| Projection methods based on wavelet functions combine optimal convergence rates with algorithmic eeciency. The proofs in this paper utilize the approximation properties of wavelets and results from the general theory of regularization methods. Moreover, adaptive strategies can be incorporated still leading to optimal convergence rates for the resulting algorithms. The so-called wavelet-vaguel...

The methods applied to regularization of the ill-posed problems can be classified under “direct” and “indirect” methods. Practice has shown that the effects of different regularization techniques on an ill-posed problem are not the same, and as such each ill-posed problem requires its own investigation in order to identify its most suitable regularization method. In the geoid computations witho...

In this paper we reduce a free boundary problem from heat transfer to a weakly Singular Volterra  integral equation of the first kind. Since the first kind integral equation is ill posed, and an appropriate method for such ill posed problems is based on wavelets, then we apply the Chebyshev wavelets to solve the integral equation. Numerical implementation of the method is illustrated by two ben...

Ill posed quadratic optimization frequently occurs in control and inverse problems and are not covered by the Lax-Milgram-Riesz theory. Typically small changes in the input data can produce very large oscillations on the output. We investigate the conditions under which the minimum value of the cost function is finite and we explore the ‘hidden connection’ between the optimization problem and t...

A new approach to solving linear ill-posed problems is proposed. The approach consists of solving a Cauchy problem for a linear operator equation and proving that this problem has a global solution whose limit at infinity solves the original linear equation.

A parameter of an econometric model is identified if there is a one-to-one or many-to-one mapping from the population distribution of the available data to the parameter. Often, this mapping is obtained by inverting a mapping from the parameter to the population distribution. If the inverse mapping is discontinuous, then estimation of the parameter usually presents an illposed inverse problem. ...

In this paper we consider nonlinear ill-posed problems with piecewise constant or strongly varying solutions. A class of nonlinear regularization methods is proposed, in which smooth approximations to the Heavyside function are used to reparameterize functions in the solution space by an auxiliary function of levelset type. The analysis of the resulting regularization methods is carried out in ...

In the Bayesian approach to ill-posed inverse problems, regularization is imposed by specifying a prior distribution on the parameters of interest and Markov chain Monte Carlo samplers are used to extract information about its posterior distribution. The aim of this paper is to investigate the convergence properties of the random-scan random-walk Metropolis (RSM) algorithm for posterior distrib...