Convergence of Lebenberg-Marquard method for the Inverse Problem with an Interior Measurement

TL;DR

This paper proves the convergence of the Levenberg-Marquardt method for an inverse problem in magnetic resonance elastography, demonstrating its applicability to reconstruct unknown moduli from interior measurements.

Contribution

It establishes the convergence of the Levenberg-Marquardt method for scalar inverse problems using interior data, with a general proof applicable to similar PDE coefficient reconstructions.

Findings

Convergence is proved via the tangential cone condition.

Numerical tests validate the method on a two-layer model.

The approach applies broadly to similar inverse PDE problems.

Abstract

The convergence of Levenberg-Marquard method is discussed for the inverse problem to reconstruct the storage modulus and loss modulus for the so called scalar model by single interior measurement. The scalar model is the most simplest model for data analysis used as the modeling partial differential equation in the diagnosing modality called the magnetic resonance elastography which is used to diagnose for instance lever cancer. The convergence of the method is proved by showing that the measurement map which maps the above unknown moduli to the measured data satisfies the so called the tangential cone condition. The argument of the proof is quite general and in principle can be applied to any similar inverse problem to reconstruct the unknown coefficients of the model equation given as a partial differential equation of divergence form by one single interior measurement. The performance of the method is numerically tested for the two layered picewise homogneneous scalar model in a rectangular domain.