Numerical solution of an ill-posed Cauchy problem for a quasilinear parabolic equation using a Carleman weight function

Abstract

This is the first publication in which an ill-posed Cauchy problem for a quasilinear PDE is solved numerically by a rigorous method. More precisely, we solve the side Cauchy problem for a 1-d quasilinear parabolc equation. The key idea is to minimize a strictly convex cost functional with the Carleman Weight Function in it. Previous publications about numerical solutions of ill-posed Cauchy problems were considering only linear equations.

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

“…1. We point out that, compared with previous publications [5,11,12,16,17,18,19] on the topic of this paper, a significantly new element of Theorems 3.3-3.5 is that now the existence of the global minimum v min is asserted rather than assumed. This became possible because of results of convex analysis of section 2.…”

“…In addition, we now need to prove the Lipschitz continuity of the Frechét derivative of our cost functional, which was not done in those previous works. These factors, in turn mean that proofs of main theorems here are different from their analogs in [17,19]. So, we prove the corresponding theorems below.…”

“…More precisely, the existence of a minimizer of a weighted Tikhonov functional is proved here rather than assumed as in [17]. Although similar assumptions of works of the second author [5,11,12,16,18,19] concerning Coefficient Inverse Problems (CIPs) can also be eliminated the same way, we are not doing this here for brevity. In addition to the theory, we present results of some numerical experiments in which we solve an ill-posed problem for a 1-D quasilinear parabolic equation with the lateral Cauchy data.…”

“…Those results of the convex analysis require us to change the previous scheme of the method. More precisely, while the previous scheme of [17,19] works with non-zero Cauchy data, we now need to have zero Cauchy data. We obtain them via "subtracting" the nonzero Cauchy data from the sought for solution.…”

Globally Strictly Convex Cost Functional for a 1-D Inverse Medium Scattering Problem with Experimental Data

“…On the other hand, we consider in the current paper, so as in [24,25,26,33], the fundamental solution of the corresponding PDE. The differences between the fundamental solutions of those PDEs and solutions satisfying non-vanishing conditions cause quite significant differences between [24, 25, 26, 33] and [6, 7, 10, 29] of corresponding versions of the GCM of the second type.Recently, the idea of the GCM of the second type was extended to the case of ill-posed Cauchy problems for quasilinear PDEs, see the theory in [28] and some extensions and numerical examples in [4,30].CIPs of wave propagation are a part of a bigger subfield, Inverse Scattering Problems (ISPs). ISPs attract a significant attention of the scientific community.…”