An Iterative Direct Sampling Method for Reconstructing Moving Inhomogeneities in Parabolic Problems
Bangti Jin, Fengru Wang, Jun Zou
TL;DR
This work introduces the Iterative Direct Sampling Method (IDSM) for reconstructing moving inhomogeneities in parabolic inverse problems from limited boundary data. By linking boundary measurements to the inclusion through a semi-linear forward map and constructing a probing kernel via a low-rank-augmented resolvent, IDSM directly images inclusions and handles nonlinear and mixed-type anomalies. The method operates on time segments with a fixed-point iteration to address nonlinearity, employs an $L^{2}$ inner product for stability, and updates a low-rank kernel on the fly, achieving robust reconstructions with only a handful of PDE solves per data set. Numerical experiments across merging/splitting, mixed-type, nonlinear, and fading/diminishing inclusions demonstrate accurate tracking under noise and limited data, underscoring the approach’s practical potential for time-dependent parabolic imaging.
Abstract
We propose in this work a novel iterative direct sampling method for imaging moving inhomogeneities in parabolic problems using boundary measurements. It can efficiently identify the locations and shapes of moving inhomogeneities when very limited data are available, even with only one pair of lateral Cauchy data, and enjoys remarkable numerical stability for noisy data and over an extended time horizon. The method is formulated in an abstract framework, and is applicable to linear and nonlinear parabolic problems, including linear, nonlinear, and mixed-type inhomogeneities. Numerical experiments across diverse scenarios show its effectiveness and robustness against the data noise.