召集人：包刚（浙江大学）、程晋（复旦大学）、张波（中国科学院数学与系统科学研究院）

时间：2024.08.18—2024.08.24

会议日程

**Monday, August ****19****th**

**Session Chair**：**Bo Zhang**

**09:10-09:30**** Workshop ****Opening**

**Session Chair**：**Jin Cheng**

09:30-10:30 PDE Perspective on Many-body Problems in Quantum Optics

**Speaker****：****John C Schotland, Yale University**

10:30-11:00 Coffee & Tea break

11:00-12:00 Boundary integral equation solvers for layered-medium scattering problems

**Speaker****：****Tao Yin, Academy of Mathematics and Systems Science**

12:00-14:30 Lunch

**Session Chair**：** J****ianwei Ma **

14:30-15:30 On an inverse problem for elastic wave equations

**Speaker****：****Yixian Gao, Northeast Normal University**

15:30-16:00 Coffee & Tea break

16:00-17:00 Time-harmonic scattering from Dirichlet/Neumann periodic curves with local perturbations

**Speaker****：****Guanghui Hu, Nankai University**

17:50-20:00 Dinner

**Tuesday, August 20th**

**Session Chair**：**Gang Bao**

**09:30-10:30 **Variational Inference for Statistical Inverse Problems

**Speaker****：****Junxiong Jia, Xian Jiaotong University**

10:30-11:00 Coffee & Tea break

**11:00-12:00 **Inverse scattering of multiple particles based on the time reversal model

**Speaker****：****Jun Lai, Zhejiang University**

12:00-14:30 Lunch

**Session Chair**：**Yanfei Wang**

**14:30-15:30 **Stability for inverse random source problems

**Speaker****：****Peijun Li, Academy of Mathematics and Systems Science**

15:30-16:00 Coffee & Tea break

**16:00-17:00 **Stability analysis and reconstruction method for random surface scattering problems

**Speaker****：****Yiwen Lin, Shanghai Jiaotong University**

17:50-20:00 Dinner

** **

**Wednesday, August 21th**

**Free discussion**

** **

**Thursday, ****August**** 2****2nd**

**Session Chair**：**Jijun Liu**

**09:30-10:30 **Acoustic inversion using resonant perturbations

**Speaker****：****Mourad Sini, Johann Radon Institute for Computational and Applied Mathematics (RICAM)**

10:30-11:00 Coffee & Tea break

**11:00-12:00** Multiscale Modeling and Computation of Nano-Optics

**Speaker****：****Di Liu, Michigan State University**

12:00-14:30 Lunch

**Session Chair**：**Peijun Li**

**14:30-15:30 **The numerical methods for the forward and inverse problems of the ocean waveguide

**Speaker****：****Keji Liu, Shanghai University of Finance and Economics**

15:30-16:00 Coffee & Tea break

**16:00-17:00 **Parametric polynomial preserving recovery of data on manifolds and its application to optimal transportation

**Speaker****：****Guozhi Dong, Central South University**

17:50-20:00 Dinner

** **** **

**Friday, August 23****rd**

**Session Chair**：**Guanghui Hu**

**09:****00****-10:****0****0 **A theory of computational resolution limit and a unified approach to super-resolution

**Speaker****：****Hai Zhang, ****HongKong ****University of ****Science and Technology**

10:00-10:30 Coffee & Tea break

**1****0****:****3****0-1****1****:****3****0 **A direct imaging method for reconstructing a locally rough interface from phaseless total-field data

**Speaker****：****Haiwen Zhang, ****Academy of Mathematics and Systems Science**

11:30-14:00 Lunch

** **

报告信息

**John C Schotland, Zhao and Ji Professor of Mathematics, Yale University**

**Title：PDE Perspective on Many-body Problems in Quantum Optics**

**Abstract:** Quantum optics is the quantum theory of the interaction of light and matter. In this talk, I will describe a real-space formulation of quantum electrodynamics. The goal is to understand the propagation of nonclassical states of light in systems consisting of many atoms. In this setting, there is a close relation to kinetic equations for nonlocal PDEs with random coefficients.

**Guozhi Dong, Central South University**

**Title: Parametric polynomial preserving recovery of data on manifolds and its application to optimal transportation **

**Abstract:**Numerical differentiation suffers from unboundedness of differential operators, particularly when data contain error or noise. In this talk, I will introduce the technique called parametric polynomial preserving recovery which is capable of improving the accuracy of gradient recovery for numerical data on discretized manifolds. Applications of the gradient recovery technique to dynamic formulation of optimal transport on surfaces will be highlighted.

**Yixian Gao, Northeast Normal University **

** **

**Title: On an inverse problem for elastic wave equations**

**Abstract:**This talk focuses on an inverse problem associated with the elastic wave equation. We establish the unique identifying results in simultaneously determining both the unknown density and the internal sources for the plate wave equation from the passive boundary measurement. For the elastic density function, we establish the local Lipschitz stability for low frequencies and the logarithmic stability estimate for high frequencies, respectively.

**Guanghui Hu, Nankai University**

** **

**Title: Time-harmonic scattering from Dirichlet/Neumann periodic curves with local perturbations **

**Abstract:**This talk is concerned with model analysis of time-harmonic scattering by locally perturbed periodic curves of Dirichlet/Neumann kind. The scattering interface is supposed to be given by a non-self-intersecting Lipschitz curve. We prove new well-posedness results for scattering of plane waves at a propagative wave number. In such a case there exist guided waves to the unperturbed problem, which are also known as Bounded States in the Continuity (BICs) in physics. In this paper uniqueness of the forward scattering follows from an orthogonal constraint condition enforcing on the total field to the unperturbed scattering problem. This constraint condition, which is also valid under the Neumann boundary condition, is derived from the singular perturbation arguments.

**Junxiong Jia, Xian Jiaotong University**

** **

**Title: Vari****ational Inference for Statistical Inverse Problems**

**Abstract：**As a general uncertainty quantification framework of inverse problems of partial differential equations (IPofPDEs), the Bayesian inverse method has attracted many researchers' attention. One of the critical obstacles to applying the Bayesian inverse approach is how to efficiently compute the statistical quantities (e.g., posterior mean and variances). Similar difficulties also meet in the investigations of uncertainty quantification of machine learning models. In the machine learning community, the researchers proposed the variational inference (VI) approach, which balances accuracy and efficiency. However, due to the infinite-dimensional formulation of the IPofPDEs, there are few investigations on VI approaches to IPofPDEs. In this talk, we briefly introduce the main ideas of VI methods. Then we construct the infinite-dimensional mean-field based VI approach for general linear problems and the infinite-dimensional transformation based VI approach for nonliear inverse problems. Both of the classical and neural network related VI methods are discussed under the circumstances of solving IPofPDEs.

**Jun Lai, Zhejiang University **

** **

**Title: Inverse scattering of multiple particles based on the time reversal model**

**Abstract: **This talk is concerned with the so-called DORT method that uses the eigenfunctions of time-reversal operator to determine the locations of small scatterers. We give a rigorous mathematical justification for the DORT method under different boundary conditions for elastic waves. Extension to inverse scattering in layered medium is also given. Utilizing the imaging result as an initial guess, a Bayesian inversion scheme is proposed to reconstruct the shape of multiple buried extended scatterers more accurately. The method is efficient, accurate and does not need a good initial guess.

** **

**Peijun Li, Academy of Mathematics and Systems Science**

**Title:Stability for inverse random source problems**

**Abstract：**In the field of inverse problems, the estimation of an unknown source term from indirect observations is a fundamental challenge. Random sources add another level of complexity to this problem due to their uncertainties. In this talk, we will focus on the stability estimates for inverse random source problems of wave equations. An overview will be provided on the existing results for estimating the stability of the solution in deterministic settings, and our recent findings will be presented for the stochastic cases.

**Yiwen Lin, Shanghai Jiaotong University**

** **

**Title:Stability analysis and reconstruction method for random surface scattering problems**

**Abstract:**In this talk, we will present a framework for the proof of a priori bounds explicitly with respect to frequencies for random surface scattering problems. By introducing a variable transform, the variational formulation in a random domain is reduced to that in a definite domain with random medium. Combining the stability result for the deterministic case, Pettis measurability theorem and Bochner’s Theorem further yield the stability for random scattering problems. Besides, an MCCUQ reconstruction method is proposed for solving the inverse random surface scattering problem. Numerical results will demonstrate the reliability and efficiency of the proposed method.

**Mourad Sini, ****Johann Radon Institute for Computational and Applied Mathematics (RICAM)****, Austria**

** **

**Title: Acoustic inversion using resonant perturbations**

**Abstract:**We will present recent results derived in the framework of inverse problems appearing in imaging modalities using resonant contrast agents.A standard family of inverse problems in waves-based imaging use wave fields measured remotely to recover internal variations of the material properties (as the acoustic, optic, elastic and thermal ones). Usually this needs to use multiple interogations (i.e. multiple emitters-receivers).

In recent years, it is proposed in the engineering literature to inject small-scaled contrast agents into the objectto image and then perform the experiments. With such perturbations, one needs only few interogations. The price to pay is to use multiple perturbations (i.e. contrast agents).However, we have different advantages as a better stability and the locality of the reconstruction, i.e. we can do the imaging only, locally, where it is needed, and have access to 'explicit' reconstruction formulas.

At the mathematical analysis level, the main advantage is that such contrast agents resonate at certain frequencies.Such resonant frequencies are well understood and can be characterized via the two main operators appearing in the underlying wave-propagators, namely the Newtonian type and the Magnetization type operators.

The general features of our findings are as follows:

1. Using time-harmonic measurements, in a band of frequencies, we can recover the already mentioned resonances. These resonances have the explicit signature of the material properties of the object (as the speed of propagation) which allows to reconstruct them.

2. Using time-domain measurements, in a large enough time, we can recover the internal values of the travel time function. Using the related Eikonal equation, for instance, we can reconstruct the related speed of propagation.

In the talk, we will use the acoustic model (i.e. ultrasound tomography) to highlight these features. But, if time allows, we will also briefly discuss the related results in optical tomography and opto-acoustic tomography (i.e. photo-acoustics).

**Di Liu, Michigan State University**

**Title:Multiscale Modeling and Computation of Nano-Optics**

**Abstract:**We present a multiscale modeling and computational scheme for optical- mechanical responses of nanostructures. The multi-physical nature of the problem is a result of the interaction between the electromagnetic (EM) field, the molecular motion, and the electronic excitation. To balance accuracy and complexity, we adopt the semi-classical approach that the EM field is described classically by the Maxwell equations, and the charged particles follow the Schr ̈oidnger equations quantum mechanically. To overcome the numerical challenge of solving the high dimensional multi-component many- body Schr ̈odinger equations, we further simplify the model with the Ehrenfest molecular dynamics to determine the motion of the nuclei, and use the Time- Dependent Current Density Functional Theory (TD-CDFT) to calculate the excitation of the electrons. This leads to a system of coupled equations that computes the electromagnetic field, the nuclear positions, and the electronic current and charge densities simultaneously. In the regime of linear responses, the resonant frequencies initiating the out-of-equilibrium optical-mechanical responses can be formulated as an eigenvalue problem. A self-consistent multiscale method is designed to deal with the well separated space scales. The isomerization of Azobenzene is presented as a numerical example.

**Keji Liu, Shanghai University of Finance and Economics **

** **

**Title: The numerical methods for the forward and inverse problems of the ocean waveguide**

**Abstract: **In this talk, we will present the identifications of scattering inhomogeneities in a 3D shallow ocean waveguide, which are motivated by applications in ocean acoustics. Some direct-type imaging methods (DIMs) will be presented to identify the marine point sources and medium objects from the observation data, and the key component of the DIMs is an imaging functional whose indicator property is quantitatively characterized. The DIMs can generate reliable initial estimates of submerged inhomogeneities, which advanced inversion methods can then utilize to accurately determine their physical properties.

**Tao Yin, Academy of Mathematics and Systems Science **

** **

**Title: Boundary integral equation solvers for layered-medium scattering problems**

**Abstract:**This talk will introduce our recent works on developing fast and high-order boundary integral equation solvers for solving the wave scattering problems in a layered-medium. Both the windowed Green function method and the perfectly-matched-layer based integral equation method will be discussed as well as the regularization for singular integral operators and the numerical discretization relying on a Chebyshev-based rectangular-polar solver. Numerical examples will be presented to show the accuracy and efficiency of the methods.

**Hai Zhang, HongKong ****University of ****Science and Technology**

** **

**Title: A theory of computational resolution limit and a unified approach to super-resolution**

**Abstract: **It is well-known that the resolution of optical imaging system is fundamentally limited by the optical wavelength. Based on this, Rayleigh proposed the Rayleigh criterion on the minimum resolvable distance between two point sources, the so called Rayleigh limit. Although widely used in the practice, this limit is not so useful for images that are subject to elaborated data processing. To remedy this, we develop a theory of computational resolution limit to characterize the fundamental resolution limit from the approximation theory point of view. The theory can be used to explain the phase transition phenomenon in the reconstruction problem. It is also extended to another parameter estimation problem in the Gaussian mixture models. In the second part, we will present a model-based super-resolution method and apply it to three concrete models including the point source models, signals with finite rate of innovation, and signals with continuous forms.

**Haiwen Zhang, Academy of Mathematics and Systems Science**

** **

**Title: A direct imaging method for reconstructing a locally rough interface from phaseless total-field data**

**Abstract: **This talk is concerned with the problem of inverse scattering of time-harmonic acoustic plane waves by a two-layered medium with a locally rough interface in two dimensions. A direct imaging method is proposed to reconstruct the locally rough interface from the phaseless total-field data measured on the upper half of the circle with a large radius at a fixed frequency. The presence of the locally rough interface poses challenges in the theoretical analysis of the imaging methods. To address these challenges, a technically involved asymptotic analysis is provided for the relevant oscillatory integrals involved in the imaging methods, based mainly on the techniques and results in our recent work on the uniform far-field asymptotics of the scattered field for acoustic scattering in a two-layered medium. Finally, extensive numerical experiments are conducted to demonstrate the feasibility and robustness of our imaging algorithm.