scholarly journals A Numerical Solution Using an Adaptively Preconditioned Lanczos Method for a Class of Linear Systems Related with the Fractional Poisson Equation

2008 ◽  
Vol 2008 ◽  
pp. 1-26 ◽  
Author(s):  
M. Ilić ◽  
I. W. Turner ◽  
V. Anh
Keyword(s):  
Linear System ◽  
Linear Systems ◽  
Poisson Equation ◽  
Matrix Function ◽  
Fractional Power ◽  
Analytic Solutions ◽  
Linear System Of Equations ◽  
Exact Analytic Solutions ◽  
The Matrix ◽  
Symmetric Positive Definite Matrices

This study considers the solution of a class of linear systems related with the fractional Poisson equation (FPE) (−∇2)α/2φ=g(x,y) with nonhomogeneous boundary conditions on a bounded domain. A numerical approximation to FPE is derived using a matrix representation of the Laplacian to generate a linear system of equations with its matrix A raised to the fractional power α/2. The solution of the linear system then requires the action of the matrix function f(A)=A−α/2 on a vector b. For large, sparse, and symmetric positive definite matrices, the Lanczos approximation generates f(A)b≈β0Vmf(Tm)e1. This method works well when both the analytic grade of A with respect to b and the residual for the linear system are sufficiently small. Memory constraints often require restarting the Lanczos decomposition; however this is not straightforward in the context of matrix function approximation. In this paper, we use the idea of thick-restart and adaptive preconditioning for solving linear systems to improve convergence of the Lanczos approximation. We give an error bound for the new method and illustrate its role in solving FPE. Numerical results are provided to gauge the performance of the proposed method relative to exact analytic solutions.

scholarly journals H-Matrices in Fuzzy Linear Systems

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
H. Saberi Najafi ◽  
S. A. Edalatpanah
Keyword(s):  
Linear System ◽  
Theoretical Analysis ◽  
Linear Systems ◽  
Numerical Experiments ◽  
Coefficient Matrix ◽  
System Of Equations ◽  
Linear System Of Equations ◽  
Fuzzy Linear System ◽  
Fuzzy Linear Systems

We consider a class of fuzzy linear system of equations and demonstrate some of the existing challenges. Furthermore, we explain the efficiency of this model when the coefficient matrix is an H-matrix. Numerical experiments are illustrated to show the applicability of the theoretical analysis.

scholarly journals Statistical applications for equivariant matrices

2001 ◽  
Vol 25 (1) ◽  
pp. 53-61
Author(s):  
S. H. Alkarni
Keyword(s):  
Linear Systems ◽  
Fourier Transforms ◽  
Toeplitz Matrices ◽  
Computational Cost ◽  
Special Kind ◽  
Circulant Matrices ◽  
Stationary Processes ◽  
Linear System Of Equations ◽  
The Matrix ◽  
A Finite Group

Solving linear system of equationsAx=benters into many scientific applications. In this paper, we consider a special kind of linear systems, the matrixAis an equivariant matrix with respect to a finite group of permutations. Examples of this kind are special Toeplitz matrices, circulant matrices, and others. The equivariance property ofAmay be used to reduce the cost of computation for solving linear systems. We will show that the quadratic form is invariant with respect to a permutation matrix. This helps to know the multiplicity of eigenvalues of a matrix and yields corresponding eigenvectors at a low computational cost. Applications for such systems from the area of statistics will be presented. These include Fourier transforms on a symmetric group as part of statistical analysis of rankings in an election, spectral analysis in stationary processes, prediction of stationary processes and Yule-Walker equations and parameter estimation for autoregressive processes.

scholarly journals New iterative methods for dense linear systems

2021 ◽  
Vol 293 ◽  
pp. 02013
Author(s):  
Jinmei Wang ◽  
Lizi Yin ◽  
Ke Wang
Keyword(s):  
Linear System ◽  
Linear Systems ◽  
Iterative Methods ◽  
Parallel Implementation ◽  
System Of Equations ◽  
Systems Of Equations ◽  
Linear System Of Equations ◽  
Dense Linear System ◽  
Linear Systems Of Equations

Solving dense linear systems of equations is quite time consuming and requires an efficient parallel implementation on powerful supercomputers. Du, Zheng and Wang presented some new iterative methods for linear systems [Journal of Applied Analysis and Computation, 2011, 1(3): 351-360]. This paper shows that their methods are suitable for solving dense linear system of equations, compared with the classical Jacobi and Gauss-Seidel iterative methods.

scholarly journals Optimal design of optical analog solvers of linear systems

2021 ◽  
Vol 2 (6) ◽  
Author(s):  
Kthim Imeri
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Optimal Design ◽  
Linear System ◽  
Linear Systems ◽  
Asymptotic Expansions ◽  
Physical Method ◽  
System Of Equations ◽  
Linear System Of Equations ◽  
Optical Analog

AbstractIn this paper, given a linear system of equations $$\mathbf {A}\, \mathbf {x}= \mathbf {b}$$ A x = b , we are finding locations in the plane to place objects such that sending waves from the source points and gathering them at the receiving points solves that linear system of equations. The ultimate goal is to have a fast physical method for solving linear systems. The issue discussed in this paper is to apply a fast and accurate algorithm to find the optimal locations of the scattering objects. We tackle this issue by using asymptotic expansions for the solution of the underlying partial differential equation. This also yields a potentially faster algorithm than the classical BEM for finding solutions to the Helmholtz equation.

scholarly journals Solving rank one revised linear systems by the scaled ABS method

2004 ◽  
Vol 46 (2) ◽  
pp. 225-236
Author(s):  
Keyvan Amini
Keyword(s):  
Mathematical Programming ◽  
Linear System ◽  
General Solution ◽  
Linear Systems ◽  
Matrix Coefficient ◽  
Active Set ◽  
Constraint Matrix ◽  
Active Set Strategies ◽  
The Matrix ◽  
Rank One

AbstractIn mathematical programming, an important tool is the use of active set strategies to update the current solution of a linear system after a rank one change in the constraint matrix. We show how to update the general solution of a linear system obtained by use of the scaled ABS method when the matrix coefficient is subjected to a rank one change.

scholarly journals Properweak regular splitting and its application to convergence of alternating iterations

Filomat ◽  
2018 ◽  
Vol 32 (19) ◽  
pp. 6563-6573 ◽  
Author(s):  
Debasisha Mishra
Keyword(s):  
Linear System ◽  
Convergence Rate ◽  
Iterative Method ◽  
Linear Systems ◽  
Iterative Scheme ◽  
Convergence Theory ◽  
System Of Equations ◽  
Comparison Result ◽  
Linear System Of Equations ◽  
Regular Splitting

Theory of matrix splittings is a useful tool for finding the solution of a rectangular linear system of equations, iteratively. The purpose of this paper is two-fold. Firstly, we revisit the theory of weak regular splittings for rectangular matrices. Secondly, we propose an alternating iterative method for solving rectangular linear systems by using the Moore-Penrose inverse and discuss its convergence theory, by extending the work of Benzi and Szyld [Numererische Mathematik 76 (1997) 309-321; MR1452511]. Furthermore, a comparison result is obtained which ensures the faster convergence rate of the proposed alternating iterative scheme.

scholarly journals Exact solution for the Poisson field in a semi-infinite strip

2017 ◽  
Vol 473 (2200) ◽  
pp. 20160908
Author(s):  
Yossi Cohen ◽  
Daniel H. Rothman
Keyword(s):  
Poisson Equation ◽  
Analytic Solutions ◽  
Length Scale ◽  
Infinite Strip ◽  
Physical Processes ◽  
The Poisson Equation ◽  
Exact Analytic Solutions ◽  
Square Root Singularity ◽  
Poisson Networks ◽  
Poisson Field

The Poisson equation is associated with many physical processes. Yet exact analytic solutions for the two-dimensional Poisson field are scarce. Here we derive an analytic solution for the Poisson equation with constant forcing in a semi-infinite strip. We provide a method that can be used to solve the field in other intricate geometries. We show that the Poisson flux reveals an inverse square-root singularity at a tip of a slit, and identify a characteristic length scale in which a small perturbation, in a form of a new slit, is screened by the field. We suggest that this length scale expresses itself as a characteristic spacing between tips in real Poisson networks that grow in response to fluxes at tips.

CRYSTALLINE GRAVITY

2009 ◽  
Vol 24 (18n19) ◽  
pp. 3243-3255 ◽  
Author(s):  
GERARD 't HOOFT
Keyword(s):  
Finite Number ◽  
Lorentz Group ◽  
Degrees Of Freedom ◽  
Holonomy Group ◽  
Discrete Subgroup ◽  
Analytic Solutions ◽  
Space Time ◽  
Exact Analytic Solutions ◽  
Finite Domains ◽  
Dimensional Crystal

Matter interacting classically with gravity in 3+1 dimensions usually gives rise to a continuum of degrees of freedom, so that, in any attempt to quantize the theory, ultraviolet divergences are nearly inevitable. Here, we investigate a theory that only displays a finite number of degrees of freedom in compact sections of space-time. In finite domains, one has only exact, analytic solutions. This is achieved by limiting ourselves to straight pieces of string, surrounded by locally flat sections of space-time. Next, we suggest replacing in the string holonomy group, the Lorentz group by a discrete subgroup, which turns space-time into a 4-dimensional crystal with defects.

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