摘要
偏微分方程反问题是一个重要的数学研究领域,覆盖了偏微分方程、泛函分析、非线性分析、优化算法和数值分析等不同的数学分支,在介质成像、遥感遥测和图像处理等当代重要的工程领域有广泛的应用.基于问题的不适定性,求解这类问题需要引进正则化思想.但是由于模型的复杂性和广泛性,很难建立统一的正则化框架.本文旨在对几类重要的偏微分方程反问题的研究给出一个系统的总结.在阐明偏微分方程反问题起源和特点的基础上,对以电阻抗成像、波场逆散射和介质热成像为应用背景的三类重要的偏微分方程反问题,系统阐述了核心研究问题、已有结果和方法、未来重要的研究方向.最后从反演方法有效实现的角度,对影响偏微分方程反问题数值求解精度和误差估计的主要因素给出了分析.
Inverse problems for partial differential equations(PDEs)are of great importance in the areas of applied mathematics,which cover different mathematical branches including PDEs,functional analysis,nonlinear analysis,optimizations,regularization and numerical analysis.These problems have found wide applications in many important engineering areas such as media imaging,remote sensing and image processing.Due to the nature of ill-posedness of such kinds of problems,the techniques of regularization should be applied for efficiently solving these problems.However,it is very hard to establish a unified framework for inverse problems of PDEs,due to the variety and complexity of the problems.This paper aims to give an overview on several important inverse problems of PDEs models.Based on the systematic recalls on the origins and specialities of inverse problems for PDEs,we focus on three kinds of PDEs models for inverse problems:electrical impedance tomograph,inverse wave scattering,thermal imaging.The crucial problems of fundamental interests,existing results and methods as well as further possible research directions are reviewed.Moreover,we also give a systematic analysis of numerical methods for solving inverse problems for PDEs.
作者
程晋
刘继军
张波
Jin Cheng;Jijun Liu;Bo Zhang
出处
《中国科学：数学》
CSCD
北大核心
2019年第4期643-666,共24页
Scientia Sinica：Mathematica
基金
国家自然科学基金(批准号:11331004
91730304
11421110002
11726503和91630309)资助项目
关键词
偏微分方程
反问题
不适定性
正则化
稳定性
数值解
partial differential equations
inverse problems
ill-posedness
regularization
stability
numerics
