We propose a deterministic-statistical method for an inverse source problem using multiple frequency limited aperture far field data. The direct sampling method is used to obtain a disc such that it contains the compact support of the source. The Dirichlet eigenfunctions of the disc are used to expand the source function. Then the inverse problem is recast as a statistical inference problem for the expansion coefficients and the Bayesian inversion is employed to reconstruct the coefficients. The stability of the statistical inverse problem with respect to the measured data is justified in the sense of Hellinger distance. A preconditioned Crank-Nicolson (pCN) Metropolis-Hastings (MH) algorithm is implemented to explore the posterior density function of the unknowns. Numerical examples show that the proposed method is effective for both smooth and non-smooth sources given limited-aperture data.
翻译:我们建议使用多频率限制孔径远远的实地数据,对反源问题采用确定-统计方法。直接取样方法用于获取盘片,使其包含源的紧凑支持。盘的 Dirichlet 机能被用来扩大源函数。然后,反向问题被重新表述为扩展系数的统计推论问题,而巴耶斯反向问题被用来重建系数。测量数据方面的统计反向问题,从Hellinger距离的意义上讲,其稳定性是有道理的。采用了一种先决条件的Crank-Nicolson(PCN)Metropolis-Hasting(MH)算法,以探索未知物的远端密度函数。数字实例显示,根据有限孔径数据,拟议的方法对光滑源和非浮源都有效。