Solution Of Linear Systems Research Articles

The accurate and efficient solutions of linear systems for generalized sign regular matrices with certain signature

In this paper, we consider how to accurately solve the linear system whose coefficient matrix is a generalized sign regular (GSR) matrix with signature (1,…,1,−1). A new algorithm with O(n2) complexity is presented to solve the GSR linear system, provided that parameterization matrices of coefficient matrices are available. We illustrate that no subtraction-cancellation occurs in the computations of the algorithm, which guarantees that all the solution components are computed with a desirable accuracy. An error analysis and numerical experiments are presented to confirm the high accuracy.

An Optimized Schwarz Method for the Optical Response Model Discretized by HDG Method.

An optimized Schwarz domain decomposition method (DDM) for solving the local optical response model (LORM) is proposed in this paper. We introduce a hybridizable discontinuous Galerkin (HDG) scheme for the discretization of such a model problem based on a triangular mesh of the computational domain. The discretized linear system of the HDG method on each subdomain is solved by a sparse direct solver. The solution of the interface linear system in the domain decomposition framework is accelerated by a Krylov subspace method. We study the spectral radius of the iteration matrix of the Schwarz method for the LORM problems, and thus propose an optimized parameter for the transmission condition, which is different from that for the classical electromagnetic problems. The numerical results show that the proposed method is effective.

Physics‐based iterative scheme for computing antenna array embedded element patterns

AbstractRigorously accounting for mutual coupling effects in antenna array synthesis, generally requires that embedded element patterns (EEPs) be determined. The method of moments (MoM) is typically used. Multiple linear system solutions are required, corresponding to every excitation vector associated with an EEP. This paper presents the direct coupling technique (DCT) to obtain the MoM solution for arrays with disjoint elements, by iteratively solving a series of localized subproblems, one associated with each array element. It utilizes a physics‐based, direct‐coupling approximation to incorporate global coupling into the local problems. Local matrices do not change between iterations or with excitation vectors, hence they can be inverted as a preprocessing step, allowing for efficient solution of multiple excitation configurations. Numerical results demonstrate rapid convergence. The convergence rate can be controlled via local problem domain sizing. Since the DCT is ideally suited to parallelization, it is applicable to very large arrays.

SIMD vectorization for simultaneous solution of locally varying linear systems with multiple right-hand sides

Developments in numerical simulation of flows and high-performance computing influence one another. More detailed simulation methods create a permanent need for more computational power, while new hardware developments often require changes to the software to exploit new hardware features. This dependency is very pronounced in the case of vector-units which are featured by all modern processors to increase their numerical throughput but require vectorization of the software to be used efficiently. We study the vectorization of a simulation method that exhibits an inherent level of vector-parallelism. This is of particular interest as SIMD operations will hopefully be available with std::simd in a future C++ standard. The simulation method considered here results in the simultaneous solution of multiple sparse linear systems of equations which only differ by their main diagonal and right-hand sides. Such structure arises in the simulation of unsteady flow in turbomachinery by means of a frequency domain approach called harmonic balance.

Perturbation analysis of fully fuzzy linear systems1

This paper considers the perturbation analysis of a class of fully fuzzy linear systems in which the coefficient matrix is a positive fuzzy matrix. The original fuzzy linear systems is extended into a brand new and simple crisp matrix equation using an embedding method. By discussing the perturbation of the extended crisp linear equation, the paper completes the perturbation analysis of the original fuzzy linear system. There are three cases of perturbation are analysed and the respective relative error bounds for solutions of fuzzy linear system are derived. Some numerical examples are given to illustrated our obtained results.

A Variational Quantum Linear Solver Application to Discrete Finite-Element Methods.

Finite-element methods are industry standards for finding numerical solutions to partial differential equations. However, the application scale remains pivotal to the practical use of these methods, even for modern-day supercomputers. Large, multi-scale applications, for example, can be limited by their requirement of prohibitively large linear system solutions. It is therefore worthwhile to investigate whether near-term quantum algorithms have the potential for offering any kind of advantage over classical linear solvers. In this study, we investigate the recently proposed variational quantum linear solver (VQLS) for discrete solutions to partial differential equations. This method was found to scale polylogarithmically with the linear system size, and the method can be implemented using shallow quantum circuits on noisy intermediate-scale quantum (NISQ) computers. Herein, we utilize the hybrid VQLS to solve both the steady Poisson equation and the time-dependent heat and wave equations.

Generalized TASE-RK methods for stiff problems

A family of Time-Accurate and Stable Explicit (TASE) methods for the numerical integration of Initial Value Problems in stiff Ordinary Differential Equations (ODEs) y′(t)=f(t,y) was recently introduced in [1]. The main idea was to make local extrapolation of a stabilized Euler method. More recently, in [3] a similar approach by considering the stabilization of arbitrary explicit Runge-Kutta methods (TASE-RK) was analyzed. In this case the explicit Runge-Kutta method integrates a transformed ODE obtained by multiplying the vector field f(t,y) by a certain operator which approximates the identity mapping up to a given order p. The main inconvenience of both approaches is that to reach order p the solution of p2 linear systems plus the evaluation of p derivatives are required per integration step.In order to substantially reduce the computational costs of the former approaches in the linear system solution, but maintaining the good accuracy and stability properties, a new family of TASE-RK methods which allow to introduce a few more free parameters are considered. The formulation of the methods was conceived to be implemented not only in sequential mode but it admits parallelism in a straightforward way. Furthermore, since these methods are linearly implicit, connections to the class of W-methods [19] are properly established. The order conditions for the new class of methods are widely studied by using the rooted tree theory. For p=3,4, new methods with p sequential stages and order p are derived and compared on semidiscrete 1D and 2D Partial Differential Equations (PDEs) to those in [1,3] and other standard Rosenbrock and W-methods in the literature.

A Deterministic Kaczmarz Algorithm for Solving Linear Systems

We propose a new deterministic Kaczmarz algorithm for solving consistent linear systems . Basically, the algorithm replaces orthogonal projections with reflections in the original scheme of Stefan Kaczmarz. Building on this, we give a geometric description of solutions of linear systems. Suppose is , we show that the algorithm generates a series of points distributed with patterns on an -sphere centered on a solution. These points lie evenly on lower-dimensional spheres , with the property that for any , the midpoint of the centers of is exactly a solution of . With this discovery, we prove that taking the average of points on any effectively approximates a solution up to relative error , where characterizes the eigengap of the orthogonal matrix produced by the product of reflections generated by the rows of . We also analyze the connection between and , the condition number of . In the worst case , while for random matrices on average. Finally, we prove that the algorithm indeed solves the linear system , where is the lower-triangular matrix such that . The connection between this linear system and the original one is studied. The numerical tests indicate that this new Kaczmarz algorithm has comparable performance to randomized (block) Kaczmarz algorithms.

Solution curve for linear control systems on Lie groups

The purpose of this paper is to describe explicitly the solution for linear control system on Lie groups. Furthermore, we present some applications when a linear control systems has an inner derivation and we study some aspects of controllability.

Mean Oscillation Gradient Estimates for Elliptic Systems in Divergence Form with VMO Coefficients

We consider gradient estimates for H1 solutions of linear elliptic systems in divergence form partial _{alpha }(A_{ij}^{alpha beta } partial _{beta } u^{j}) = 0. It is known that the Dini continuity of coefficient matrix A = (A_{ij}^{alpha beta }) is essential for the differentiability of solutions. We prove the following results:(a) If A satisfies a condition slightly weaker than Dini continuity but stronger than belonging to VMO, namely that the L2 mean oscillation ωA,2 of A satisfiesXA,2:=lim supr→0r∫r2ωA,2(t)t2expC∗∫tRωA,2(s)sdsdt<∞,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$ X_{A,2} := \\limsup\\limits_{r\\rightarrow 0} r {{\\int \\limits }_{r}^{2}} \\frac {\\omega _{A,2}(t)}{t^{2}} \\exp \\left (C_{*} {{\\int \\limits }_{t}^{R}} \\frac {\\omega _{A,2}(s)}{s} ds\\right ) dt < \\infty , $\\end{document} where C∗ is a positive constant depending only on the dimensions and the ellipticity, then ∇u ∈ BMO.(b) If XA,2 = 0, then ∇u ∈ VMO.(c) Finally, examples satisfying XA,2 = 0 are given showing that it is not possible to prove the boundedness of ∇u in statement (b), nor the continuity of ∇u when nabla u in L^{infty } cap VMO.

Editorial: Tensor numerical methods and their application in scientific computing and data science

Editorial: Tensor numerical methods and their application in scientific computing and data science

On convergence rates of Kaczmarz-type methods with different selection rules of working rows

The Kaczmarz method is a classical while effective iteration method for solving very large-scale consistent systems of linear equations, and the randomized Kaczmarz method is an important and valuable variant of the Kaczmarz method. By theoretically analyzing and numerically experimenting several criteria typically adopted in the non-randomized and the randomized Kaczmarz method for selecting the working row, we derive sharper upper bounds for the convergence rates of some of the correspondingly induced Kaczmarz-type methods including those with respect to the maximal residual, maximal distance, and distance selection rules of the working row, and, for this whole suite of iteration methods consisting of the Kaczmarz methods with respect to the uniform, non-uniform, residual, distance, maximal residual, and maximal distance selection rules of the working row, we reveal their comparable relationships in terms of both mean-squared distance and mean-squared error, and show their computational effectiveness and numerical robustness based upon implementing a large number of test examples. Here the mean-squared distance is defined as the mean-value of the squared Euclidean norm of the current update increment of the iteration, and the mean-squared error is defined as the mean-value of the squared Euclidean norm of the current error that is the difference between the current iterate and the true solution of the target linear system.

The Differential Transform Method as an Effective Tool to Solve Implicit Hessenberg Index-3 Differential-Algebraic Equations

Differential-algebraic equations (DAEs) are important tools to model complex problems in various application fields easily. Those DAEs with an index-3, even the linear ones, are known to cause problems when solving them numerically. The present article proposes a new algorithm together with its multistage form to efficiently solve a class of nonlinear implicit Hessenberg index-3 DAEs. This algorithm is based on the idea of applying the differential transform method (DTM) directly to the DAE without applying the traditional index reduction methods, which can be complex and often result in violations of the DAE constraints. Also, to deal with the nonlinear terms in the DAE, we approximate them using the Adomian polynomials. This new idea has given us a simple and efficient algorithm, which involves the solution of linear algebraic systems except for the initial recursion terms. This algorithm is easy to implement in Maple or Mathematica. Furthermore, to enlarge the interval of convergence of the power series solution obtained from the DTM, an algorithm for the multistage DTM is also given. Both algorithms are applied to solve two examples of highly nonlinear implicit index-3 Hessenberg DAEs. Numerical results show that the DTM can determine the exact solution in convergent power series, while the multistage DTM can compute accurate numerical solutions over large intervals.

The Lawson‐Hanson algorithm with deviation maximization: Finite convergence and sparse recovery

SummaryThe Lawson‐Hanson with Deviation Maximization (LHDM) method is a block algorithm for the solution of NonNegative Least Squares (NNLS) problems. In this work we devise an improved version of LHDM and we show that it terminates in a finite number of steps, unlike the previous version, originally developed for a special class of matrices. Moreover, we are concerned with finding sparse solutions of underdetermined linear systems by means of NNLS. An extensive campaign of experiments is performed in order to evaluate the performance gain with respect to the standard Lawson‐Hanson algorithm. We also show the ability of LHDM to retrieve sparse solutions, comparing it against several ‐minimization solvers in terms of solution quality and time‐to‐solution on a large set of dense instances.

Acceleration of Nuclear Reactor Simulation and Uncertainty Quantification Using Low-Precision Arithmetic

In recent years, interest in approximate computing has been increasing significantly in many disciplines in the context of saving energy and computation cost by trading off on the quality of numerical simulation. The hardware acceleration based on the low-precision floating-point arithmetic is anticipated by the upcoming generation of microprocessors and code compilers and has already proven to be beneficial for weather and climate modelling and neural network training. The present work illustrates the application of low-precision arithmetic for the nuclear reactor core uncertainty analysis. We studied the performance of an elementary transient reactor core model for the arbitrary precision of the floating-point multiplication in a direct linear system solver. Using this model, we calculated the reactor core transients initiated by the control rod ejection taking into account the uncertainty of the model input parameters. Then, we evaluated the round-off errors of the model outputs for different precision levels. The comparison of the round-off errors and the model uncertainty showed the model could be run using a 15-bit floating-point number precision with an acceptable degradation of the result’s accuracy. This precision corresponds to a gain of about 6× in the bit complexity of the linear system solution algorithm, which can be actualized in terms of reduced energy costs on low-precision hardware.

Bit-complexity of classical solutions of linear evolutionary systems of partial differential equations

We study the bit-complexity intrinsic to solving the initial-value and (several types of) boundary-value problems for linear evolutionary systems of partial differential equations (PDEs), based on the Computable Analysis approach. Our algorithms are guaranteed to compute classical solutions to such problems approximately up to error 1/2n, so that n corresponds to the number of reliable bits of the output; bit-cost is measured with respect to n. Computational Complexity Theory allows us to prove in a rigorous sense that PDEs with constant coefficients are algorithmically ‘easier’ than general ones. Indeed, solutions to the latter are shown (under natural assumptions) computable using a polynomial number of memory bits, and we prove that the complexity class PSPACE is in general optimal; while the case of constant coefficients can be solved in #P—also essentially optimally so: the Heat Equation ‘requires’ #P1. Our algorithms raise difference schemes to exponential powers, efficiently: we compute any desired entry of such a power in #P, provided that the underlying exponential-sized matrices are circulant of constant bandwidth. Exponentially powering modular two-band circulant matrices is established even feasible in P; and under additional conditions, also the solution to certain linear PDEs becomes polynomial time computable.

Symmetry results of solutions for elliptic systems with linear couplings

This paper is devoted to study the symmetry and monotonicity of positive solutions for linear coupling elliptic systems in a ball in R N . Using the Alexandrov–Serrin method of moving planes combined with the strong maximum principle, we prove that the solutions of elliptic systems with linear couplings in a ball are symmetric w.r.t. 0 and radially decreasing. For our problems, the tangential gradient of solutions and the coupling conditions play important roles in using the moving plane method. Our results on the symmetry of solutions are further research based on the existence of solutions in [1].

Fast computational E-field dosimetry for transcranial magnetic stimulation using adaptive cross approximation and auxiliary dipole method (ACA-ADM)

Transcranial Magnetic Stimulation (TMS) is a non-invasive brain stimulation technique that uses a coil to induce an electric field (E-field) in the brain and modulate its activity. Many applications of TMS call for the repeated execution of E-field solvers to determine the E-field induced in the brain for different coil placements. However, the usage of solvers for these applications remains impractical because each coil placement requires the solution of a large linear system of equations. We develop a fast E-field solver that enables the rapid evaluation of the E-field distribution for a brain region of interest (ROI) for a large number of coil placements, which is achieved in two stages. First, during the pre-processing stage, the mapping between coil placement and brain ROI E-field distribution is approximated from E-field results for a few coil placements. Specifically, we discretize the mapping into a matrix with each column having the ROI E-field samples for a fixed coil placement. This matrix is approximated from a few of its rows and columns using adaptive cross approximation (ACA). The accuracy, efficiency, and applicability of the new ACA approach are determined by comparing its E-field predictions with analytical and standard solvers in spherical and MRI-derived head models. During the second stage, the E-field distribution in the brain ROI from a specific coil placement is determined by the obtained rows and columns in milliseconds. For many applications, only the E-field distribution for a comparatively small ROI is required. For example, the solver can complete the pre-processing stage in approximately 4 hours and determine the ROI E-field in approximately 40 ms for a 100 mm diameter ROI with less than 2% error enabling its use for neuro-navigation and other applications. Highlight: We developed a fast solver for TMS computational E-field dosimetry, which can determine the ROI E-field in approximately 40 ms for a 100 mm diameter ROI with less than 2% error.

Resolvent analysis of a finite wing in transonic flow

Shock waves interacting with turbulent boundary layers on wings can result first in self-sustained flow unsteadiness and eventually in structural vibration. Due to its importance to modern wing design and aircraft certification, the transonic flow physics continue to be investigated intensively. Herein we focus the discussion on three main aspects. First, we assess a practical implementation of an iterative resolvent algorithm in the linear harmonic incarnation of an industrial computational fluid dynamics code for computing optimal forcing and response modes. This heavily relies on the efficient solution of large sparse linear systems of equations. Second, we showcase its application as a predictive tool to detect transonic buffet flow unsteadiness, well before a global stability analysis can first identify its dynamics through weakly damped eigenmodes, using the NASA common research model at wind-tunnel conditions. Third, we discuss its ability to uncover modal physics, not identifiable through global stability analysis, revealing higher-frequency wake and wingtip vortex modes while shedding some light on the elusive finite wing equivalent of the aerofoil buffet mode. We demonstrate that earlier computational limitations of resolvent analysis, when solving the truncated singular value decomposition using matrix-forming methods with direct matrix factorisation, have been overcome ready for industrial use.

Parameters and Infinite Solutions of Linear Equation Systems

Resumen En el presente trabajo investigamos la utilidad didáctica de usar parámetros como variables de naturaleza dual (variables activas o inactivas), así como la actualización de cinco profesores mexicanos en este saber. En esta actualización se analizaron sistemas de ecuaciones lineales (SEL) con dos ecuaciones y tres incógnitas que tienen infinitas soluciones. Para ello propusimos, vía internet, cinco artefactos semióticos: 1. una definición, 2. la inclusión del parámetro en el SEL, 3. el cálculo de soluciones, 4. representaciones gráficas y 5. la aplicación a un problema en contexto. Los resultados revelaron la pertinencia de la introducción del parámetro mediante este proceso y, además, los profesores lograron un control sobre las soluciones válidas de entre las infinitas del sistema planteado. Para algunos profesores, la estrategia y el papel del parámetro fue sólo un método más que se agrega a los ya conocidos; sin embargo, para otros, permitió dar una explicación a la relación entre la variable y el parámetro, así como la posibilidad del uso controlado de las infinitas soluciones.