Akaike's Bayesian information criterion (ABIC) has been widely used in geophysical inversion and beyond. However, little done to investigate its statistical aspects. We present an alternative derivation of the marginal distribution measurements, whose maximization directly leads invention ABIC by Akaike. show that is statistically estimate variance measurements prior maximizing measurements. Th...

Many inverse problems are concerned with the estimation of non-negative parameter functions. In this paper, in order to obtain stable approximate solutions ill-posed linear operator equations a Hilbert space setting, we develop two novel non-negativity preserving iterative regularization methods. They based on fixed point iterations combination preconditioning ideas. contrast projected Landwebe...

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 ...