While experimental design for well-posed inverse linear problems has been well studied, covering a vast range of well-established design criteria and optimization algorithms, its ill-posed counterpart is a rather new topic. The ill-posed nature of the problem entails the incorporation of regularization techniques. The consequent non-stochastic error introduced by regularization, needs to be tak...

Many theoretical and practical problems in science involve acquisition of data via indirect observations of a model or phenomena. Naturally, the observed data are determined by the physical properties of the model sought, as well as by the physical laws that govern the problem, however, they also depend on the experimental configuration. Unlike the model and the physical laws, the latter can be...

Many problems of physical chemistry belong to the class of inverse problems, in which from known experimental data of the object we need to determine some of its properties based on a certain model connecting these properties with measured characteristics. Inverse problems typically lead to mathematical models that are not well-posed in the sense of Hadamard, i.e. to the ill-posed problems. Thi...

Inverse problems focus on the problem of determining parameters and data inherent in the mathematical model of a physical or biological phenomenon from measurements of the observable data. Such problems are almost invariably ill-posed in the sense that in general existence, uniqueness or continuous dependence on the data is no longer true. In this chapter we have chosen three “canonical” exampl...

Within this chapter we present recent results on sparse recovery algorithms for inverse and ill-posed problems, i.e. we focus on those inverse problems in which we can assume that the solution has a sparse series expansion with respect to a preassigned basis or frame. The presented approaches to approximate solutions of inverse problems are limited to iterative strategies that essentially rely ...

The inverse kinematics problem for redundant manipulators is ill-posed and nonlinear. There are two fundamentally different issues which result in the need for some form of regularization; the existence of multiple solution branches (global ill-posedness) and the existence of excess degrees of freedom (local illposedness). For certain classes of manipulators, learning methods applied to input-o...

Nonparametric estimation of a structural cointegrating regression model is studied. As in the standard linear cointegrating regression model, the regressor and the dependent variable are jointly dependent and contemporaneously correlated. In nonparametric estimation problems, joint dependence is known to be a major complication that affects identification, induces bias in conventional kernel es...

Nonparametric estimation of a structural cointegrating regression model is studied. As in the standard linear cointegrating regression model, the regressor and the dependent variable are jointly dependent and contemporaneously correlated. In nonparametric estimation problems, joint dependence is known to be a major complication that affects identification, induces bias in conventional kernel es...

An adjoint based functional optimization technique in conjunction with the spectral stochastic finite element method is proposed for the solution of an inverse heat conduction problem in the presence of uncertainties in material data, process conditions and measurement noise. The ill-posed stochastic inverse problem is restated as a conditionally well-posed L2 optimization problem. The gradient...