# Landweber-Kaczmarz for parameter identification in time-dependent inverse problems: All-at-once versus Reduced version

## TL;DR

This paper compares All-at-once and Reduced Landweber-Kaczmarz methods for parameter identification in time-dependent inverse problems, analyzing their performance under continuous and discrete observation scenarios.

## Contribution

It introduces a loping Landweber-Kaczmarz iteration and compares two modeling approaches for time-dependent inverse problems.

## Key findings

- The loping strategy improves convergence.
- All-at-once and Reduced models have different advantages.
- The method handles both continuous and discrete data effectively.

## Abstract

In this study, we consider a general time-space system, whose model operator and observation operator are locally Lipschitz continuous, over a finite time horizon and parameter identification by using Landweber-Kaczmarz regularization. The problem is investigated in two different modeling settings: An All-at-once and a Reduced version, together with two observation scenarios: continuous and discrete observations. Segmenting the time line into several subintervals leads to the idea of applying the Kaczmarz method. A loping strategy is incorporated into the method to yield the loping Landweber-Kaczmarz iteration.