Search results for "Linear equation"

Systems of Linear Equations

2016

A linear equation in \(\mathbb {R}\) in the variables \(x_1,x_2,\ldots ,x_n\) is an equation of the kind:

Partially Implicit Runge-Kutta Methods for Wave-Like Equations

2014

Runge-Kutta methods are used to integrate in time systems of differential equations. Implicit methods are designed to overcome numerical instabilities appearing during the evolution of a system of equations. We will present partially implicit Runge-Kutta methods for a particular structure of equations, generalization of a wave equation; the partially implicit term refers to this structure, where the implicit term appears only in a subset of the system of equations. These methods do not require any inversion of operators and the computational costs are similar to those of explicit Runge-Kutta methods. Partially implicit Runge-Kutta methods are derived up to third-order of convergence. We ana…

A Lagrange Multiplier Based Domain Decomposition Method for the Solution of a Wave Problem with Discontinuous Coefficients

2008

In this paper we consider the numerical solution of a linear wave equation with discontinuous coefficients. We divide the computational domain into two subdomains and use explicit time difference scheme along with piecewise linear finite element approximations on semimatching grids. We apply boundary supported Lagrange multiplier method to match the solution on the interface between subdomains. The resulting system of linear equations of the “saddle-point” type is solved efficiently by a conjugate gradient method.

Asymptotical Convergence Evaluation for a Parabolic Problem Arising in Circuit Theory

1990

Non-linear Local Polynomial Regression Multiresolution Methods Using $$\ell ^1$$-norm Minimization with Application to Signal Processing

2015

Harten’s Multiresolution has been developed and used for different applications such as fast algorithms for solving linear equations or compression, denoising and inpainting signals. These schemes are based on two principal operators: decimation and prediction. The goal of this paper is to construct an accurate prediction operator that approximates the real values of the signal by a polynomial and estimates the error using \(\ell ^1\)-norm in each point. The result is a non-linear multiresolution method. The order of the operator is calculated. The stability of the schemes is ensured by using a special error control technique. Some numerical tests are performed comparing the new method with…

Generalized Harnack inequality for semilinear elliptic equations

2015

Abstract This paper is concerned with semilinear equations in divergence form div ( A ( x ) D u ) = f ( u ) , where f : R → [ 0 , ∞ ) is nondecreasing. We introduce a sharp Harnack type inequality for nonnegative solutions which is a quantified version of the condition for strong maximum principle found by Vazquez and Pucci–Serrin in [30] , [24] and is closely related to the classical Keller–Osserman condition [15] , [22] for the existence of entire solutions.

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

2018

[EN] 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 106 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…

Motion of the wave-function zeros in spin-boson systems.

1995

In the analytic Bargmann representation associated with the harmonic oscillator and spin coherent states, the wave functions considered as consisting of entire complex functions can be factorized in terms of their zeros in a unique way. The Schr\"odinger equation of motion for the wave function is turned to a system of equations for the zeros of the wave function. The motion of these zeros as a nonlinear flow of points is studied and interpreted for linear and nonlinear bosonic and spin Hamiltonians. Attention is given to the study of the zeros of the Jaynes-Cummings model and to its finite analog. Numerical solutions are derived and dicussed.

Color decomposition of multi-quark one-loop QCD amplitudes

2014

In this talk we discuss the color decomposition of tree-level and one-loop QCD amplitudes with arbitrary numbers of quarks and gluons. We present a method for the decomposition of partial amplitudes into primitive amplitudes, which is based on shuffle relations and is purely combinatorial. Closed formulae are derived, which do not require the inversion of a system of linear equations.

Multiple solutions for strongly resonant Robin problems

2018

We consider nonlinear (driven by the p†Laplacian) and semilinear Robin problems with indefinite potential and strong resonance with respect to the principal eigenvalue. Using variational methods and critical groups, we prove four multiplicity theorems producing up to four nontrivial smooth solutions.