高维扩散方程源项反问题及其在数值微分中的应用

Inverse problems of diffusion equations and numerical differentiation are the hotspots in the study of inverse problems, and have a broad background in the practical application. But There are few references on inverse problems of multi-dimensional diffusion equations and, numerical differentiation methods based on the direct and inverse problems of diffusion equations. In this project we mainly study: (1) inverse problems of multi-dimensional parabolic equations for determining the space-dependent source and their applications in numerical differentiation; (2) inverse problems of fractional diffusion equations for identifying the space-dependent source and their applications in numerically fractional differentiation; (3) inverse problems of multi-dimensional diffusion equations for determining the space and time dependent source. We will use some theories and methods, such as regularization methods of ill-posed problems, partial differential equations, Sobolev space, calculus of variations, the finite element method and the finite difference, to study the posedness and numerical inversion algorithms of inverse problems. The uniqueness and stability of solutions for the inverse source problems in the muliti-dimensional space will be obtained together with several stably numerical inversion methods; some numerical differentiation methods based on the direct and inverse problems will be proposed together with the uniqueness and stability of solutions. Implementation of this project will enrich the theory of ill-posed problems and computational methods, and promote inverse problems of mathematical physics to be better used in practical science and engineering researches.

反问题研究领域中，扩散方程反问题与数值微分都是研究的热点，且有着广泛的应用背景，但对高维扩散方程反问题以及基于扩散方程正反问题的数值微分算法的研究并不多。本项目主要研究：（1）高维抛物型方程源项 反问题及其在整数阶数值微分中的应用；（2）分数阶扩散方程源项 反问题及其在分数阶数值微分中的应用；（3）重建时空依赖源项 的高维扩散方程反问题。本项目将利用不适定问题的正则化、偏微分方程、Sobolev空间、变分法、有限元法、有限差分方法等理论与方法来研究反问题解的适定性与数值反演算法；采取理论分析、算法构造与数值模拟相结合的技术路线，获得若干高维空间中源项反问题解的唯一性、稳定性与稳定的反演算法；提出若干基于扩散方程正反问题的数值微分算法，给出其唯一性和稳定性结果。本项目的实施将丰富一般反问题的理论与计算方法，促进数学物理反问题的广泛应用。

本项目主要研究抛物型方程的源项反问题及相关反问题，完成了预定的基本研究目标。本项目获得重建仅依赖空间变量源项的扩散方程反问题的数值算法，给出了反问题解的唯一性与稳定性，在此基础上给出了同时反演空间依赖源项与初始条件的数值反演算法及其解的唯一性和正则化解的收敛性；获得了基于扩散方程正反问题的高维整数阶数值微分问题的唯一性、稳定性和数值算法；获得了扩散方程中仅依赖于时间的源项反问题唯一性和稳定性理论结果和数值反演算法；也获得了若干与本项目相关的反问题的数值反演算法与理论分析结果。研究成果将为如环境污染控制、图像处理、医学检测等应用学科领域提供理论基础和算法支撑。本项目至今为止完成论文18篇，其中SCI收录论文13篇；四年内培养研究生5名。