Mini-workshop on inverse problems

发布时间:2019年11月27日 作者:Hyeonbae Kang   消息来源:    阅读次数:[]



10:00-11:00, Sanghyeon Yu (Korea University)

Title1: Surface plasmons of geometrically singular nanostructures

Abstract: Surface plasmons, which are optical resonances supported by metallic nanostructures, are extremely useful for manipulating light on the nanoscale. When the nanostructures have geometric singularities such as corners and two close-to-touching surfaces, their surface plasmons exhibit extreme light concentration and singular spectral shifts. In this talk, we shall discuss our recent works (S. Yu and H. Ammari SIAM Review 60 (2018), and S. Yu and H. Ammari PNAS (2019)) on understanding of these singular plasmons and their applications to the effective design of plasmonic metamaterials. This talk is based on joint works with Habib Ammari (ETH).

11:00-12:00, Lingyun Qiu (Tsinghua University)

Title2: The inverse problem for wave equations based on optimal transportation theory and deep learning

Abstract: The wave-based inverse problems aiming at geological imaging is often accompanied by nonlinearity, significant ill-posedness, and high computational cost. It usually uses the iterative method based on gradient descent to solve the corresponding optimization problem and is easy to fall into local minimums. Therefore, improving the convexity of the objective function of the optimization problem, avoiding the local minimization, and improving the efficiency of the large-scale inversion are the key issues in the study of the inverse problem of the wave equation.

First, with the aid of optimal transport theory, we propose a new data transformation method, improve the stability and convexity of the inverse problem, and expand the convergence radius of the inverse problem. At the same time, we also study the noise in the data and the inaccuracy of the measurement to the disturbance of the optimal transport metric, and systematically propose a fast and stable solution. Second, convolutional neural networks are also used to improve the performance and accuracy of the algorithm further. The network architecture we use is modified from the deep class aware model. This model does not use a fully connected layer, so fewer training samples are needed in the training step, and less memory and lower arithmetic complexity are required in the inference step. We use only a small amount of representative data during the training of the network, and the trained model can be used to extract the main features from the high-dimensional intermediate results automatically. This approach provides a novel platform for data-driven inversion/imaging methods.

邱凌云博士,现任清华大学丘成桐数学科学中心助理教授,于2013年在美国普渡大学数学系获得博士学位。在加入清华大学之前,其曾在2015年至2018年就职于PGS (Petroleum Geo-Services)位于美国休斯敦的全球研发总部,从事地震波反演问题的研究工作。2013年至2015年,邱凌云博士在明尼苏达大学的IMA(Institute for Mathematics and its Applications)和埃克森美孚位于美国新泽西州的研究与工程中心(ExxonMobils Research and Engineering Technology Center)担任联合职位博士后。邱博士的主要研究兴趣包括非线性反问题的分析与计算、最优输运理论、正则化方法、最优化问题的迭代算法以及深度学习在反问题上的应用。

12:00-14:00, Lunch

15:00-16:00, Wenjia Jing(Tsinghua University)

Title3: Layer Potential Techniques in homogenization of perforated domains

Abstract: We present some recent results on periodic layer potentials, and their applications in solving the cell problems associated to homogenization of Dirichlet problems on periodically perforated domains. In particular, it can be used to quantify the rate of convergence of certain rescaled cell problems, and, as a result, yield quantitative homogenization result and new correctors.

荆文甲,清华大学丘成桐数学科学中心助理教授。2006年本科毕业于北京大学,2011年博士毕业于哥伦比亚大学。2011-2013年于巴黎高等师范学院从事博士后工作,2013-2016年芝加哥大学“L.E. Dickson Instructor”。

16:00-17:00, Jianliang Li (Changsha University of Science & Technology)

Title4: Inverse random source problems for time-harmonic acoustic and elastic waves


This talk concerns the random source problems for the time-harmonic acoustic and elastic wave equations in two and three dimensions. The goal is to determine the compactly supported external force from the radiated wave field measured in a domain away from the source region. The source is assumed to be a microlocally isotropic generalized Gaussian random function such that its covariance operator is a classical pseudo-differential operator. Given such a distributional source, the direct problem is shown to have a unique solution by using an integral equation approach and the Sobolev embedding theorem. For the inverse problem, we demonstrate that the amplitude of the scattering field averaged over the frequency band, obtained from a single realization of the random source, determines uniquely the principle symbol of the covariance operator. The analysis employs asymptotic expansions of the Green functions and microlocal analysis of the Fourier integral operators associated with the Helmholtz and Navier equations.

李建樑,20096月本科毕业于中国农业大学理学院,20147月获中国科学院大学理学博士学位。20147月起任长沙理工大学寶盈娱乐APP讲师。201711月-201811月受国家留学基金委资助赴美国普渡大学数学系访问一年。主要研究领域为反散射问题的理论与数值方法、随机反散射问题的理论研究。主持国家自然科学基金青年项目一项,湖南省教育厅一般项目一项。在《Inverse Problems in Science and Engineering》、《Applicable Analysis》、《Computers and Mathematics with applications》、《SIAM Journal on Imaging Sciences》、《SIAM Journal on Mathematical Analysis》发表论文六篇。


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