A Journey on Computational and Learning Methods for Large-Scale Inverse Problems

Abstract: Inverse problems are ubiquitous in many fields of science such as engineering, biology, medical imaging, atmospheric science, and geophysics. Three emerging challenges on obtaining relevant solutions to large-scale and data-intensive inverse problems are ill-posedness of the problem, large dimensionality of the parameters, and the complexity of the model constraints. In this talk we take a journey on the fascinating world of inverse problems by discussing potential challenges and recent contributions on deterministic and learning methods. In particular, in the first part of the talk we discuss efficient methods for computing solutions to dynamic inverse problems, where both the quantities of interest and the forward operator may change at different time instances. We consider large-scale ill-posed problems that are made more challenging by their dynamic nature and, possibly, by the limited amount of available data per measurement step.

To remedy these difficulties, we apply efficient regularization methods that enforce simultaneous regularization in space and time (such as edge enhancement at each instant and proximity at consecutive time instants) and achieve this with low computational cost and enhanced accuracy.

In the remainder of the talk, we focus on computationally efficient methods that learn optimal lp and lq norms for Lp-Lq regularization and learn optimal parameters for regularization matrices defined by covariance kernels.

Numerical examples from a wide range of applications, such as tomographic reconstruction and image deblurring are used to illustrate the effectiveness of the described approaches.