Linearized Inverse Schrödinger Potential Problem at a Large Wavenumber

Abstract

We investigate recovery of the (Schrödinger) potential function from many boundary measurements at a large wavenumber. By considering such a linearized form, we obtain a Hölder type stability which is a big improvement over a logarithmic stability in low wavenumbers. Furthermore we extend the discussion to the linearized inverse Schrödinger potential problem with attenuation, where an exponential dependence of the attenuation constant is traced in the stability estimate. Based on the linearized problem, a reconstruction algorithm is proposed aiming at the recovery of the Fourier modes of the potential function. By choosing the large wavenumber appropriately, we verify the efficiency of the proposed algorithm by several numerical examples.

Increasing stability in the linearized inverse Schrödinger potential problem with power type nonlinearities

“…A classical result in [1] shows that if the wavenumber k = 0 in (1.1) the stability of the inverse Schrödinger potential problem is logarithmic. When the wavenumber is sufficiently large, increasing stability with respect to the wavenumber k has been observed and well documented, starting with [15] and with many further results given in [19,17,18] for (1.1) or its linearized form. These results are often stated as stability estimates involving a Hölder term and a logarithmic term which goes to zero as the wavenumber goes to infinity.…”

“…We emphasize that the stability estimate O( It can be viewed as the advantage of the quadratic nonlinearity term when we solve the nonlinear inverse problems (1.2) with m = 2. A clear numerical evidence will be provided in Section 4 and one can stably recover the Fourier coefficients with |ξ| ≤ 3k whereas in [18] one can only recover those with |ξ| ≤ 2k. Such advantages highly depend on the sophisticatedly selected complex exponential functions and the modified Calderón identity (3.3) considered above.…”

“…The obtained increasing stability is better than the standard one in the linear Schrödinger potential problem which highlights the advantage of the nonlinearity term in solving inverse problems. Noticing that both linearized DtN maps D m 0 Λ c and Λ c can be numerically approximated, we extend the reconstruction algorithm in [18] to the inverse Schrödinger potential problem with quadratic and general nonlinearity terms in Section 4, respectively. We note that one of these reconstruction algorithms is realized by the linearized DtN map Λ c with multiple wavenumbers.…”

“…Finally, for the inverse Fourier transform, a numerical quadrature rule can be constructed by a suitable choice of the weights σ i;s according to these points ξ i;s . We summarize our reconstruction algorithm below, which is similar to that in [18] but one has to solve the nonlinear Schrödinger potential problem three times at each iteration because of the quadratic nonlinearity term. Calculate the approximated linearized Neumann boundary data…”

Inverse Scattering Source Problems

“…As I mentioned before, there are many scientist and researcher have been working on inverse scattering and more specifically on inverse source problems. To expand your knowledge and further mathematical development in this field of research, please see the result authors in [29][30][31][32][33][34][35][36][37][38][39][40][41], which were discussed different aspects of the problems.…”