本科生、研究生入门阅读书籍

基础知识学习指南:

 

本科阶段学习书籍:

 

----> Richard C. Aster, Brian Borchers, Clifford H. Thurber, Parameter Estimation and Inverse Problems, Second Edition, 2013. PDF(第一、二版)PDF(第三版) (有限维理论+算法;阅读基础:高等代数,数学分析)

----> Daniela Calvetti and Erkki Somersalo, Introduction to Bayesian Scientific Computing -- Ten Lectures on Subjective Computing, Springer, 2007. (贝叶斯计算;阅读基础:高等代数,数学分析)

----> Jari Kaipio, E. Somersalo, Statistical and Computational Inverse Problems, Applied Mathematical Sciences 160, 2004. (贝叶斯理论+算法;阅读基础:高等代数,数学分析,泛函分析)

----> Christopher M. Bishop, Pattern Recognition And Machine Learning, Springer, 2006 (贝叶斯观点下的机器学习;阅读基础:高等代数,数学分析,泛函分析)

 

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研究生阶段学习书籍:

 

反问题正则化理论基础:

 

----> Thorsten Hohage, Lecture Notes on Inverse problems, 2002. PDF (复习泛函分析的同时学习经典的正则化方法)

 

统计反问题理论基础:

 

----> 严加安,测度论讲义(第二版),科学出版社,2004 (整本) PDF,课后习题答案PDF

----> J. C. Robinson, Infinite-Dimensional Dynamical Systems, An Introduction to Dissipative Prabolic PDEs and the Theory of Global Attractors, Cambridge Texts in Applied Mathematics, 2001 (阅读1--7章 复习泛函分析,了解PDE的基本理论)

----> Giuseppe Da Prato, An Introduction to Infinite-Dimensional Analysis, Springer, 2006. (阅读1--3章) 

----> Giuseppe Da Prato and Jerzy Zabczyk, Stochastic Equations in Infinite Dimensions, Second Edition, Cambridge University Press, 2014. (阅读2--4章 了解无穷维测度的基本理论)

----> Lawrence C. Evans, Partial Differential Equations, Second Edition, American Mathematical Society, 2010. (阅读5--7章 回顾线性PDE的基础)

----> E. D. nezza, G. Palatucci, E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. math. 136 (2012) 521-573 (简单学习分数次Sobolev空间)

----> Masoumeh Dashti and Andrew M. Stuart, The Bayesian Approach to Inverse Problem, Handbook of Uncertainty Quantification, Springer International Publishing Switzerland 2017. 与 A. M. Stuart, Inverse problems: A Bayesian perspective, Acta Numerica, 2010 (学习无穷维贝叶斯反演理论与算法)

 

 

统计反问题计算基础(基于有限元):

 

----> Daniela Calvetti and Erkki Somersalo, Introduction to Bayesian Scientific Computing -- Ten Lectures on Subjective Computing, Springer, 2007. (整本书)

----> 刘继军,现代数值计算方法,科学出版社,2010 (阅读1--3章了解有限元方法的基本知识)

----> Juan Carlos De los Reyes, Numerical PDE-Constrained Optimization, Springer, 2015 (PDE约束下优化问题的基本理论)

----> T. Bui-Thanh, O. Ghattas, J. Martin, G. Stadler, A computational framework for infinite-dimensional Bayesian inverse problems Part I: The linearized case, with application to global seismic inversion, SIAM J. Sci. Comput, 35(6), 2013, A2494-A2523 (学习基于有限元离散的贝叶斯反演计算基本概念

----> 学习Python编程,学习 https://uvilla.github.io/inverse17/  (熟悉网页上的程序)

 

统计反问题+机器学习(还没有书,可通过以下文章了解):

 

----> Pelip Cucker, Steve Samle, On the mathematical foundations of learning, Bulletin of The American Mathematical Society, 39(1), 2001, 1-49. 

----> Ernesto De Vito, Lorenzo Rosasco, Andrea Caponnetto et al., Learning from examples as an inverse problem, Journal of Machine Learning Research, 6, 2005, 883-904. (在核方法的框架下阐明了反问题与机器学习问题)

----> M. M. Dunlop, M. A. Girolami, A. M. Stuart, and A. L. Teckentrup; How Deep Are Deep Gaussian Processes? Journal of Machine Learning Research 19(54):1−46, 2018.

----> N. B. Kovachki, and A. M. Stuart; Ensemble Kalman Inversion: A Derivative-Free Technique For Machine Learning Tasks. Inverse Problems, Vol. 35, Number 9, (2019).

----> K. Bhattacharya, B. Hosseini, N. B. Kovachki, A. M. Stuart; Model Reduction and Neural Networks for Parametric PDEs

----> F. Hoffmann, B. Hosseini, Z. Ren, A. M. Stuart; Consistency of Semi-Supervised Learning Algorithms on Graphs: Probit and One-Hot Methods, Journal of Machine Learning Research, 21(186):1−55, 2020.

----> M. M. Dunlop, D. Slepcev, A. M. Stuart, M. Thorpe; Large Data and Zero Noise Limits of Graph-Based Semi-Supervised Learning Algorithms, Applied & Computational Harmonic Analysis, 49-2 (2020), pp. 655-697.

 

梯度流:

 

----> Filippo Santambrogio, Optimal Transport for Applied Mathematicans---Calculus of Variations, PDEs, and Modeling, Birkhauser, 2015

----> Luigi Ambrosio, Nicola Gigli, Giuseppe Savare, Gradient Flows --- in Metric Spaces and in the Space of Probability Measures, Second Edition, 2008

 

图神经网络:

 

​​​​​​​----> Yao Ma, Jiliang Tang, Deep Learning on Graphs, 2020.

 

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注:书籍和论文可以在 http://gen.lib.rus.ec/ 上下载

一些有意思的链接 (Some Interesting Links)

公开课网站:

MOOC Coursera https://www.coursera.org/    学堂在线 https://www.xuetangx.com/   

Stanford Online http://online.stanford.edu/      爱课程 http://www.icourses.cn/home/

 

有意思的书:T. Villani,  一个定理的诞生---我与菲尔兹奖的一千个日夜;Daniel Kahneman,  思考,快与慢; 

与数学有关的电影: 知者无涯---印度数学家拉马努金; 模仿游戏---阿兰.图灵;美丽心灵---约翰.纳什

 

文章推荐:

 

学习方法:

1. 怎样练习一万小时----刻意练习天才训练指南

2. 别指望灵感,还是要靠汗水 ——“创造性思维”的三个迷信

3. 最高级的想象力是不自由的

4. 笔记本就是力量

5. Myth和Truth:什么是搞科研

6. 用强力眼读书,

 

社会思考:

1. 精致的利己主义者和常青藤的绵羊