Solution Of Ill-posed Problems Research Articles

On the conditions of convergence for one class of methods used for the solution of ill-posed problems

We propose a new class of projection methods for the solution of ill-posed problems with inaccurately specified coefficients. For methods from this class, we establish the conditions of convergence to the normal solution of an operator equation of the first kind. We also present additional conditions for these methods guaranteeing the convergence with a given rate to normal solutions from a certain set.

The impact of temporal regularization on estimates of the BOLD hemodynamic response function: A comparative analysis

In fMRI data analysis it has been shown that for a wide range of situations the hemodynamic response function (HRF) can be reasonably characterized as the impulse response function of a linear and time invariant system. An accurate and robust extraction of the HRF is essential to infer quantitative information about the relative timing of the neuronal events in different brain regions. When no assumptions are made about the HRF shape, it is most commonly estimated using time windowed averaging or a least squares estimated general linear model based on either Fourier or delta basis functions. Recently, regularization methods have been employed to increase the estimation efficiency of the HRF; typically these methods produce more accurate HRF estimates than the least squares approach [Goutte, C., Nielsen, F.A., Hansen, L.K., 2000. Modeling the Haemodynamic Response in fMRI Using Smooth FIR Filters. IEEE Trans. Med. Imag. 19(12), 1188–1201.]. Here, we use simulations to clarify the relative merit of temporal regularization based methods compared to the least squares methods with respect to the accuracy of estimating certain characteristics of the HRF such as time to peak (TTP), height (HR) and width (W) of the response. We implemented a Bayesian approach proposed by Marrelec et al. [Marrelec, G., Benali, H., Ciuciu, P., Pelegrini-Issac, M., Poline, J.-B., 2003. Robust Estimation of the Hemodynamic Response Function in Event-Related BOLD fMRI Using Basic Physiological Information. Hum. Brain Mapp. 19, 1–17., Marrelec, G., Benali, H., Ciuciu, P., Poline, J.B. Bayesian estimation of the hemodynamic of the hemodynamic response function in functional MRI. In: R. F, editor; 2001; Melville. p 229–247.] and its deterministic counterpart based on a combination of Tikhonov regularization [Tikhonov, A.N., Arsenin, V.Y., 1977. Solution of ill-posed problems. Washington DC: W.H. Winston.] and generalized cross-validation (GCV) [Wahba, G., 1990. Spline Models for Observational Data. Philadelphia: SIAM.] for selecting the regularization parameter. The performance of both methods is compared with least square estimates as a function of temporal resolution, color and strength of the noise, and the type of stimulus sequences used. In almost all situations, under the considered assumptions (e.g. linearity, time invariance and smooth HRF), the regularization-based techniques more accurately characterize the HRF compared to the least-squares method. Our results clarify the effects of temporal resolution, noise color, and experimental design on the accuracy of HRF estimation.

Solution of ill-posed problems on sets of functions convex along all lines parallel to coordinate axes

In the paper we consider linear ill-posed problems on sets of functions convex upwards or downwards along all lines that belong to a functions' domain and are parallel to coordinate axes. A regularizing algorithm is constructed such that an approximate solution tends to the exact one uniformly of some subsets of the domain. The algorithms to estimate an error of finite dimensional approximation and to find a lower and an upper functions that bound all approximation solutions are provided. As a model example, an inverse problem for a two-dimensional heat conduction equation is solved.

Electric conductivity measurement of oxides melts

Notwithstanding on variety of existing processes of induction skull melting of oxides, electrical and thermal properties of oxides melts are practically unknown. From the other hand, liquid-phase synthesis of Hi-Tech oxides materials such as new ceramics, monocrystals and glasses, requires knowledge of melts properties up to 3700 K. Inasmuch as the main physical property, which influence to the melt power consumption at induction heating is electrical conductivity then study of electrical conductivity temperature dependence is necessary. The principle and technical realization of the electrical conductivity measurement of melts are presented. The approach is based on induction melting in cold crucible technology and inverse electromagnetic problem solution. Analysis of the method accuracy via input data errors is presented. For increasing of Ill-posed problem solution was used over determination of set of equations. Conductivity of aluminum oxide melts for the temperature up to 3223 K is shown too.

Estimation of discontinuous solutions of ill-posed problems via adaptive grid regularization

Estimation of discontinuous solutions of ill-posed problems via adaptive grid regularization

Estimation of discontinuous solutions of ill-posed problems via adaptive grid regularization

Estimation of discontinuous solutions of ill-posed problems via adaptive grid regularization

Strong convergence results for hemivariational inequalities

The purpose of this paper is to study a regularization method of solutions of ill-posed problems involving hemivariational inequalities in Banach spaces. Under the assumption that the hemivariational inequality is solvable, a strongly convergent approximation procedure is designed by means of the so-called Browder-Tikhonov regularization method. Our results generalize and extend the previously known theorems.

Comparing different types of approximators for choosing the parameters in the regularization of ill-posed problems

Regularized approximations to the solutions of ill-posed problems typically vary from over-smoothed, inaccurate reconstructions to under-smoothed and unstable solutions as the regularization parameter varies about its optimal value. It thus makes sense to compare two (or more) regularized approximations, and seek the parametervalues where the distance between the approximations is a minimum. This paper advances the theory and practice of this methodology. The method appears to workvery well and be very stable, particularly in the presence of extreme error levels.

On the use of Genetic Algorithms for Solving Ill-Posed Problems

In this article we investigate the use of genetic algorithms for regularizing ill-posed problems. Various approaches of genetic algorithms are considered in order to deal with the numerical problems caused by the instability of the solution of ill-posed problems. Two test examples, a simple one-dimensional ill-posed problem and the Cauchy problem for the Laplace equation are considered in order to illustrate the performance and limitations of genetic algorithms when applied to inverse and ill-posed problems. The combination of genetic algorithms with other regularization methods such as the function specification method is also investigated.

EXAFS investigations of Cu–Mg–O compound

Abstract The interest to systems containing copper oxide is connected with the problem of high-temperature superconductivity because of the closeness of its basic physical properties and properties of superconductor mother Cu-compounds. In this work, EXAFS study of the Cu 0.2 Mg 0.8 O compound is presented. A new iterative algorithm of the solution of ill-posed problem on determining three partial pair correlation functions from one EXAFS-data set near the Cu K-edge is described. The results of X-ray scattering study of a given sample show a presence of a single phase with the MgO structure and a lattice parameter of 4.219 A instead of 4.208 A for pure MgO. From the EXAFS investigations, we find the local distortion of the lattice. We revealed that the short range order differs both from a hypothetical alloy with the MgO structure and from copper oxide.

Regularization for Curve Representations: Uniform Convergence for Discontinuous Solutions of Ill-Posed Problems

This paper is concerned with a new approach for regularizing problems with discontinuous solutions: regularization for curve representations. The idea of this approach is to represent a (discontinous) function as a curve with parameterization (a(t),b(t)). A combination with nonlinear Tikhonov regularization then yields uniform convergence of the regularized solutions. The method is applied to deblurring and denoising problems in signal processing. A numerical example for a deblurring problem is presented.

On finite-dimensional approximation of solutions of ill-posed problems

We show that the modified method for finite-dimensional approximation of solutions of Fredholm integral equations of the first kind presented in this paper is more economical than traditional methods for finite-dimensional approximation.

Asymptotics of orthogonal polynomials and the numerical solution of ill-posed problems

The paper reviews the impact of modern orthogonal polynomial theory on the analysis of numerical algorithms for ill-posed problems. Of major importance are uniform bounds for orthogonal polynomials on the support of the weight function, the growth of the extremal zeros, and asymptotics of the Christoffel functions.

On the optimization of projection-iterative methods for the approximate solution of ill-posed problems

We consider a new version of the projection-iterative method for the solution of operator equations of the first kind. We show that it is more economical in the sense of amount of used discrete information.

Complexity of projective methods for the solution of ill-posed problems

We consider the problem of finite-dimensional approximation for solutions of equations of the first kind and propose a modification of the projective scheme for solving ill-posed problems. We show that this modification allows one to obtain, for many classes of equations of the first kind, the best possible order of accuracy for the Tikhonov regularization by using an amount of information which is far less than for the standard projective technique.

Expectation Minimum (EM): A New Principle for the Solution of Ill-Posed Problems in Photothermal Science

The expectation-minimum (EM) principle is a new strategy for recovering robust solutions to the ill-posed inverse problems of photothermal science. The expectation-minimum principle uses the addition of well-characterized random noise to a model basis to be fitted to the experimental response by linear minimization or projection techniques. The addition of noise to the model basis improves the conditioning of the basis by many orders of magnitude. Multiple projections of the data onto the basis in the presence of noise are averaged, to give the solution vector as an expectation value which reliably estimates the global minimum solution for general cases, while the conventional approaches fail. This solution is very stable in the presence of random error on the data. The expectation-minimum principle has been demonstrated in conjunction with several projection algorithms. The nature of the solutions recovered by the expectation minimum principle is nearly independent of the minimization algorithms used and depends principally on the noise level set in the model basis.

The Mollification Method and the Numerical Solution of Ill-Posed Problems (Diego A. Murio)

The Mollification Method and the Numerical Solution of Ill-Posed Problems (Diego A. Murio)

Discussion of the Paper by Vardi and Lee

Discussion of the Paper by Vardi and Lee

On the solution of ill-posed problems with weakly dependent data errors

On the solution of ill-posed problems with weakly dependent data errors

Effect of the incompressibility condition on the stability of the solutions of ill-posed problems of the theory of elasticity

One considers the energy functional with density depending only on the first invariant of the strain tensor. Under natural constraints on the density and in the absence of exterior actions, the zero displacements yield a minimum to the energy functional both for the compressible and the incompressible media. In the paper one proves the isolation of this minimum for incompressible media and its nonisolation for compressible media.