摘要
反问题是一个重要的数学研究领域,其在医学成像、地震勘探成像、图像处理和天气预报等众多工程技术领域有着广泛的应用.基于反问题的不适定性,人们引入正则化思想求解这类问题,得到参数的一个近似估计.随着计算能力的提升,在医学成像和勘探成像等领域,人们不再满足于获取待估参数的一个合理估计,而是试图综合经验知识和观测数据的不确定性信息,给出待估参数不确定性的完整刻画.为了实现这一目标,反问题被转化为Bayes统计推断问题,进而发展出了Bayes反演理论与数值算法.不同于经典的统计学研究,反问题研究中待估参数与观测数据是由复杂的数学模型(如偏微分方程)联系起来的,因而需要引入新的思路和新的数学理论.本文聚焦于针对无限维反问题建立的无限维Bayes反演理论,从先验测度构造、Bayes适定性、有限元离散、统计抽样算法和统计大样本理论等方面梳理现有的研究工作,旨在阐明无限维Bayes反演方法的基本研究思路、核心研究问题、已有结果和方法以及未来可能的研究方向.
Inverse problems constitute a significant area of mathematical research,with extensive applications across various engineering and technical fields such as medlical imaging,seismic exploration imaging,image processing,and weather forecasting.Owing to the ill-posedness of inverse problems,the concept of regularization is introduced to solve these problems,resulting in an approximate estimation of the parameters.With the advancement of computational capabilities,people in fields like medical and exploration imaging are no longer satisfied with obtaining a reasonable estimate of the parameters to be estimated.Instead,they attempt to integrate empirical knowledge and uncertainty information of observational data to provide a complete characterization of the uncertainty of the parameters to be estimated.To achieve this goal,people transform inverse problems into Bayesian statistical inference problems,leading to the development of Bayesian inversion theory and numerical algorithms.Unlike classical statistical research,in inverse problem research,the parameters to be estimated and the observational data are connected by complex mathematical models(e.g.,partial differential equations),thus necessitating the introduction of new ideas and mathematical theories.In this paper,we focus on the infinitedimensional Bayesian inversion theory established for inverse problems and organize existing research work from aspects such as prior measure construction,Bayesian well-posedness,finite element discretization,statistical sampling algorithms,and statistical large-sample theory.The aim is to clarify the basic research ideas,core research issues,existing results and methods of infinite-dimensional Bayesian inversion methods,and potential future research directions.
作者
贾骏雄
孟德宇
张远祥
Junxiong Jia;Deyu Meng;Yuanxiang Zhang
出处
《中国科学:数学》
北大核心
2025年第8期1649-1688,共40页
Scientia Sinica:Mathematica
基金
国家自然科学基金(批准号:12322116,12271428,12226004和12326606)
国家重点研发计划(批准号:2023YFC3503400)资助项目。
关键词
反问题
无限维
BAYES
方法
离散不变算法
变分推断
后验收缩率估计
inverse problems
infinite-dimensional Bayesian methods
discretization-invariant algorithms
variational interence
posteriori contraction estimates
