Extended Sampling Method in Inverse Scattering

A new sampling method for inverse scattering problems is proposed to process far field data of one incident wave. As the linear sampling method, the method sets up ill-posed integral equations and uses the (approximate) solutions to reconstruct the target. In contrast, the kernels of the associated integral operators are the far field patterns of sound soft balls. The measured data is moved to right hand sides of the equations, which gives the method the ability to process limit aperture data. Furthermore, a multilevel technique is employed to improve the reconstruction. Numerical examples show that the method can effectively determine the location and approximate the support with little a priori information of the unknown target

Existence of entire solutions to the Lagrangian mean curvature equations in supercritical phase

In this paper, we establish the existence and uniqueness theorem of entire solutions to the Lagrangian mean curvature equations with prescribed asymptotic behavior at infinity. The phase functions are assumed to be supercritical and converge to a constant in a certain rate at infinity. The basic idea is to establish uniform estimates for the approximating problems defined on bounded domains and the main ingredient is to construct appropriate subsolutions and supersolutions as barrier functions. We also prove a nonexistence result to show the convergence rate of the phase functions is optimal

Deterministic-Statistical Approach for an Inverse Acoustic Source Problem using Multiple Frequency Limited Aperture Data

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

Photocatalytic Removal of Organics over BiVO4-Based Photocatalysts

Organic compounds, such as organic dyes and phenols, are the main pollutants in wastewater. In the past years, a large number of studies on the fabrication and photocatalytic organics degradation of BiVO4 and its related materials have been reported in the literature. In this chapter, we shall focus on the advancements in the synthesis and photocatalytic applications of several kinds of BiVO4-based photocatalysts: (i) well-defined morphological BiVO4 photocatalysts, (ii) porous BiVO4 photocatalysts, (iii) heteroatom-doped BiVO4 photocatalysts, (iv) BiVO4-based heterojunction photocatalysts, and (v) supported BiVO4 photocatalysts. We shall discuss the structure–photocatalytic performance relationship of the materials and the involved photocatalytic degradation mechanisms. In addition, we also propose the research trends and technologies for practical applications of the BiVO4-based photocatalytic materials