Carleman weight functions for a globally convergent numerical method for ill-posed Cauchy problems for some quasilinear PDEs

Abstract

In a series of publications of the second author, including some with coauthors, globally strictly convex Tikhonov-like functionals were constructed for some nonlinear ill-posed problems. The main element of such a functional is the presence of the Carleman Weight Function. Compared with previous publications, the main novelty of this paper is that the existence of the regularized solution (i.e. the minimizer) is proved rather than assumed. The method works for both ill-posed Cauchy problems for some quasilinear PDEs of the second order and for some Coefficient Inverse Problems. However, to simplify the presentation, we focus here only on ill-posed Cauchy problems. Along with the theory, numerical results are presented for the case of a 1-D quasiliear parabolic PDE with the lateral Cauchy data given on one edge of the interval (0,1).

Convexification of a 3-D coefficient inverse scattering problem

“…2. We omit the proof of Theorem 3 below since, by Lemma 2.1 of [2], Theorem 3 follows immediately from Theorem 2. In addition, we omit the proof of Theorem 6 since Theorem 2.1 of [2] implies that Theorem 6 follows immediately from Theorem 2.…”

“…The main reason of this was the lack of some theorems at that time, which would ensure a proper behavior of iterates. These theorems were first proved in [2].…”

“…After [2], a number of works on the convexification was published by the first author with coauthors, in which the theory is combined with numerical results, see, e.g. [16,26,27,28,29].…”

“…The latter does not allow a numerical implementation, see [3] for a similar conclusion regarding a different CIP. In addition, since [23] was published before [2], then uniqueness and existence of the minimizer as well as the global convergence of the gradient projection method are not proven in [23]. Besides, numerical studies were not conducted in [23].…”

“…Thus, the local minima do not exist. Furthermore, as stated above, starting from the publication [2], all works about the convexification contain theorems, which claim the existence and uniqueness of the minimizer of J λ on the set B (d) and convergence of the gradient projection method of the minimization of J λ to that minimizer, if starting from an arbitrary point of B (d) . Next, as long as the level of the noise in the data tends to zero, those minimizers converge to the correct solution of the corresponding CIP.…”