In this talk on variational regularization for ill-posed nonlinear problems, we discuss the impact of utilizing an oversmoothing penalty term. This means that the searched-for solution of the considered nonlinear operator equation does not belong to the domain of definition of the penalty functional. In the past years, such variational regularization has been investigated comprehensively in Hilbert scales. Our present results extents those results to Banach scales. This new study includes convergence rates results for a priori and a posteriori choices of the regularization parameter, both for H\"older-type smoothness and low order-type smoothness. An illustrative example intends to indicate the specific characteristics of non-reflexive Banach spaces.

In this talk, we will introduce few simplified iterative regularization methods, in a Banach space setting, to obtain stable approximate solution of nonlinear ill-posed operator equation. We will discuss convergence analysis of these methods under suitable non linearity conditions. Numerical examples will be demonstrated to show applicability of these methods to practical problems.

Motivated by the necessity to identify stochastic parameters in a wide range of stochastic partial differential equations, this talk will focus on an abstract inversion framework for stochastic inverse problems. The stochastic inverse problem will be posed as a convex stochastic optimization problem. The essential properties of the solution maps and the solvability of the inverse problem will be discussed. Convergence rates for the stochastic inverse problem without requiring the so-called smallness condition will be presented. We will discuss an application of the abstract framework to estimate stochastic Lam\'e parameters in the system of linear elasticity. We will present numerical results to show the feasibility and efficacy of the developed framework.