基于贝叶斯观点的分数阶扩散方程反问题研究

项目名称： 基于贝叶斯观点的分数阶扩散方程反问题研究

项目编号： No.11501270

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 张远祥

作者单位： 兰州大学

项目金额： 18万元

中文摘要： 本项目旨在研究贝叶斯框架下分数阶扩散方程相关反问题的求解。我们的研究内容与目标是：（1）针对分数阶扩散方程反问题的具体特征，利用正问题的正则性估计，从理论上分析条件后验概率分布的适定性和相容性；（2）基于贝叶斯反演中条件后验概率分布的理论分析，利用近年来发展的各种不确定性量化方法，设计出快速求解分数阶扩散方程反问题的数值算法，最后利用若干数值例子加以验证。本项目的研究结果将提供一个探索分数阶扩散方程反问题的新视角，对反常扩散、流变学以及粘弹性力学等众多科学技术领域的发展具有重要的理论意义。

中文关键词： 分数阶扩散方程反问题；正则化参数选择；贝叶斯正则化；后验相容性；不确定性量化

英文摘要： This project studies the solution of inverse problems related to fractional diffusion equations based on the Bayesian perspective. Our research contents and objects are as follows. (1) According to the specific characteristics of inverse problems for fractional diffusion equations, we will theoretically analyze the well-posedness and consistency of the conditional posterior probability distributions by means of the regularity of related direct problems. (2) Based on the theory analysis of the conditional posterior probability distributions in Bayesian approach, we will develop fast numerical algorithms for the solution of inverse problems for the fractional diffusion equations by making use of sorts of uncertainty quantification methods developed in recent years. Finally, we will give some numerical examples to illustrate the practicability of the proposed algorithms. The research results of this project will provide a new angle of view for the exploration of the inverse problems for fractional diffusion equations, and be of some theoretical significance in the development of a number of scientific areas such as anomalous diffusion, rheology and viscoelastic mechanics, etc.

英文关键词： Inverse problems for fractional diffusion equation;Choice of regularization parameter;Bayesian regularization;Posterior consistency;Uncertainty quantification