Convexification of restricted Dirichlet-to-Neumann map

Abstract

By our definition, "restricted Dirichlet-to-Neumann map" (DN) means that the Dirichlet and Neumann boundary data for a Coefficient Inverse Problem (CIP) are generated by a point source running along an interval of a straight line. On the other hand, the conventional DN data can be generated, at least sometimes, by a point source running along a hypersurface. CIPs with the restricted DN data are nonoverdetermined in the n−D case with n ≥ 2. We develop, in a unified way, a general and a radically new numerical concept for CIPs with restricted DN data for a broad class of PDEs of the second order, such as, e.g. elliptic, parabolic and hyperbolic ones. Namely, using Carleman Weight Functions, we construct globally convergent numerical methods. Hölder stability and uniqueness are also proved. The price we pay for these features is a well acceptable one in the Numerical Analysis: we truncate a certain Fourier-like series with respect to some functions depending only on the position of that point source. At least three applications are: imaging of land mines, crosswell imaging and electrical impedance tomography.

On the travel time tomography problem in 3D

“…For an integer N ≥ 1 consider the N × N matrix A N = (d m,n ) N −1 m,n=0 . Then this matrix is invertible [18]. We now present some details which were not discussed previously.…”

“…It follows from (3.2)-(3.5) that every function f ∈ L 2,N (0, π) can be uniquely determined from its first derivative without a knowledge of any initial condition f (a 0 ) for any point a 0 ∈ (0, π) . In fact, this is the reason why this basis was originally introduced in [18]. However, even though the function D N f is sufficiently close to the function f in the L 2 (0, π) −norm for sufficiently large values of N, this does not imply that functions…”

“…Therefore, both here and in [18,20,23,24, ? ], when we recover functions of L 2,N (0, π) from their derivatives without knowledge of initial conditions, we work only in the approximate sense of the orthogonal projection operator D N :…”

Travel time tomography with formally determined incomplete data in 3D

“…We now reproduce a special orthonormal basis {Ψ n (α)} ∞ n=0 in L 2 (0, 1) , which was constructed in [16]. This basis has the following two properties:…”

“…We now describe the basis of [16]. Consider the set of functions {ξ n (α)} ∞ n=0 = {(α + a) n e α } ∞ n=0 , where a = const.…”