Workshop on Inverse Scattering Problems

Date:Saturday, September 25, 2021

Venue:Room 200-9, Run Run Shaw Science Building, Yuquan Campus, Zhejiang University

A Lipschitz stability for the inverse problems of elastic wave equation 

Yixian Gao, Northeast Normal University 

This work is concerned with the time-domain elastic scattering problems in a bounded domain. A explicit reconstruct formula for the mass density is given via the boundary measurements model using Dirichlet-to-Neumann operator. The reconstruction is mainly based on the modified boundary control method and incorporates features from the complex geometric optics solutions approach. We also explain the elasticity wave equation is stable observability by establishing a Carleman estimate. Furthermore, we state the reconstruction formula Lipschitz stability with the system is stable observability. 

An interior inverse scattering problem in elasticity 

Fang Zeng, Chongqing University  

We consider an interior inverse scattering problem of reconstructing the shape of an elastic cavity. First of all, we show a reciprocity relation for the scattered elastic field and a uniqueness theorem for the inverse problem . Then we employ the decomposition method to determine the boundary of the cavity and present some convergence results. Numerical examples are provided to show the viability of the method.

Some research on the weak Galerkin finite element methods 

Yanli Chen, Northeastern University 

In this talk, we will discuss the weak finite element method for general elliptic problems, the Stokes problems and Navier-Stokes problems. Our main work is as follows. 1. A new weak finite element schemes is presented for the convectiondiffusion equations. 2. The H^1-superconvergence of order k+2 is derived for the pure elliptic problem in divergence form if the weak finite element space pair W_{k,k+1}(K)~[P_{k+1}(K)]^d is used. 3. For several weak finite element methods, we establish some efficient and reliable a posteriori error estimates of residual type, in the discrete $H^1$- norm. 4. For the Stokes and Navier-Stokes problems, we give a weak finite element scheme which satisfies the inf-sup condition without adding any stability term and prove the optimal order error estimate. 

Inverse random source problems for wave equations 

Xu Wang, Academy of Mathematics and Systems Science 

In this talk, inverse random source problems for acoustic and electromagnetic waves will be introduced. The unknown random source is assumed to be a microlocally isotropic Gaussian random field with its covariance operator being a classical pseudo-differential operator. The wellposedness of the direct problem in the distribution sense as well as the regularity of the solution is given for the case that the random source is extremely rough and should be interpreted as a distribution. The strength of the random source, involved in the principal symbol of its covariance operator, is shown to be uniquely determined by a single realization of the magnitude of the wave field averaged over the frequency band with probability one.