Iterated quasi-reversibility method applied to elliptic and parabolic data completion problems

We study the iterated quasi-reversibility method to regularize ill-posed elliptic and parabolic problems: datacompletion problems for Poisson's and heat equations. We define an abstract setting to treat both equations at once. We demonstrate theconvergence of the regularized solution to the exact one, and propose a strategy to deal with noise on the data. We present numerical experimentsfor both problems: a two-dimensional corrosion detection problem and the one-dimensional heat equation with lateral data. In both cases,the method proves to be efficient even with highly corrupted data.