Determining a Random Schrödinger Equation with Unknown Source and Potential

Abstract

We are concerned with the direct and inverse scattering problems associated with a time-harmonic random Schrödinger equation with unknown source and potential terms. The well-posedness of the direct scattering problem is first established. Three uniqueness results are then obtained for the corresponding inverse problems in determining the variance of the source, the potential and the expectation of the source, respectively, by the associated far-field measurements. First, a single realization of the passive scattering measurement can uniquely recover the variance of the source without the a priori knowledge of the other unknowns. Second, if active scattering measurement can be further obtained, a single realization can uniquely recover the potential function without knowing the source. Finally, both the potential and the first two statistic moments of the random source can be uniquely recovered with full measurement data. The major novelty of our study is that on the one hand, both the random source and the potential are unknown, and on the other hand, both passive and active scattering measurements are used for the recovery in different scenarios.

Inverse random source scattering for the Helmholtz equation in inhomogeneous media

“…Another interesting problem is to consider that both the medium and the source are random functions. Similar problems for the Schrödinger equation were investigated in [22,23]. The Helmholtz equation is more difficult because of the coupling of the medium with the wavenumber.…”

Finite/fixed-time synchronization for complex networks via quantized adaptive control

“…It is brought to our attention that the synchronization control problem in our study is closely related to inverse problems for differential equations, see e.g. [3,4,21,22,28,29] in the deterministic setting and [14,15] in the random setting. It would be interesting for us to consider inverse problem techniques to the synchronization control problem in our future study.…”

On recent progress of single-realization recoveries of random Schrödinger systems

“…The single-realization recovery has been studied in the literature. In this paper we mainly focus on [8,[18][19][20][21][22][23][24][25].…”

“…In [21], the authors consider the direct and inverse scatterings for (1.1)-(1.3) with a deterministic potential and a random source. The random source is a generalized Gaussian random field with local mean value function and local variance function, which are assumed to be bounded and compactly supported.…”