Xiaodong Liu

Xiaodong Liu, Jiguang Sun

The inverse scattering problems have been popular for the past thirty years. While very successful in many cases, progress has lagged when only {\em limited-aperture} measurement is available. In this paper, we perform some elementary study to recover data that can not be measured directly. To be precise, we aim at recovering the {\em full-aperture} far field data from {\em limited-aperture} measurement. Due to the reciprocity relation, the multi-static response matrix (MSR) has a symmetric structure. Using the Green's formula and single layer potential, we propose two schemes to recover {\em full-aperture} MSR. The recovered data is tested by a recently proposed direct sampling method and the factorization method. The numerical results show the possibility to recover, at least partially, the missing data and consequently improve the reconstruction of the scatterer.

Yucheng Xing, Xiaodong Liu, Xin Wang

With the development of artificial intelligence, more and more attention has been put onto generative models, which represent the creativity, a very important aspect of intelligence. In recent years, diffusion models have been studied and proven to be more reasonable and effective than previous methods. However, common diffusion frameworks suffer from controllability problems. Although extra conditions have been considered by some work to guide the diffusion process for a specific target generation, it only controls the generation result but not its process. In this work, we propose a new adaptive framework, $\textit{Adaptively Controllable Diffusion (AC-Diff) Model}$, to automatically and fully control the generation process, including not only the type of generation result but also the length and parameters of the generation process. Both inputs and conditions will be first fed into a $\textit{Conditional Time-Step (CTS) Module}$ to determine the number of steps needed for a generation. Then according to the length of the process, the diffusion rate parameters will be estimated through our $\textit{Adaptive Hybrid Noise Schedule (AHNS) Module}$. We further train the network with the corresponding adaptive sampling mechanism to learn how to adjust itself according to the conditions for the overall performance improvement. To enable its practical applications, AC-Diff is expected to largely reduce the average number of generation steps and execution time while maintaining the same performance as done in the literature diffusion models.

Xiaodong Liu, Bo Zhang

This paper is concerned with the inverse problem of scattering of time-harmonic acoustic waves by an inhomogeneous penetrable obstacle in a piecewise homogeneous medium. The well-posedness of the direct problem is first established by using the integral equation method. We then proceed to establish two tools that play an important role for the inverse problem: one is a mixed reciprocity relation and the other is a priori estimates of the solution on some part of the interfaces between the layered media. For the inverse problem, we prove in this paper that both the penetrable interfaces and the possible inside inhomogeneity can be uniquely determined from a knowledge of the far field pattern for incident plane waves.