H\"older stability of quantitative photoacoustic tomography based on partial data

TL;DR

This paper proves a H"older stability estimate for reconstructing diffusion and absorption coefficients in photoacoustic tomography from partial internal data, enabling local reconstructions even with limited boundary information.

Contribution

It establishes the first H"older stability result for partial data in quantitative photoacoustic tomography, including stability estimates and numerical validation.

Findings

H"older stability holds in subregions with partial internal data

Stability exponent approaches a positive constant as subregion shrinks

Numerical experiments confirm local reconstruction for smooth and discontinuous media

Abstract

We consider the reconstruction of the diffusion and absorption coefficients of the diffusion equation from the internal information of the solution obtained from the first step of the inverse photoacoustic tomography (PAT). In practice, the internal information is only partially provided near the boundary due to the high absorption property of the medium and the limitation of the equipment. Our main contribution is to prove a H\"older stability of the inverse problem in a subregion where the internal information is reliably provided based on the stability estimation of a Cauchy problem satisfied by the diffusion coefficient. The exponent of the H\"older stability converges to a positive constant independent of the subregion as the subregion contracts towards the boundary. Numerical experiments demonstrates that it is possible to locally reconstruct the diffusion and absorption coefficients for smooth and even discontinuous media.