課程大綱

課程資訊

課程名稱

數值逆問題
Numerical Inverse Problem

開課學期

102-2

授課對象

理學院應用數學科學研究所

授課教師

薛克民

課號

MATH5430

課程識別碼

221 U6300

班次

學分

全/半年

半年

必/選修

選修

上課時間

星期一7,8(14:20~16:20)星期四@(~)

上課地點

天數305天數305

備註

總人數上限：30人

Ceiba 課程網頁

http://ceiba.ntu.edu.tw/1022MATH5430_

課程簡介影片

核心能力關聯

本課程尚未建立核心能力關連

課程大綱

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課程概述

Inverse problems are the opposites of the direct problems. Informally, in a direct problem one finds an effect from a cause, and in an inverse problem one is given the effect and wants to recover the cause.
Representative examples of inverse problems arise in many important applications, ranging from biomedical imaging to seismic prospecting and beyond. The aim of this course is to discuss both the underlying mathematics and the computational methods used to solve inverse problems.
Sample topics to be discussed are:
1. Ill-posedness and regularization
2. Numerical optimization tools
3. Parameter identification
4. Total variation regularization
5. Sparse solutions to linear system of equations

課程目標

1. Introduce sample mathemaical models for inverse problems
2. Devise numerical techniques for numerical computational of inverse problems

課程要求

修課學生需具備基本的線性代數與計算數學之知識

預期每週課後學習時數

Office Hours

另約時間

指定閱讀

Overview of inverse problems (an essay written by P. Argoul; available
at ceiba bulletin)

參考書目

1. Jennifer L. Mueller and Samuli Siltanen, Linear and nonlinear inverse problems with
practical applications, SIAM 2012 (on reserved at Math. library, NTU)
2. Curtis R. Vogel, Computational Methods for Inverse Problems, SIAM 2002 (e-book,
available at NTU
library system)
3. Per Christian Hansen, Discrete Inverse Problems: Insight and Algorithms, 2009 SIAM (e-
book,
available at NTU library system)
4. Richard C. Aster, B. Borchers, and Clifford H. Thurber, Parameter estimation and
inverse problems, 2nd Edition, Elsevier 2011 (e-book available at the authors webpage, see
share resources)
5. Inverse problems (lecture notes given by S. Siltanen, available at Ceiba bulletin)
6. An introduction to inverse problems (lectures given by M. Sambridge, available at Ceiba
bulletin)
7. Inverse problems in geophysics (lecture notes given by R. M. Richardson and G. Zandt,
available at Ceiba bulletin)

評量方式
(僅供參考)

No.	項目	百分比	說明
1.	Homework	100%	Four homework sets in total (25 point each)

課程進度

週次	日期	單元主題
第18週	06/16	Term project presentation
第1-1週	02/17	Course overview
第1-2週	02/20	Tikhonov regularization of linear ill-posed problems
第3-1週	03/03	Exercise session (Astro/Math bldg 3F)
第3-2週	03/06	Sparse & total variation regularization I
第4-1週	03/10	Numerical optimization: review
第4-2週	03/13	Numerical optimization: Quasi-Newton method
第5-1週	03/17	Quasi-Newton (Cont.)
第5-2週	03/20	BFGS derivation
第6-1週	03/24	l1 magic
第6-2週	03/27	Primal-dual problem
第7-1週	03/31	Duality & log. barrier
第7-2週	04/03	溫書假
第8-1週	04/07	Fast l1-min algorithms
第8-2週	04/10	FISTA
第9-1週	04/14	出國開會
第9-2週	04/17	出國開會
第10-1週	04/21	Bregman algorithm
第10-2週	04/24	Bregman algorithm (Cont.)
第11-1週	04/28	Term project presentation
第12-2週	05/08	Sparse dynamics for PDEs
第13-1週	05/12	Minimum rank problem
第14-1週	05/19	sub-differential of matrix norm
第14-2週	05/22	Parameter identification: Adjoint method
第16-1週	06/02	端午節
第16-2週	06/05	Adjoint method (Cont.)