Newton-Kantorovich method for three-dimensional potential inverse scattering problem and stability of the hyperbolic Cauchy problem with time-dependent data

Abstract

A special version of the Newton-Kantorovich method is applied to the three-dimensional potential inverse scattering problem in the time domain. The related hyperbolic Cauchy problem with data on the side of the time cylinder is solved by the quasi-reversibility method, and a new stability theorem is established by Carleman-type estimates. The geometrical convergence of the Newton-Kantorovich method, used here, is also established.

Numerical solution of the problem of computational time reversal in a quadrant

“…In the case of the quadrant the data (1.4) were given on finite parts of coordinate axis, and they were given on finite parts of coordinate planes in the case of the octant. Using previous results of [9,7,5], he has proven the Lipschitz stability estimate for this problem, has shown its connection with the refocusing of time reversed wave fields and has proposed a convergent numerical method. Note that the Lipschitz stability in an unbounded domain such as quadrant or octant is rather surprising, since, unlike the bounded domain case, a large part of the energy never reaches the surface (curve in 2-D) where measurements of the wave field and its normal derivative are taken.…”

“…Since (1.4) is the Cauchy data, then the Inverse Problem 1 is a particular case of the so-called Cauchy problem for the hyperbolic equation with the lateral data. Uniqueness and stability results for this problem are obtained via Carleman estimates and can be found in, e.g., books of Klibanov and Timonov [9] and Lavrentiev, Romanov and Shishatskii [12], as well as in papers of Klibanov and Malinsky [7], Kazemi and Klibanov [5] and Klibanov [6]. The Hölder stability estimate was established in [12] and the stronger Lipschitz stability estimate for bounded domains was proven in [9,7,5].…”

“…Uniqueness and stability results for this problem are obtained via Carleman estimates and can be found in, e.g., books of Klibanov and Timonov [9] and Lavrentiev, Romanov and Shishatskii [12], as well as in papers of Klibanov and Malinsky [7], Kazemi and Klibanov [5] and Klibanov [6]. The Hölder stability estimate was established in [12] and the stronger Lipschitz stability estimate for bounded domains was proven in [9,7,5]. Klibanov [6] has studied both ϕ-and ψ-problems for a general hyperbolic equation (1.1) with variable coefficients (including hyperbolic inequalities) for n = 2, 3, assuming that the domain D is either a quadrant in the 2-D case or an octant in the 3-D case.…”

“…We observe the latter in numerical experiments (Sections 5 and 6). We also mention that previous Lipschitz stability estimates of [9,7,5] were obtained for the case when the lateral Cauchy data are given at the boundary of a bounded domain and solution is sought for inside this domain. Thus, they imply a high degree of refocusing of time reversed wave fields propagating in bounded domains.…”

Four variations in global Carleman weights applied to inverse and controllability problems

“…The product of this matrix and a vector fi eld v is exactly cos θ(v − v ⊥ ) + T θ v ⊥ , expression which appears in the defi nition (21) of r . The main steps in the deduction of such inequality are taken from [35,23] following a well known technique due to Bukhgeim and Klibanov [11], [29]. Roughly speaking, the technique in this case consists in reducing the problem to a source inverse problem for the perturbed equation around u(q) and then take its time derivative in order to obtain a quasi-observability inequality after use of the Carleman inequality.…”