DETERMINING A RANDOM SCHRODINGER EQUATION WITH UNKNOWN SOURCE AND POTENTIAL

Li, Jingzhi1; Liu, Hongyu2; Ma, Shiqi1,2

2019

10.1137/18M1225276

SIAM JOURNAL ON MATHEMATICAL ANALYSIS   影响因子和分区

0036-1410

1095-7154

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.

random Schrodinger equation inverse scattering passive/active measurements asymptotic expansion ergodicity

SCI ; EI

英语

第一 ; 通讯

Hong Kong RGC grants[12302017] ; Hong Kong RGC grants[12301218]

Mathematics

Mathematics, Applied

WOS:000483950800022

SIAM PUBLICATIONS

20194807753871

Inverse problems ; Recovery

Mathematics:921

MATHEMATICS

Web of Science

1.Southern Univ Sci & Technol, Dept Math, Shenzhen, Peoples R China
2.Hong Kong Baptist Univ, Dept Math, Kowloon, Hong Kong, Peoples R China

Li, Jingzhi,Liu, Hongyu,Ma, Shiqi. DETERMINING A RANDOM SCHRODINGER EQUATION WITH UNKNOWN SOURCE AND POTENTIAL[J]. SIAM JOURNAL ON MATHEMATICAL ANALYSIS,2019,51(4):3465-3491.

Li, Jingzhi,Liu, Hongyu,&Ma, Shiqi.(2019).DETERMINING A RANDOM SCHRODINGER EQUATION WITH UNKNOWN SOURCE AND POTENTIAL.SIAM JOURNAL ON MATHEMATICAL ANALYSIS,51(4),3465-3491.

Li, Jingzhi,et al."DETERMINING A RANDOM SCHRODINGER EQUATION WITH UNKNOWN SOURCE AND POTENTIAL".SIAM JOURNAL ON MATHEMATICAL ANALYSIS 51.4(2019):3465-3491.