# Infinite-dimensional inverse problems with finite measurements

## TL;DR

This paper develops a framework for solving infinite-dimensional inverse problems with finite measurements, establishing stability, uniqueness, and a convergent reconstruction algorithm applicable to various nonlinear ill-posed problems.

## Contribution

It introduces a general theory for finite measurement inverse problems in infinite dimensions, including a globally convergent Landweber iteration-based algorithm.

## Key findings

- Stability estimates hold with finite measurements for Lipschitz stable problems.
- The framework applies to nonlinear problems like electrical impedance tomography.
- A practical reconstruction algorithm is proposed and analyzed.

## Abstract

We present a general framework to study uniqueness, stability and reconstruction for infinite-dimensional inverse problems when only a finite-dimensional approximation of the measurements is available. For a large class of inverse problems satisfying Lipschitz stability we show that the same estimate holds even with a finite number of measurements. We also derive a globally convergent reconstruction algorithm based on the Landweber iteration. This theory applies to nonlinear ill-posed problems such as electrical impedance tomography, inverse scattering and quantitative photoacoustic tomography, under the assumption that the unknown belongs to a finite-dimensional subspace.