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报告承办单位: 数学与统计学院

报告题目:  Bayesian approach to a nonlinear inverse problem for time-space fractional diffusion equation

报告内容:  

The inverse problems for fractional differential equations has become a promising research area because of its wide applications in many scientific and engineering fields. Particularly, the correct orders of fractional derivatives are hard to know as they are usually determined by experimental data and contain non-negligible uncertainty, therefore, the research on inverse problems involving the orders are in necessary. Furthermore, the problems involving the inversion of fractional orders are essentially nonlinear, classical methods may hard to provide satisfying approximations and fail to capture the relevant uncertainty, a natural way to solve such inverse problems is through Bayesian approach. In this talk, we consider an inverse problem of simultaneously recovering the source function and the orders of both time and space fractional derivatives for time-space fractional diffusion equation. The problem will be formulated in the Bayesian framework, where the solution is the posterior distribution incorporating the prior information about the unknown and the noisy data. Under the considered infinite dimensional function space setting, we prove that the corresponding Bayesian inverse problem is well-defined based upon proving the continuity of the forward mapping. In addition, we also prove that the posterior distribution depends continuously on the data with respect to the Hellinger distance. Moreover, we adopt the iterative regularizing ensemble Kalman method to provide numerical implementation to the considered inverse problem for one dimensional case, the numerical results shed light on the viability and efficiency of the method.

报告人姓名:  张远祥

报告人所在单位: 兰州大学数学与统计学院

报告人职称/职务及学术头衔:    副教授

报告时间:  2020年10月30日 14:30－15:10

报告方式: 理科楼A-419 

报告人简介:  张远祥副教授2012年获兰州大学博士学位。于2011-2012年赴英国华威大学数学系进行博士联合培养。张远祥副教授研究兴趣为偏微分方程反问题的贝叶斯理论及快速算法。主持国家自然科学基金项目1项，在Inverse Problems等杂志上发表论文17篇。