Ill-conditioned System Of Linear Equations Research Articles

Hybrid neuro-genetic based method for solving ill-posed inverse problem occurring in synthesis of electromagnetic fields

Fredholm Integral Equation of the First Kind (FIEFK) is an example of ill-posed problems. Solving this type of equation using conventional methods of discretization often leads to an ill-conditioned system of linear equations. This paper deals with the numerical solution for the FIEFK occurring in the synthesis of the electromagnetic fields. To tackle this problem, we propose a hybrid method based on Genetic Algorithms (GAs) and Artificial Neural Networks. The method consists of two major steps. The first step is to find an initial solution by utilizing a GA, and the second is to refine the solution using a regularized neural network. Experimental results prove the efficiency of our proposed method in comparison with a previous work.

A note on the pin-pointing solution of ill-conditioned linear system of equations

Volokh and Vilnay [Appl. Math. Lett. 13 (2000), pp. 119–124] proposed a method for computing an accurate solution of nearly singular linear systems of equations. The method is processed in two stages. Their method uses the truncated singular value decomposition of the initial coefficient matrix at the first stage and the Gaussian elimination procedure for well-conditioned reduced system of linear equations at the second stage. In this note, we propose a multistage approach based on the Volokh and Vilnay's method. Some numerical results are given to show the efficiency of the proposed method.

A numerical method for solving a nonlinear inverse parabolic problem

This study is motivated by a desire to obtain a numerical approach combining the use of the finite difference method with the solution of ordinary differential equation proposed for the determination of unknown coefficient in an inverse heat conduction problem. To regularize the resultant ill-conditioned linear system of equations, we apply the Tikhonov regularization method to obtain the stable numerical approximation to the solution.

THE METHOD OF FUNDAMENTAL SOLUTIONS FOR AN INVERSE INTERNAL BOUNDARY VALUE PROBLEM FOR THE BIHARMONIC EQUATION

In this paper, an inverse internal boundary value problem associated to the biharmonic equation is considered. The problem consists of determining unknown boundary conditions from extra interior measurements. The method of fundamental solutions (MFS) is used to discretize the problem and the resulting ill-conditioned system of linear equations is solved using the Tikhonov regularization technique. It is shown that, unlike the least-squares method, the MFS-regularization numerical technique produces stable and accurate numerical solutions for an appropriate choice of the regularization parameter given by the L-curve criterion.

A numerical technique for solving IHCPs using Tikhonov regularization method

This study is intended to provide a numerical algorithm for solving a one-dimensional inverse heat conduction problem. The given heat conduction equation, the boundary conditions, and the initial condition are presented in a dimensionless form. The numerical approach is developed based on the use of the solution to the auxiliary problem as a basis function. To regularize the resultant ill-conditioned linear system of equations, we apply the Tikhonov regularization method to obtain the stable numerical approximation to the solution.

A DISCREPANCY PRINCIPLE FOR THE SOURCE POINTS LOCATION IN USING THE MFS FOR SOLVING THE BHCP

Based on the discrepancy principle, we develop in this paper a new method of choosing the location of source points to solve the backward heat conduction problem (BHCP) by using the method of fundamental solutions (MFS). The standard Tikhonov regularization technique with the L curve method for an optimal regularized parameter is adopted for solving the resultant highly ill-conditioned system of linear equations. Numerical verifications of the proposed computational method are presented for both the one-dimensional and the two-dimensional BHCP.

Local tomography troposphere model over mountains area

The term GNSS meteorology refers to the utilization of the Global Navigation Satellite System's (GNSS) radio signals to derive information about the state of the troposphere. GNSS tomography allows to resolve the spatial structure and temporal behavior of the tropospheric water vapor. This paper presents the verification of GNSS tomography over dense local GNSS network. The paper addresses the problem of obtaining a stable tomographic solution from an ill-conditioned system of linear equations. The main interests are in suitable horizontal and vertical resolution in given conditions. Here the Moore–Penrose pseudo inverse of variance–covariance matrix is used. The minimum constraints solution is obtained with no additional assumptions. The results are validated with the help of simulated weather conditions. Three various scenarios are tested. As general output of this paper the optimal model construction scheme is presented with possible further improvements. The verification of the tomography model based on the local GPS KARKONOSZE, situated in the Karkonosze mountains area in Poland.

Reconstruction of a moving boundary from Cauchy data in one-dimensional heat equation

In this article, we propose a new numerical method for determining a moving boundary from Cauchy data in a one-dimensional heat equation where an initial temperature is not required. The numerical scheme is based on the use of fundamental solutions of the heat equation as basis functions. In order to regularize the ill-conditioned linear system of equations resulting by collocating boundary data, we successfully apply the Tikhonov regularization with the generalized cross-validation parameter choice rule to obtain a stable numerical approximation to the moving boundary. The uniqueness of the moving boundary is also proved under some assumptions.

Preconditioned Nullspace Method for the Two-Dimensional Oseen Problem

The Oseen problem, which arises in the simulation of the time-dependent Navier-Stokes equations for incompressible fluid flow, leads to indefinite, nonsymmetric and possibly ill-conditioned linear systems of equations. This paper presents a method for obtaining a reduced linear system from the original system, which is then solved by the preconditioned BiCGStab method. The system reduction is obtained through an efficient implicit representation of a basis of discretely divergence-free functions, also known as the nullspace method. We will show a close relationship between this preconditioned nullspace method and the BFBt-preconditioned pressure Schur complement problem arising in typical block preconditioners. We will use this relationship to analyze the spectrum of the preconditioned reduced system. We will also present numerical tests for the two-dimensional Oseen problem to illustrate the performance of this method, which is similar to the performance of the BFBt-preconditioned pressure Schur complement solver. In particular, it is robust w.r.t. the strength of the viscosity for simple constant wind but not for more complicated flows, and it shows only a slight dependence on the mesh size.

Wilkinson’s iterative refinement of solution with automatic step-size control for linear system of equations

In a recent paper [X.Y. Wu, R. Shao, Yiran Zhu, New iterative improvement of solution for an ill-conditioned system of linear equations based on a linear dynamic system, Comput. Math Appl. 44 (2002) 1109–1116], the authors analyzed the dynamic behavior for Wilkinson’s iterative improvement of solution and proposed a new iterative improvement of solution for an ill-conditioned system of linear equations based on a linear dynamic system. It was claimed that Wilkinson’s iterative improvement of solution can be regarded as the explicit Euler method with fixed step size h = 1 for the dynamic system. In this paper, the iterative improvement of solution with automatic step-size control is presented for linear system of algebraic equations. The convergence of the iterative procedure is shown for both stationary and non-stationary iteration, when the procedure is thought of as an iterative method. Numerical experiments are accompanied in comparison with Wilkinson’s iterative improvement.

Stochastic Matrices and Lp Norms : New Algorithms for Solving Ill-conditioned Linear Systems of Equations

We propose new iterative algorithms for solving a system of linear equations, possibly singular and inconsistent, presenting outstanding performances regarding ill-conditioning and error propagation. The basis of our approach is constructing with the l 1 norm, a preconditioning matrix C (an approximation of a generalized inverse of the matrix) such that the preconditioned matrix CA is stochastic. This property allows us to retrieve, in an original way, the Schultz-Hotelling-Bodewig's algorithm of iterative refinement of the approximate inverse of a matrix. The approach, valid for non-negative matrices, is then generalized to any complex, rectangular matrix. We are then able to compute a generalized inverse of any matrix and this inverse is fit for use in classical solving schemes such as : Richardson-Tanabe, Schultz-Hotelling-Bodewig, preconditioned conjugate gradients and also in the Kaczmarz scheme (that we have generalized using l p norms). Regarding the obtained results on pathological well-known test-cases such as Hilbert and Nakasaka matrices, some of the proposed algorithms are empirically shown to be more efficient than the known classical techniques.

Error analysis for the predictor–corrector process relating to ill-conditioned linear system of equations

In [Xinyuan Wu, An effective iterative procedure process for large scale linear system of equations, Appl. Math. Comput. 180 (1) (2006) 160–166; Xinyuan Wu, Note on predictor–corrector process for ill-conditioned linear system of equations, Appl. Math. Comput., in press, doi:10.1016/j.amc.2006.05.075], the author proposed predictor–corrector processes for large scale linear system of algebraic equations with ill-conditioned coefficient matrix, however, the error analysis for the predictor–corrector processes relating to ill-conditioned linear system of equations has not been given. This paper presents the results on error analysis for the predictor–corrector process proposed in Wu (in press).

Note on predictor–corrector process for ill-conditioned linear system of equations

In [Xinyuan Wu, An effective iterative procedure process for large scale linear system of equations, Appl. Math. Comput., in press], the author proposed a predictor–corrector process for large scale linear system of algebraic equations with ill-conditioned coefficient matrix. Basing on the same principle, this article introduces another iterative process which simplifies and reduces the computational procedure. Numerical experiments are included to show the efficiency of the process.

Transient inverse design of radiative enclosures for thermal processing of materials

This work presents a methodology to solve transient inverse design of radiative enclosures for heating processes that require refined temperature control. The proposed methodology is applied to find the heat input to a heater located at the top of a three-dimensional enclosure that can satisfy a prescribed time-dependent temperature curve on a surface located at the base of the enclosure. The process is governed by radiative exchanges between diffuse, gray surfaces. This problem is described by an ill-conditioned system of linear equations, which is regularized by the truncated singular value decomposition (TSVD) method. The inverse analysis led to a heat input in the heater that assured, within an error less than 1.0%, both uniformity and the correct magnitude of the design surface temperature in every instant of the process.

Regulation of a human thermal environment by inverse method

This work proposes a method for designing a human thermal environment. The aim is to determine the thermal surroundings (wall and air temperatures) of a complex enclosure that can provide a desired human thermal state. Such a mathematical process is called inverse modeling described by an ill-conditioned system of linear equations. This problem is regularized by means of the Tikhonov method. The system becomes nonlinear when convective phenomena are considered. The inverse analysis leads to optimal surrounding conditions that, within an acceptable error level, ensure both the temperature and heat flux required on the human body to maintain thermal comfort conditions. An application to driver in his car is presented.

Inverse radiative design in human thermal environment

This work investigates inverse boundary design for radiative heat transfer applied to human thermal environment. The problem consists in finding the temperatures of certain surfaces in a complex configuration around the human being that satisfy both the temperature and the heat flux prescribed on his body. Such a mathematical procedure is called inverse modeling which is described by an ill-conditioned system of linear equations based on the absorption factors method. The solution is obtained by regularizing the system of equations by the Tikhonov method. As a result we obtain optimized conditions for a complex human thermal system.

A fundamental solution method for inverse heat conduction problem

In this paper, we develop a new meshless and integration-free numerical scheme for solving an inverse heat conduction problem. The numerical scheme is developed based on the use of the fundamental solution as a radial basis function. To regularize the resultant ill-conditioned linear system of equations, we apply successfully both the Tikhonov regularization technique and the L-curve method to obtain a stable numerical approximation to the solution. The approach is readily extendable to solve high-dimensional problems under irregular domain.

New iterative improvement of a solution for an Ill-conditioned system of linear equations based on a linear dynamic system

In this paper, the analysis of the dynamic system for iterative improvement of a solution is discussed, and the new iterative improvement of a solution is proposed based on the dynamic system. We have proved that the new iterative improvement of the solution is convergent unconditionally. The numerical experiments illustrate that the new iterative improvement of the solution is more effective for an ill-conditional system of linear equations.

Regularized boundary element solution for an inverse boundary value problem in linear elasticity

AbstractIn this paper the Tikhonov regularization method is analysed for the ill‐conditioned system of linear equations which arises from the discretization of an ill‐posed boundary value problem in linear elasticity. The choice of the regularization parameter is achieved according to the L‐curve method. Then the regularization method is numerically implemented using the boundary element method (BEM). Copyright © 2002 John Wiley & Sons, Ltd.

Lanczos-Based Exponential Filtering for Discrete Ill-Posed Problems

We describe regularizing iterative methods for the solution of large ill-conditioned linear systems of equations that arise from the discretization of linear ill-posed problems. The regularization is specified by a filter function of Gaussian type. A parameter μ determines the amount of regularization applied. The iterative methods are based on a truncated Lanczos decomposition and the filter function is approximated by a linear combination of Lanczos polynomials. A suitable value of the regularization parameter is determined by an L-curve criterion. Computed examples that illustrate the performance of the methods are presented.