Inverse problems are typically ill-posed or ill-conditioned and require regularization. Tikhonov regularization is a popular approach and it requires an additional parameter called the regularization parameter that has to be estimated. The χ method introduced by Mead in [8] uses the χ distribution of the Tikhonov functional for linear inverse problems to estimate the regularization parameter. H...

We address discrete nonlinear inverse problems with weighted least squares and Tikhonov regularization. Regularization is a way to add more information to the problem when it is ill-posed or ill-conditioned. However, it is still an open question as to how to weight this information. The discrepancy principle considers the residual norm to determine the regularization weight or parameter, while ...

The rst part of this paper studies a Levenberg-Marquardt scheme for nonlinear inverse problems where the corresponding Lagrange (or regularization) parameter is chosen from an inexact Newton strategy. While the convergence analysis of standard implementations based on trust region strategies always requires the invertibility of the Fr echet derivative of the nonlinear operator at the exact solu...

For joint estimation of state variables and unknown parameters, adaptive observers usually assume some persistent excitation (PE) condition. In practice, the PE condition may not be satisfied, because underlying recursive problem is ill-posed. To remedy lack condition, inspired by ridge regression, this paper proposes a regularized observer with enhanced parameter adaptation gain. Like in typic...

For good image quality using ultrasound inverse scattering, one alternately solves the well-posed forward scattering equation for an estimated total field and the ill-posed inverse scattering equation for the desired object property function. In estimating the total field, error or noise contaminates the coefficients of both matrix and data of the inverse scattering equation. Previous work on i...

generally, the presence of noise in geophysical measurements is inevitable and depending on the type and the level it affects the results of geophysical studies. so, denoising is an important part of the processing of geophysical data. on the other hand, geophysicists make inferences about the physical properties of the earth interior based on the indirect measurements (data) collected at or ne...

The weILposedr7ess of the problems is not always guaranteed in inverse problems, unlike the forward problems. Dnts, a number of methods for giving wellposedrjess hm?e been studied in mathematical fields. In the ,field qf neural! networks, the network inversion method. for solving inverse problems was proposed; it is useflll but does not dissolute the ill-posedness of inverse problems. To overco...

Instead of the Tikhonov regularization method which with a scalar being the regularization parameter, Liu et al. [1] have proposed a novel regularization method with a vector as being the regularization parameter. As a continuation we further propose an optimally scaled vector regularization method (OSVRM) to solve the ill-posed linear problems, which is better than the Tikhonov regularization ...

Estimation of the distribution function f and potential of hot stellar systems from kinematical data is discussed. When the functional forms of f and are not speciied a priori, accurate estimation of either function requires very high quality data : either accurate \line prooles" at radii extending well beyond an eeective radius, or large samples (N > 10 3) of discrete radial velocities. Estima...