分数阶扩散方程反向问题的正则化理论与算法研究

项目名称： 分数阶扩散方程反向问题的正则化理论与算法研究

项目编号： No.11426117

项目类型： 专项基金项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 程浩

作者单位： 江南大学

项目金额： 3万元

中文摘要： 当前不适定问题研究的核心和主流是恢复解对数据的连续依赖性，其理论基础是正则化。然而每一种正则化方法在处理具体问题中总有它的优点和不足，很难找到一种万能的方法去解决所有的问题。本项目对分数阶扩散方程反向问题进行研究，给出最优性分析结果、最优正则化方法和迭代型正则化方法，结合基于误差界的偏差原理、平衡性法则、单调差法则等后验参数选取法则，给出不同的构造策略和证明方法，建立完善的理论估计和提供稳定高效的数值算法。项目的重点是提出构造的迭代型正则化方法，给出先验和后验的误差估计，特别是在高维数、变系数上的有所突破。

中文关键词： 不适定问题；正则化；参数选取规则；误差估计；数值计算

英文摘要： The current research of ill-posed problem is to recover the continuous dependence on the data for the solution, which is the core and mainstream, and its theory is based on the regularization. However, each kind of the regularization method in dealing with specific problem has its limits, it is difficult to find a universal method to solve all the problems. The aims of this project is to obtain the optimal analysis, construct the optimal and iterative regularization methods, and give the posterior parameter selection rules for the backward problem in fractional diffusion equation. The focus of the project is to propose the iterative regularization method, and establish the theoretical estimation and provide effective algorithm. We will study the backward problem of the fractional diffusion equation is gradually deepening, especially in high dimension, variable coefficient and contains nonlinear term strives for breakthrough.

英文关键词： Ill-posed problem；Regularization；Parameter choice rule；Error estimate；Numerical explement