Regularization for time-dependent inverse problems: Geometry of Lebesgue-Bochner spaces and algorithms

TL;DR

This paper explores regularization techniques for time-dependent inverse problems within Lebesgue-Bochner spaces, analyzing their geometric properties and demonstrating their application to dynamic tomography.

Contribution

It introduces two regularization methods in Lebesgue-Bochner spaces and investigates their geometric properties, enabling improved handling of time-dependent inverse problems.

Findings

Lebesgue-Bochner spaces are smooth of power type.

Tikhonov regularization can be implemented with different regularities for time and space.

Both methods are tested on dynamic computerized tomography data.

Abstract

We consider time-dependent inverse problems in a mathematical setting using Lebesgue-Bochner spaces. Such problems arise when one aims to recover a function from given observations where the function or the data depend on time. Lebesgue-Bochner spaces allow to easily incorporate the different nature of time and space. In this manuscript, we present two different regularization methods in Lebesgue Bochner spaces: 1. classical Tikhonov regularization in Banach spaces 2. temporal variational regularization by penalizing the time-derivative In the first case, we additionally investigate geometrical properties of Lebesgue Bochner spaces. In particular, we compute the duality mapping and show that these spaces are smooth of power type. With this we can implement Tikhononv regularization in Lebesgue-Bochner spaces using different regularities for time and space. We test both methods using the example of dynamic computerized tomography.