flows in natural and mixed convection over unstructured grids

In the present paper we propose a new numerical methodology for the solution of 2D non-isothermal incompressible flows for natural and mixed convection in irregular geometries. The governing equations are the Incompressible Navier-Stokes equations and the energy conservation equation. Fluid velocity and temperature are coupled in the buoyancy term of the momentum equations according to the Oberbeck-Boussinesq approximation. The governing equations are discretized over unstructured triangular meshes satisfying the Delaunay property. Thanks to the Oberbeck-Boussinesq hypothesis, the flow and energy problems are solved in an uncoupled way, and two fractional time step procedures are sequentially applied to solve each problem. The prediction steps of both procedures are solved applying a Marching in Space and Time (MAST) numerical Eulerian scheme, which explicitly handles the non-linear terms in the momentum equations of the fluid problem, and allows numerical stability for Courant numbers greater than one. An analytical solution is applied for the prediction thermal problem. The correction steps of the two fractional time step procedures involve the solution of large linear systems, whose matrices are sparse and symmetric and have the “M” property if the mesh satisfies the Delaunay condition. This allows a well performing condition number. The matrix coefficients are constant in time, so that they are calculated and factorized only once, before the simulations loop starts, saving a lot of computational time. We present four numerical applications. For the first test, an analytical solution is available, and this makes it possible to analyze the spatial convergence order of the numerical solver. In the other applications, we investigate the capability of the proposed algorithm to handle irregular geometries, and we compare the computed results with experimental data and the outputs of literature models. A study of the required computational costs is also presented.

On rigorous integration of continuous piecewise linear systems

In this work, rigorous methods to compute solutions of continuous piecewise linear systems are presented. A general technique, based on integrating a perturbed linear system, is described to compute enclosures of trajectories crossing planes where the vector field in not smooth. Algorithms to compute enclosures of trajectories passing arbitrarily close to unstable equilibria are proposed. The usefulness of the methods is shown by proving the existence of trapping regions enclosing three different types of chaotic attractors existing in the Chua’s circuit.

Survey of Eight Modern Methods of Hamiltonian Mechanics

Here we describe eight new methods, arisen in the last 60 years, to study solutions of a Hamiltonian system with n degrees of freedom. The first six of them are intended for systems with small parameters or without them. The methods allow to find families of periodic solutions and families of invariant n-dimensional tori by means of analytic computation near a stationary solution, near a periodic solution and near an invariant torus, using the corresponding normal form of a Hamiltonian. Then we can continue the founded families by means of numerical computation. In a Hamiltonian system without parameters, only periodic solutions and invariant n-dimensional tori form one-parameter families. The last two methods are intended for systems with not small parameters, which do not depend on time. They allow computing sets of parameters, which guarantee the stability of some solutions for linear (method seven) and nonlinear (method eight) systems. We do not consider chaotic behaviors, but only regular ones.

Drazin-Star and Star-Drazin Inverses of Bounded Finite Potent Operators on Hilbert Spaces

The aim of this work is to extend to bounded finite potent endomorphisms on arbitrary Hilbert spaces the notions of the Drazin-Star and the Star-Drazin of matrices that have been recently introduced by D. Mosić. The existence, structure and main properties of these operators are given. In particular, we obtain new properties of the Drazin-Star and the Star-Drazin of a finite complex matrix. Moreover, the explicit solutions of some infinite linear systems on Hilbert spaces from the Drazin-Star inverse of a bounded finite potent endomorphism are studied.

Adaptive cross approximation for E-field-guided non-invasive magnetic brain stimulation

Abstract Introduction: During transcranial magnetic stimulation (TMS), a coil placed on the scalp driven by a low-frequency pulse is used to modulate brain activity. The TMS induced E-field has been shown to correlate well with stimulated cortical region. As a result, E-field information is essential for the determining where to place the coil and the necessary pulse intensity. A bottleneck to E-field informed TMS is that FEM requires the solution of a large linear systems of equations for each coil placement, making repeated execution of FEM for finding optimum coil placement impractical. Objectives: The goal is to develop a practical tool for E-field dosimetry that allows determination of E-fields in real-time, thereby, enabling the fast computation of optimum coil placement. Methods: We arrange E-field results into a matrix with each column consisting of E-field samples determined from a single coil placement via FEM. Its rows consist of the E-field at a single coil position for all possible coil placements and can be sampled using our previously proposed auxiliary dipole method (ADM). We use the adaptive cross approximation (ACA) to estimate the whole matrix from a few sample columns and rows. Results: This method can evaluate the E-field distribution in a 2 cm diameter ROI from for ∼1000 coil positions and 360 coil orientations with error <2% in under 10 and 20 simulations in a spherical and MRI-derived head model, respectively. For the MRI-derived head model, this method obtained all the fields in <20 minutes. Conclusion: The proposed method enables the fast computation of optimal coil placement and pulse intensity, as well as, it provides E-field distribution inside the ROI for each coil placement.View Large Image Figure ViewerDownload Hi-res image Download (PPT)View Large Image Figure ViewerDownload Hi-res image Download (PPT)View Large Image Figure ViewerDownload Hi-res image Download (PPT)View Large Image Figure ViewerDownload Hi-res image Download (PPT) Keywords: TMS, Low-rank approximation, Coil placement optimizaiton, E-field dosimetry

Solution of Linear System of Volterra Integral Equations of the First Kind via Generalized Inverse

In this research, we have presented and discussed an algorithm for analytical method to solve linear system of Volterra integral equations of the first kind ( LVIE1st ). Algorithm of the analytical method for this system based on the Laplace transform and generalized inverse. The proposed algorithm has been applied to some examples to demonstrate the efficiency and simplicity.

High‐order implicit time‐stepping with high‐order central essentially‐non‐oscillatory methods for unsteady three‐dimensional computational fluid dynamics simulations

AbstractThis article develops high‐order implicit time‐stepping methods combined with the fourth‐order central essentially‐non‐oscillatory (CENO) scheme for stiff three‐dimensional computational fluid dynamics problems having disparate characteristic time scales. Both aerodynamic and magnetohydrodynamic problems are considered on three‐dimensional multiblock body‐fitted grids with hexahedral cells. Several implicit time integration methods of third‐ and fourth‐order accuracy are considered, including the multistep backward differentiation formulas (BDF4), multistage explicitly singly diagonally implicit Runge‐Kutta (ESDIRK4), and Rosenbrock‐type methods (ROS34POW2). The resulting nonlinear algebraic system of equations is solved via a preconditioned Jacobian‐free inexact Newton–Krylov method with additive Schwarz preconditioning using block‐based incomplete LU decomposition. The performance of the high‐order implicit time‐stepping methods on smooth and stiff problems is compared with a standard fourth‐order explicit Runge‐Kutta (RK4) method. It is shown that the Rosenbrock methods, despite their advantage of only requiring the solution of linear systems, have significant drawbacks in terms of robustness issues for highly nonlinear compressible flows. The implicit BDF4 and ESDIRK4 methods are found to be much more efficient than the explicit fourth‐order RK4 method for a stiff resistive magnetohydrodynamic (MHD) problem discretized with the fourth‐order CENO method. When applied to the problem of vortex shedding governed by the Navier–Stokes equations, an A‐stable ESDIRK4 scheme proved to be the more robust and accurate implicit time‐marching scheme and was able to offer significant speedup compared with the RK4 method. Initial results are also shown for high‐order implicit time integration applied to two problems with discontinuities. The current study represents the first to achieve high‐order implicit time integration for MHD, enabling large time steps and substantial speedups for stiff MHD problems with high‐order accuracy, and it also represents the first to establish high‐order implicit time integration for high‐order CENO in space.

ABS-Based Direct Method for Solving Complex Systems of Linear Equations

Efficient solution of linear systems of equations is one of the central topics of numerical computation. Linear systems with complex coefficients arise from various physics and quantum chemistry problems. In this paper, we propose a novel ABS-based algorithm, which is able to solve complex systems of linear equations. Theoretical analysis is given to highlight the basic features of our new algorithm. Four variants of our algorithm were also implemented and intensively tested on randomly generated full and sparse matrices and real-life problems. The results of numerical experiments reveal that our ABS-based algorithm is able to compute the solution with high accuracy. The performance of our algorithm was compared with a commercially available software, Matlab’s mldivide (\) algorithm. Our algorithm outperformed the Matlab algorithm in most cases in terms of computational accuracy. These results expand the practical usefulness of our algorithm.

Commutative d-torsion K-theory and its applications

Commutative d-torsion K-theory is a variant of topological K-theory constructed from commuting unitary matrices of order dividing d. Such matrices appear as solutions of linear constraint systems that play a role in the study of quantum contextuality and in applications to operator-theoretic problems motivated by quantum information theory. Using methods from stable homotopy theory, we modify commutative d-torsion K-theory into a cohomology theory that can be used for studying operator solutions of linear constraint systems. This provides an interesting connection between stable homotopy theory and quantum information theory.

Asymptotic solutions of (q(qy′)′)′+py′+ry=0

We are concerned with the third-order linear differential equation (q(qy′)′)′+py′+ry=0 defined on an interval [t0,∞). We derive asymptotic solutions for cases characterized by p being “small” and “large”, respectively. Our main tool is a result by Eastham on the asymptotic solution of linear differential systems.

Analytical solution of problem of shielding low-frequency magnetic field by thin-walled cylindrical screen in presence of cylinder

The analytical solution of boundary value problem describing the process of penetration of low-frequency magnetic field through thin-walled cylindrical screen with cylindrical inclusion is constructed by use of approximate boundary conditions. The source of the field is a thin thread of infinitely small length with an infinitely small cross-section where current circulates. Thread is located in a plane which is perpendicular to axis of cylindrical screen, in outer region with respect to a screen. Initially the potential of initial magnetic field is represented as spherical harmonic functions, then using addition theorems connecting spherical and cylindrical harmonic functions, it became as cylindrical harmonic functions superposition. Secondary potentials of magnetic field are also presented as superposition of cylindrical harmonic functions in three-dimensional space. It is shown that the solution of formulated boundary value problem is reduced to the solution of linear algebraic equations system for coefficients included in the representation of secondary fields. The influence of some aspects of the problem on the value of the screening coefficient of an external magnetic field when passing through a cylindrical copper screen in the presence of a cylindrical inclusion is studied numerically. Calculation results are presented in graphs form. Obtained results can be used to shield technical devices and biological objects against the effects of magnetic fields to provide ecological surrounding of operating electrical installations and devices.

Formula omitted]-multigrid with partial smoothing: An efficient preconditioner for discontinuous Galerkin discretizations with modal bases

Multigrid methods are efficient tools for the solution of linear systems arising in the discretization of partial differential equations and in other applications. While traditional multigrid methods often exhibit optimal or near-optimal computational cost, their application to advanced discretizations and complex geometries is challenging. In this work, we combine p-multigrid methods with algebraic multigrid (AMG) techniques to obtain scalable preconditioners for modal discontinuous Galerkin discretizations on complex, three-dimensional domains. We introduce the notion of partial smoothers, which update only a subset of the unknowns at each level of the multigrid hierarchy. Using partial smoothing, we incorporate hierarchical scale separation (HSS), a two-level p-multigrid-like technique with a more complete separation of high-order and low-order unknowns, into the existing p-multigrid framework. The resulting p-multigrid methods with partial smoothing are a generalization of existing HSS methods, offering tradeoffs between p-multigrid convergence and computational performance. In numerical experiments, we show that combined p-multigrid-AMG preconditioners can significantly accelerate large-scale simulations of pore scale flow, and we demonstrate that partial smoothing can further reduce computational cost by as much as 23%.

Hypothetical Learning Trajectory: To Determine The Solution of Linear Equation System in Elementary Row Operation

The background of this research is the learning obstacles that arise on the topic of determining the solution of a system of linear equations (SLE) in applying elementary row operations (ERO). From these findings, it is necessary to formulate a hypothetical learning trajectory (HLT) that can guide the flow of thinking in designing didactic situations in the learning process. HLT expect to anticipate learning obstacles (LO) that arise. This research is a preliminary design of a series of didactical design research. The steps taken in compiling the HLT are 1) reviewing the lesson plan, the methods used to convey the material to determine SST solutions, and the textbooks used; 2) reviewing students' thinking processes conceptually and procedurally regarding theoretically determining SLE solutions; 3) reviewing design plans didactic. HLT is formulated into five types of student responses. It integrates into didactic situations by explaining the relevance of the elimination method in high school with three operations on ERO and then by explaining matrix material before SPL material. From this didactic situation, it can be formulated in teaching material that can accommodate emerging LO.

Double parameter splitting (DPS) iteration method for solving complex symmetric linear systems

Complex linear equation systems appear in many branches of mathematical sciences and have many applications in science and engineering. Therefore, there have been developed many methods to solve them. Here a new method based on decomposition of coefficient matrix for the solution of a positive symmetric linear system is studied. Our method uses two parameters. We obtain a domain on which this new method converges without any condition. We will also discuss how to calculate optimal parameters. Moreover we will prove that there exists a region in R2 such that DPS method is faster than CRI method (Wang et al. (2017) [48]). Finally some test problems will be given and simulation results will be reported to support the theoretical results.

Refinement of Extended Accelerated Over-Relaxation Method for Solution of Linear Systems

Given any linear stationary iterative methods in the form z^(i+1)=Jz^(i)+f, where J is the iteration matrix, a significant improvements of the iteration matrix will decrease the spectral radius and enhances the rate of convergence of the particular method while solving system of linear equations in the form Az=b. This motivates us to refine the Extended Accelerated Over-Relaxation (EAOR) method called Refinement of Extended Accelerated Over-Relaxation (REAOR) so as to accelerate the convergence rate of the method. In this paper, a refinement of Extended Accelerated Over-Relaxation method that would minimize the spectral radius, when compared to EAOR method, is proposed. The method is a 3-parameter generalization of the refinement of Accelerated Over-Relaxation (RAOR) method, refinement of Successive Over-Relaxation (RSOR) method, refinement of Gauss-Seidel (RGS) method and refinement of Jacobi (RJ) method. We investigated the convergence of the method for weak irreducible diagonally dominant matrix, matrix or matrix and presented some numerical examples to check the performance of the method. The results indicate the superiority of the method over some existing methods.

Drug–disease associations prediction via Multiple Kernel-based Dual Graph Regularized Least Squares

Predicting associations in drug–disease network provides effective information for the drug repositioning. Therefore, it is an important task to develop an effective drug–disease association prediction method. In this paper, we propose a model, called the Multiple Kernel-based Dual Graph Regularized Least Squares Model (MKDGRLS), to predict potential drug–disease associations. First, we calculate the multiple kernels of drug and disease spaces respectively. Moreover, we utilize multiple kernels and related Laplacian regularization terms to construct MKDGRLS. Finally, we solve the object function of MKDGRLS by the Alternating Least squares algorithm (ALSA), for predicting drug–disease associations. Simultaneously, we design a variant MKDGRLS-A, which is added one dimension as the bias for compensate the inexact solution of linear systems. Our proposed approach has better prediction performance than existing prediction tools under two types of cross validation on the three real drug–disease association network datasets. The results of the case studies of the two diseases prove that our model can effectively predict new associations. We also test MKDGRLS on six real-world networks and it achieves better performance than other methods. In conclusion, our research work can effectively discover potential drug–disease associations, and can provide effective help for related drug repositioning work.

Machine learning for optimal selection of sparse triangular system solvers on GPUs

Many numerical algorithms for science and engineering applications require the solution of sparse triangular linear systems (sptrsv) as their most costly stage. For this reason, considerable research has been dedicated to produce efficient implementations for almost all high performance computing platforms. In the case of graphics processing units (GPUs), there are several strategies to perform this operation, which translate into a handful of different routines. In general, it is difficult to establish a priori which is the best routine for a given problem, and thus, an automatic procedure able to select the best solver for each matrix can entail large performance benefits.This work extends a previous effort, in which we relied on machine learning techniques to predict the best sptrsv routine for each matrix, by improving both the accuracy and the speed of the selection procedure. Specifically, we focus on the most efficient machine learning techniques regarding the speed of their training and prediction stages; evaluate the artificial generation of sparse matrices to expand our dataset; and propose heuristics to compute approximations of some expensive features. The experimental results show that we can strongly improve the runtime of our procedure without compromising the quality results.

Finite-time stability of linear stochastic fractional-order systems with time delay

This paper focuses on the finite-time stability of linear stochastic fractional-order systems with time delay for alpha in (frac{1}{2},1). Under the generalized Gronwall inequality and stochastic analysis techniques, the finite-time stability of the solution for linear stochastic fractional-order systems with time delay is investigated. We give two illustrative examples to show the interest of the main results.