会议日程表

2019年04月13日，周六，上午，数学楼2-1会议室：

8:30—9:00	开幕式
1、主持人：张波
时间	报告人	题目
9:00—9:40	程晋	TBA
9:40—10:20	武海军	Adaptive FEM for diffraction grating problems
10:20-10:40	茶歇
2、主持人：程晋
10:40—11:20	刘继军	Image   restoration from noisy incomplete frequency data by alternative iteration   scheme
11:20—12:00	高静怀	High   resolution inversion of seismic wavelet and reflectivity using iterative deep   neural networks

2019年04月13日，周六，下午，数学楼2-1会议室：

3、主持人：刘继军
时间	报告人	题目
2:30--3:00	胡广辉	Inverse time-harmonic scattering from rigid elastic bodies   with a single incoming wave
3:00--3:30	鲁汪涛	A numerical mode matching method for wave scattering   problem in a layered medium with a stratified inhomogeneity
3:30—4:00	陈冲	Joint image reconstruction and motion estimation   in spatiotemporal medical imaging
4:00-4:30	茶歇
4、主持人：高静怀
4:30--5:00	赖俊	Elastic scattering of multiple   particles and its inverse problems
5:00--5:30	蒋代军	Uniqueness analysis   and numerical methods for an inverse source problem in a time-fractional   diffusion-advection equation

2019年04月14日，周日，上午，数学楼2-1会议室：

1、主持人：胡广辉
时间	报告人	题目
9:00—9:30	李景治	Determining a random Schr{\"o}dinger equation with   unknown source and potential
9:30—10:00	郭玉坤	The Fourier method for identifying   electromagnetic sources with multi-frequency far-field data
10:00-10:30	茶歇
2、主持人：李景治
10:30—11:00	刘晓东	Data Completion for Wave Scattering Problems   and its Application
11:00—11:30	王海兵	Asymptotic expansions for the diffusion   equation with applications
11:30-12:00	闫亮	Adaptive   multi-fidelity neural networks for large-scale Bayesian inverse problems

2019年04月14日，周日，下午，数学楼2-1会议室：

3、主持人：杨家青
时间	报告人	题目
2:30--3:00	张磊	Some results about   scattering and inverse scattering by rough surfaces
3:00--3:30	孙鸿鹏	Sparse reconstructions of acoustic   source for inverse scattering problems in measure space
3:30—4:00	刘可伋	Optimal Mesh Size for   Inverse Medium Scattering Problems
4:00—4:30	茶歇
4、主持人：刘晓东
4:30—5:00	黄记祖	A preconditioned   inexact Newton algorithm for steady state lattice Boltzmann equations
5:00--5:30	陈洁	Generalized multiscale approximation of mixed finite   elements with velocity elimination for subsurface flow

报告摘要

1. 陈冲，中国科学院数学与系统科学研究院

Title: Joint Image Reconstruction and Motion Estimation in Spatiotemporal Medical Imaging

Abstract: Image reconstruction becomes challenging in spatiotemporal medical imaging, such as tomographic imaging of the heart or lungs. As an example, accounting for the unknown motion of the organs is important in PET/CT cardiac imaging when data is acquired over a relatively long period of time (often in the range of minutes). We will introduce the general strategies for spatiotemporal image reconstruction, and propose a new variational model for joint image reconstruction and motion estimation, which is investigated along a general framework that we present with shape theory. This model consists of two components, one for conducting modified static image reconstruction, and the other performs sequentially indirect image registration. For the latter, we generalize the large deformation diffeomorphic metric mapping framework into the sequentially indirect registration setting. The proposed model is compared theoretically against alternative approaches (optical flow based model and diffeomorphic motion models), and we demonstrate that the proposed model has desirable properties in terms of the optimal solution. This talk is concluded by some numerical examples in 2D space + time tomography with very sparse and/or highly noisy data.

2. 陈洁，西安交通大学

Title: Generalized multiscale approximation of mixed finite elements with velocity elimination for subsurface flow

Abstract: A frame work of the mixed generalized multiscale finite element method (GMsFEM) for solving Darcy’s law in heterogeneous media is studied in this talk. Our approach approximates pressure in multiscale function space that is between fine-grid space and coarse-grid space and solves velocity directly in the fine-grid space. To construct multiscale basis functions for each coarse-grid element, three types of snapshot space are raised. The first one is taken as the fine-grid space for pressure and the other two cases need to solve a local problem on each coarse-grid element. We describe a spectral decomposition in the snapshot space motivated by the analysis to further reduce the dimension of the space that is used to approximate the pressure. Since the velocity is directly solved in the fine-grid space, in the linear system for the mixed finite elements, the velocity matrix can be approximated by a diagonal matrix without losing any accuracy. Thus it can be inverted easily. This reduces computational cost greatly and makes our scheme simple and easy for application. Moreover, the proposed method preserves the local mass conservation property that is important for subsurface problems. Numerical examples are presented to illustrate the good properties of the proposed approach. If offline spaces are appropriately selected, one can achieve good accuracy with only a few basis functions per coarse element according to the numerical results.

3. 程晋，复旦大学

Title: 待定

Abstract:  待定

4. 高静怀，西安交通大学

Title: High resolution inversion of seismic wavelet and reflectivity using iterative deep neural networks

Abstract: In this work, we propose a deep learning based data-driven method for high resolution inversion of seismic data. The method split the inversion into two subproblems: one inverts the seismic wavelet and the other for reflectivity. Using a partially learned approach, the proposed method simultaneously estimates the wavelet and reflectivity in an alternative way, and realized by deep neural networks (DNN). Both synthetic and field data examples clearly demonstrate the advantage of the proposed method in reducing the prediction error, ensuring the sparsity of the reflectivity and improving the lateral stability.

3. 郭玉坤，哈尔滨工业大学

Title: The Fourier method for identifying electromagnetic sources with multi-frequency far-field data

Abstract: This talk is focused on the inverse problem of determining an unknown source current distribution associated with the homogeneous Maxwell system. We propose a novel non-iterative reconstruction method for solving this inverse problem from multi-frequency far-field measurements. The method is based on recovering the Fourier coefficients of the unknown source. A key ingredient of the method is to establish the relationship between the Fourier coefficients and the multi-frequency far-field data. Uniqueness and stability results are established for the proposed reconstruction method. Numerical experiments will be presented to illustrate the effectiveness and efficiency of the method.

4. 胡广辉，北京计算科学研究中心

Title: Inverse time-harmonic scattering from rigid elastic bodies with a single incoming wave

Abstract: In this talk we will review reflection principles for some elliptic partial differential equations, including harmonic equations, bi-harmonic equations, Helmholtz equations, Maxwell equations and Navier equations etc.. The reflection principle provides a non-local extension formula for these PDEs across a hyper-plane under a certain boundary condition. We will show the application of non-pointwise reflection principles to inverse time-harmonic elastic scattering with a single incoming wave. This is a joint work with Johannes Elschner.

5. 黄记祖，中国科学院数学与系统科学研究院

Title: A preconditioned inexact Newton algorithm for steady state lattice Boltzmann equations

Abstract: Most existing methods for calculating the steady state solution of the lattice Boltzmann equations are based on pseudo time stepping, which often takes a large number of time steps especially for high Reynolds number problems. To avoid the time integration, a preconditioned inexact Newton algorithm is developed to solve the steady state lattice Boltzmann equations directly. In this work, a second-order finite difference scheme is proposed and a two-level inexact Newton algorithm with a local nonlinear elimination preconditioner is employed to solve the nonlinear algebraic system. On the coarse level, an inexact Newton method with Reynolds number continuation is applied to generate the initial guess for the fine level inexact Newton solver. On the fine level, the local high nonlinearity of the system is eliminated by the nonlinear elimination preconditioner before the global Newton update. Two numerical experiments are presented to demonstrate the robustness and efficiency of the algorithm, especially for high Reynolds number.

6. 蒋代军，华中师范大学

Title: Uniqueness analysis and numerical methods for an inverse source problem in a time-fractional diffusion-advection equation

Abstract: In this study, we first establish a weak unique continuation property for time-fractional  diffusion-advection equations. The proof is mainly  based on the Laplace transform and the unique continuation properties  for elliptic and parabolice quations. The result is weaker than its parabolic counterpart in the sense that we additionally impose the homogeneous boundary condition. As a direct application, we prove the  uniqueness for an inverse problem on determining the spatial component in the source term by interior measurements. Numerically, we reformulate our inverse source problem as an optimization problem, and propose an iterative thresholding algorithm. Finally, several numerical experiments are presented to show the accuracy and efficiency of the algorithm. This is a joint work with Prof. Masahiro Yamamoto, Yikan Liu and Zhiyuan Li.

7. 赖俊，浙江大学

Title: Elastic scattering of multiple particles and its inverse problems

Abstract: Efficient algorithms for evaluating the elastic scattering from multiple particles are important in material science and seismic imaging. In this talk, we propose a fast hybrid method to evaluate the two dimensional elastic multi-particle scattering. The main ingredients of the algorithm are Helmholtz decomposition, boundary integral equations and fast multiple method. Numerical simulation for a large number of particles will be presented to demonstrate the efficiency of the algorithm. In the end, we will show its application in the inverse elastic scattering problems with phaseless far field data.

8. 李景治，南方科技大学

Title: Determining a random Schr{\"o}dinger equation with unknown source and potential

Abstract: This talk presents the direct and inverse scattering problem associated with a time-harmonic random Schr{\"o}dinger equation with a Gaussian white noise source term. We establish the well-posedness of the direct scattering problem and obtain three uniqueness results in determining the variance of the source term, the potential and the mean of the source term, sequentially, by the corresponding far-field measurements. The first one shows that a single realization of the passive scattering measurement can uniquely recover the variance of the source term, without knowing the other two unknowns. The second shows that if active scattering measurement is further used, then a single realization can uniquely recover the potential function without knowing the source term. The last one shows that if full measurements are used, then both the potential and the random source can be uniquely recovered.

9. 刘继军，东南大学

Title: Image restoration from noisy incomplete frequency data by alternative iteration scheme

Abstract: Abstract: Consider the image restoration from incomplete noisy frequency data with total variation and sparsity regularizing penalty terms. Firstly, we establish an unconstrained optimization model with different smooth approximations on the regularizing terms. Then, to weaken the amount of computations for cost functional with total variation term, the alternating iterative scheme is developed to obtain the exact solution through shrinkage thresholding in inner loop, while the nonlinear Euler equation is appropriately linearized at each iteration in exterior loop, yielding a linear system with diagonal coefficient matrix in frequency domain. Finally the linearized iteration is proven to be convergent in generalized sense for suitable regularizing parameters, and the error between the linearized iterative solution and the one gotten from the nonlinear Euler equation is rigorously estimates, revealing the essence of the proposed alternative iteration scheme. Numerical tests for different configurations show the validity of the proposed scheme, compared with some existing algorithms.

10. 刘可伋，上海财经大学

Title: Optimal Mesh Size for Inverse Medium Scattering Problems

Abstract: An optimal mesh size of the sampling region can help to reduce computational burden in practical applications. In this work, we investigate optimal choices of mesh sizes for the identifications of medium obstacles from either the far-field or near-field data in two and three dimensions. The results would have applications in the reconstruction process of inverse scattering problems.

11. 刘晓东，中国科学院数学与系统科学研究院

Title: Data Completion for Wave Scattering Problems and its Application

Abstract: The two well known properties of the inverse problems are nonlinearity and ill-posedness. Actually, in many practical application, we have only incomplete data. Difficulties will arise because of the missing data. In this talk, we will introduce some data completion techniques for limited-aperture problems and phaseless problems.

12. 鲁汪涛，浙江大学

Title: A numerical mode matching method for wave scattering problem in a layered medium with a stratified inhomogeneity

Abstract: Numerical mode matching (NMM) methods are widely used for analyzing wave propagation and scattering in structures that are piecewise uniform along one spatial direction. For open structures that are unbounded in transverse directions (perpendicular to the uniform direction), the NMM methods use the perfectly matched layer (PML) technique to truncate the transverse variables. When incident waves are specified in homogeneous media surrounding the main structure, the total field is not always outgoing, and the NMM methods rely on reference solutions for each uniform segment. Existing NMM methods have difficulty handing grazing incident waves and special incident waves related to the onset of total internal reflection, and are not very efficient at computing reference solutions for non-plane incident waves. In this paper, a new NMM method is developed to overcome these limitations. A hybrid Dirichlet-Robin boundary condition is proposed to ensure that non-propagating and non-decaying wave field components are not reflected by truncated PMLs. Exponential convergence of the PML solutions based on the hybrid Dirichlet-Robin boundary condition is established theoretically. A fast method is developed for computing reference solutions for cylindrical incident waves. The new NMM is implemented for two-dimensional structures and polarized electromagnetic waves. Numerical experiments are carried out to validate the new NMM method and to demonstrate its performance.

13. 孙鸿鹏，中国人民大学

Title: Sparse Reconstructions of Acoustic Source for Inverse Scattering Problems in Measure Space.

Abstract: This paper proposes a systematic mathematical analysis of both the direct and inverse acoustic scattering problem given the source in Radon measure space. For the direct problem, we investigate the well-posedness including the existence, the uniqueness, and the stability by introducing a special definition of the weak solution, i.e. very weak solution. For the inverse problem, we choose the Radon measure space instead of the popular L1 space to build the sparse reconstruction, which can guarantee the existence of the reconstructed solution. The sparse reconstruction problem can be solved by the semismooth Newton method in the dual space. Numerical examples are included.

14. 王海兵，东南大学

Title: Asymptotic expansions for the diffusion equation with applications

Abstract: We show the asymptotic expansions of solutions to initial boundary value problems for the diffusion equation, as the size of the domain of interest (inclusion, cavity, etc.) goes to zero. More explicitly, we consider the following two model problems. First, for the heat conduction problem by a cluster of many small cavities, we show that the dominating heat is a sum, over the number of the cavities, of the heats generated by each cavity after interacting with each other. This interaction is described through densities computable as solutions of a close, and invertible, system of time domain integral equations of a second kind. As an application of these expansions, we derive the effective heat conductivity which generates approximately the same heat as the cluster of cavities, distributed in a 3D bounded domain, with explicit error estimates in terms of that cluster. Second, for the diffusion process by a finite number of small inclusions, we analyze the asymptotic behavior of the boundary measurements weighted by the fundamental solution of a backward diffusion equation. As an application, we derive an efficient non-iterative algorithm for locating small inclusions from finite boundary measurements.

15. 武海军，南京大学

Title: Adaptive FEM for diffraction grating problems

Abstract: Several adaptive FEM algorithms are introduced for diffraction grating problems.

16. 闫亮，东南大学

Title: Adaptive multi-fidelity neural networks for large-scale Bayesian inverse problems

Abstract: Surrogate approaches are used widely for Bayesian inverse problems(BIPs) in order to improve computational efficiency. However, their application to large-scale BIPs is hindered by the curse of dimensionality and the nonlinear concentration of posterior. In this talk, we propose an adaptive multi-fidelity neural network (NN) methodology to tackle these issues. Two essential ingredients of the algorithm are: 1) a deep NN surrogate model, built offline according to the prior distribution, to alleviate the curse of dimensionality, and 2) a local composite NN adaptively constructed online using multi-fidelity data, i.e. a small set of high-fidelity data from the forward model evaluations and a larger set of inexpensive low-fidelity data from the DNN surrogate evaluations, to handle the local structure of the posterior.  Two numerical examples demonstrate the accuracy and efficiency of our approach.

17. 张磊，黑龙江大学

Title: Some results about scattering and inverse scattering by rough surfaces

Abstract: In this talk, we mainly focus on the scattering and inverse scattering problems with rough surfaces and obstacles which lie above (below) the surfaces. Some theoretical results for the direct and inverse problems will be addressed.