Stability Analysis in Magnetic Resonance Elastography II

TL;DR

This paper develops stability estimates for recovering tissue shear modulus in magnetic resonance elastography using pseudodifferential methods, improving previous results and ensuring convergence of numerical schemes.

Contribution

It introduces refined stability estimates for the inverse problem with less regularity and removes the finite dimensional kernel, enhancing practical applicability.

Findings

Proved stability estimates in 2D and 3D with reduced regularity

Showed the finite dimensional kernel can be eliminated

Established convergence of the Landweber iterative scheme

Abstract

We consider the inverse problem of finding unknown elastic parameters from internal measurements of displacement fields for tissues. In the sequel to Ammari, Waters, Zhang (2015), we use pseudodifferential methods for the problem of recovering the shear modulus for Stokes systems from internal data. We prove stability estimates in $d=2,3$ with reduced regularity on the estimates and show that the presence of a finite dimensional kernel can be removed. This implies the convergence of the Landweber numerical iteration scheme. We also show that these hypotheses are natural for experimental use in constructing shear modulus distributions.