带斜导数边界条件的偏微分方程定解问题的边界反演

Boundary reconstructions for partial differential equation models are a class of important inverse problems of mathematical physics, and have wide applications in medium imaging and nondestructive testing. The difficulty of this kind of problems consists in its nonlinearity and ill-posedness. Inverse scattering and thermal imaging are two important physical models which aim to inspect internal structure of the object medium from some information contained in wave fields and temperature distributions, respectively. Under some assumptions, they are modeled as boundary reconstruction problems for partial differential equations of elliptic and parabolic types. This project studies three inverse scattering and thermal imaging models which have specific application backgrounds. The common feature is that the descriptions of the scattering and heat conduction processes in media need to introduce the so-called oblique derivative boundary conditions, which lead to some essential difficulties for both the forward and inverse problems. In this project, we develop efficient numerical methods for solving the forward and inverse problems for partial differential equations with oblique derivative boundary conditions. We focus on non-iterative numerical methods for the boundary reconstruction problems with accuracy analysis. Further, we establish the mathematical theory and methods for making numerical reconstructions more or less accurate. This project is a typical research on mathematical theory and methods motivated by applied problems, combining mathematical modeling, analysis and computation. With the help of Green functions for oblique derivative problems, we expect to obtain some deep mathematical results in a unified framework for several kinds of inverse problems for partial differential equations.

利用偏微分方程定解问题解的信息重建区域的边界是一类重要的数学物理反问题，其研究难点在于问题的非线性性和不适定性，在介质成像和无损探伤等领域有广泛的应用。逆散射和热成像是利用波场和温度场包含的信息检测介质内部异常或缺陷的两类重要物理模型，在某些假定下，它们数学上归结为椭圆型和抛物型方程的边界反演问题。本项目研究三个具有明确应用背景的波场逆散射和热成像模型，其共同特点是需要引入斜导数边界条件来描述介质中的散射和热传导过程，这给正问题和反问题的理论分析及数值计算带来了本质的困难。本项目将发展求解带斜导数边界条件的偏微分方程正问题和反问题的数值方法，重点研究边界反演问题的非迭代数值方法并分析算法的精度，提出改变重建结果精度的理论和方法。本项目是应用问题驱动的数学理论和方法的研究，集建模、分析和计算于一体，借助于斜导数问题的格林函数，有望在统一的框架下对几类偏微分方程反问题得到比较深刻的数学结果。

利用偏微分方程定解问题解的信息重建区域的边界是一类重要的数学物理反问题，其研究难点在于问题的非线性性和不适定性，在介质成像和无损探伤等领域有广泛的应用。逆散射和热成像是利用波场和温度场包含的信息检测介质内部异常或缺陷的两类重要物理模型，在某些假定下，它们数学上归结为椭圆型和抛物型方程的边界反演问题。本项目研究几类具有明确应用背景的波场逆散射和热成像模型，主要研究内容和重要结果如下：一、对均匀手性介质中的斜入射电磁波散射，建立了正散射问题的适定性和逆散射问题的唯一性；二、对一般椭圆型方程的边界反演问题，提出了两类非迭代型区域采样算法；三、对一类具有小尺寸空腔的介质热传导问题，证明了其解的渐近性质，并将其应用于等效介质理论；四、基于抛物型方程反边值问题，提出了求解扩散光学层析成像问题的高效稳定的数值方法；五、利用位势理论，发展了抛物型方程反边值问题的线性抽样型算法，并揭示了其数学本质；六、利用拟微分算子理论，构造了扩散方程内透射问题的格林函数，进而证明了其唯一可解性，保证了上述抽样型算法的可行性。通过上述研究，项目组人员对偏微分方程的边界反演问题有了更深的理解，对这类问题的非迭代型数值方法的可行性、收敛性和稳定性等得到了深刻的数学结果，并将其用于介质热成像和扩散光学层析成像等重要科学问题。本项目的研究成果对其它数学物理反问题和最优控制问题具有重要的借鉴意义。作为项目的自然延伸，项目组人员对抛物型方程解的渐近性质及其在反问题与成像中的应用开展了初步的研究，为下一个项目的研究奠定了坚实的基础。

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