报告1：Asymptotic Analysis for Time-dependent PDE Models with Small Holes and Applications，王海兵，东南大学教授，时间8月16日14:00；

    In this talk, we consider two time-dependent PDE models with a cluster of small holes. Based on the time-domain boundary integral equation method, we derive the asymptotic expansion of the solution as the size of the holes goes to zero. Under certain geometrical constraints on the size and the minimum distance of the holes, we show that the solution is approximated by a linear combination of point-sources where the weights are given by the capacitance of each hole. A rigorous justification of the asymptotic expansion is shown under natural conditions on the cluster of holes. As an application of the asymptotic expansion, we derive, in the limit case when the holes are densely distributed and occupy a bounded domain, the equivalent effective medium that generates, approximately, the same solution as the cluster of holes. Finally, we numerically verify the asymptotic expansions by comparing the asymptotic approximations with the numerical solutions via the finite element method.

报告2：Mathematical Study on Fluorescence Diffuse Optical Tomography— Recovering the Distribution of Fluorophores Using an Approximate Substitute，孙春龙，南京航空航天大学讲师，时间：8月16日15:00。

    In this talk the time-domain fluorescence diffuse optical tomography (FDOT) is theoretically and numerically investigated based on analytical expressions for a three-space dimensional diffusion equation model (DE model). Physically the radiative transfer equation model (RTE model) is a better model to describe the physical process behind the measurement of the FDOT. We carefully analyzed the derivation of the DE model from RTE model to consider about the modelling error. Here there are two diffusion equations coupled in one of its source term. Each of them describes the emission of angularly averaged excited photon density (i.e. excited light) and that of emitted photon density (i.e. emitted light). Usually for the excited light the distribution of fluorophores in biological tissue is ignored and have the so-called linearized DE model. The emission light is analytically calculated by solving an initial boundary value problem for coupled diffusion equations in the half space. Based on the analytic expression of the solution to this initial boundary value problem, we establish an error estimate for linearizing the DE model. Our FDOT is to recover the distribution of fluorophores in biological tissue based on the linearized DE model by using the time-resolved measurement data on the boundary surface. We theoretically analyzed the identifiability of this inverse absorption problem. Further, aiming a fast and robust algorithm for our FDOT inverse problem, we identify the location of a fluorescence target by assuming that it has a regular shape such as sphere and cuboid. We call this identification the FDOT using approximate substitute. We proposed and verified our inversion strategy which is a combination of theoretical arguments and numerical arguments for an inversion, which enables to obtain a stable inversion and accelerate the speed of convergence. Its effectivity and performance were tested numerically using simulated data and experimental data obtained from ex vivo beef phantoms. (Joint work with Prof. Jijun Liu, Prof. Gen Nakamura, Prof. Goro Nishimura and Prof. Manabu Machida).

报告地点：腾讯ID：815 574 002

主办单位：数学与统计学院

欢迎光临！

    王海兵，东南大学数学学院教授，博士研究生导师，主要从事数学物理反问题的研究。现任中国数学会计算数学分会常务委员。2012年获得北海道大学和东南大学的理学博士学位，2014年获得江苏省优秀博士学位论文，2016年入选江苏高校“青蓝工程”中青年学术带头人培养对象，2018年获得江苏省工业与应用数学学会第二届“工业与应用数学奖青年奖”，2020年获得江苏省数学会第七届江苏省“数学成就奖”。已作为负责人获得三项国家自然科学基金和一项江苏省自然科学基金，在SIAM系列刊物、IP、JCP、JDE等刊物上发表学术论文三十余篇。

    孙春龙，南京航空航天大学讲师。2020年6月和8月分别获得北海道大学和东南大学博士学位，2021年入职南京航空航天大学。主要研究方向为数学物理反问题的理论分析和数值计算方法，已在Inverse Probl.，Sci. China. Math.，J. Opt. Soc. Am. A 等期刊发表学术论文10余篇，主持江苏省自然科学青年基金1项。