带非局部扩散系数的非线性抛物方程反问题及数值实现

The nonlinear parabolic equation with nonlocal diffusion coefficient plays an important role in describing the diffusion process depending on the global diffusion state, which is widely used in ecology, physics and so on. In practical situations, some parameters of this nonlinear parabolic equation cannot be measured directly, therefore the unknown ingredients are reconstructed by using some additional data of the diffusion system. This project aims to research how to reconstruct the unknown parameters in this parabolic equation effectively, stably, and quickly, which belongs to the inverse problem of nonlinear parabolic equation. For this nonlinear parabolic equation, we will study the inversion of discontinuous initial value and source term respectively, including theoretical analyses and numerical algorithms. We will consider the uniqueness with the help of well posedness of direct problem and additional data, and also establish the regularization scheme based on the idea of optimization. Since the inverse problems are nonlinear, we first transform the inverse problem to approximate quadratic optimization problem with the help of Frechlet derivative, based on the linearization idea. Then, we design numerical algorithms based on the multiscale idea and alternative iteration idea. It will be the basics of other parameters inversion of this kind equation at the same time. Therefore, it is a significant project in mathematical theory and practical application.

带非局部扩散系数的非线性抛物方程广泛用于描述依赖于全局状态的扩散过程，在生物、物理等方面有广泛的应用。在很多应用问题中，系统的某些参数无法直接测量，因而需要通过其他可测量的间接信息来确定这些未知参数。本项目旨在研究如何有效、稳定、快速地反演此类扩散系统的未知参数，数学上它是非线性抛物方程反问题。对此类方程，申请人拟研究间断初值和源项反演的问题，包括理论分析和数值算法。基于正问题的适定性和附加条件研究反问题解的唯一性，基于优化的思想构造正则化反演方案。由于反问题的非线性很强，拟利用线性化思想并借助Frechlet导数，将反问题转换为近似的二次泛函优化问题，再利用多重网格和交替迭代的思想设计高效稳定的数值算法。这两类反问题的成功解决将为此类方程其他参数的反演提供强有力的基础和启发。因此，该项目的研究具有非常重要的理论意义和明确的应用前景。

抛物方程在描述依赖于时间变化的扩散过程中有广泛的应用，例如生物种群扩散过程、热传导过程等。在很多应用问题中，受测量工具、参数性质等客观条件的限制，模型中的某些重要参数无法直接测量，因此需要利用特定形式的间接数据来有效、稳定的反演这些参数。本项目分别研究了抛物方程中非局部方程识别源项问题、非局部数据识别内边界问题、以及间断热流识别问题。对源项识别问题，本项目建立了正问题的级数形式解，并基于拟逆正则化思想，构造了源项的正则化解，在先验选取正则化参数策略下建立了对数型误差估计。对非局部数据识别内边界问题，本项目构造了Tikhonov型正则化方案，证明了极小元的唯一性，在后验正则化参数选取策略下建立了广义误差估计，并设计了最速下降迭代反演算法。对间断热流识别问题，本项目基于TV罚项、优化和交替迭代的思想设计了反演算法和建立了广义误差估计。