TY - JOUR
T1 - Determining a Random Schrödinger Equation with Unknown Source and Potential
AU - Li, Jingzhi
AU - Liu, Hongyu
AU - Ma, Shiqi
N1 - Funding Information:
The work of the first author was partially supported by the National Natural Science Foundation of China under grants 11571161 and 11731006 and by the Shenzhen Sci-Tech fund grant JCYJ20170818153840322. The work of the second author was partially supported by HKBU FRG and Hong Kong RGC grants 12302017 and 12301218.
Publisher copyright:
© 2019, Society for Industrial and Applied Mathematics
PY - 2019/8/21
Y1 - 2019/8/21
N2 - We are concerned with the direct and inverse scattering problems associated with a time-harmonic random Schrodinger 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.
AB - We are concerned with the direct and inverse scattering problems associated with a time-harmonic random Schrodinger 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.
KW - Asymptotic expansion
KW - Ergodicity
KW - Inverse scattering
KW - Passive/active measurements
KW - Random Schrodinger equation
UR - http://www.scopus.com/inward/record.url?scp=85075571610&partnerID=8YFLogxK
U2 - 10.1137/18M1225276
DO - 10.1137/18M1225276
M3 - Journal article
AN - SCOPUS:85075571610
SN - 0036-1410
VL - 51
SP - 3465
EP - 3491
JO - SIAM Journal on Mathematical Analysis
JF - SIAM Journal on Mathematical Analysis
IS - 4
ER -