Numerical Methods for Optimal Experimental Design of Ill-posed Problems Public

Abstract

Abstract
The two goals of this thesis are to develop numerical methods for solving large-scale
optimal experimental design problems efficiently and to apply optimal experimen-
tal design ideas to applications in regularization techniques and geophysics.
The thesis can be divided into three parts. In the first part, we consider the
problem of experimental design for linear ill-posed inverse problems. The mini-
mization of the objective function in the classic A-optimal design is generalized
to a Bayes risk minimization with a sparsity constraint. We present efficient al-
gorithms for applications of such designs to large-scale problems. This is done by
employing Krylov subspace methods for the solution of a subproblem required to
obtain the experiment weights. The performance of the designs and algorithms is
illustrated with a one-dimensional magnetotelluric example and an application to
two-dimensional super-resolution reconstruction with MRI data.
In the second part, we find the optimal regularization for linear ill-posed prob-
lems. We propose an optimal L2 regularization approach enabling us to obtain
inexpensive and good solutions to the inverse problem. In order to reduce the
computational cost, several sparsity patterns are added to the regularization oper-
ator. Numerical experiments will show that our optimal L2 regularization approach
provides much better results than the traditional Tikhonov regularization.
In the last part of the thesis, we design optimal placement of sources and
receivers in a CO2 injection monitoring. An optimal criteria is proposed based
on a target zone and different treatments for placing sources and receivers are
discussed.

Table of Contents

1 Introduction 1

1.1 Bayesian optimal experimental design . . . . . . . . . . . . . . . . . 3

1.1.1 A brief review of Bayesian experimental designs . . . . . . . 5

1.1.2 Numerical Challenges . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Optimal design for regularization . . . . . . . . . . . . . . . . . . . 9

1.2.1 Review of regularization . . . . . . . . . . . . . . . . . . . . 10

1.2.2 Motivation of nding the optimal regularization . . . . . . . 11

1.3 Optimal design in CO2 injection monitoring . . . . . . . . . . . . . 12

1.3.1 Background of CO2 injection . . . . . . . . . . . . . . . . . 13

1.3.2 Motivation of design for CO2 injection monitoring . . . . . . 13

1.4 Overview of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Numerical methods for A-optimal design 19

2.1 The well-posed case . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2 The sparsity control design . . . . . . . . . . . . . . . . . . . . . . . 25

2.3 Formulation for ill-posed problems . . . . . . . . . . . . . . . . . . . 27

2.3.1 The AB design . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3.2 The Api design . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3.3 More about the above designs . . . . . . . . . . . . . . . . . 31

2.4 Numerical optimization of the AB and A designs . . . . . . . . . . 33

2.4.1 Evaluating the traces in the objective function . . . . . . . . 33

2.4.2 Evaluating the derivatives . . . . . . . . . . . . . . . . . . . 35

2.4.3 Solving the linear systems . . . . . . . . . . . . . . . . . . . 37

2.4.4 Numerical optimization . . . . . . . . . . . . . . . . . . . . . 42

2.5 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.5.1 An ill-posed 1D magnetotelluric example . . . . . . . . . . . 43

2.5.2 Super-resolution . . . . . . . . . . . . . . . . . . . . . . . . . 50

3 Numerical methods for E-optimal design 55

3.1 The EB design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.2 The ETik design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.3 Numerical optimization of the EB and ETik designs . . . . . . . . . 57

3.3.1 Eigenvalue approximation . . . . . . . . . . . . . . . . . . . 57

3.3.2 Evaluating the derivatives . . . . . . . . . . . . . . . . . . . 59

3.4 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.4.1 An ill-posed 1D magnetotelluric example . . . . . . . . . . . 61

3.4.2 A borehole ray tomography example . . . . . . . . . . . . . 66

4 Optimal design for regularization 69

4.1 An optimal regularization operator . . . . . . . . . . . . . . . . . . 70

4.1.1 The rst complication: MSE is dependent on the true solution 71

4.1.2 The second complication: The computational complexity . . 74

4.2 Numerical optimization of the optimal regularization . . . . . . . . 76

4.2.1 Matrix-based derivative techniques . . . . . . . . . . . . . . 76

4.2.2 The covariance design approach . . . . . . . . . . . . . . . . 78

4.2.3 The training design approach . . . . . . . . . . . . . . . . . 81

4.2.4 Numerical Optimization . . . . . . . . . . . . . . . . . . . . 82

5 Optimal sparse regularization 83

5.1 The local diagonal pattern . . . . . . . . . . . . . . . . . . . . . . . 83

5.2 The L1 norm pattern . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.3 The Kronecker product pattern . . . . . . . . . . . . . . . . . . . . 85

5.4 Numerical optimization of different sparse patterns . . . . . . . . . 86

5.4.1 The local diagonal pattern . . . . . . . . . . . . . . . . . . . 86

5.4.2 The L1 norm pattern . . . . . . . . . . . . . . . . . . . . . . 89

5.4.3 The Kronecker product pattern . . . . . . . . . . . . . . . . 90

5.5 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.5.1 The 1D magnetotelluric problem . . . . . . . . . . . . . . . 92

5.5.2 An MRI example . . . . . . . . . . . . . . . . . . . . . . . . 96

6 Optimal design in CO2 injection monitoring 105

6.1 Crosswell array constraints . . . . . . . . . . . . . . . . . . . . . . . 106

6.2 Mathematical framework . . . . . . . . . . . . . . . . . . . . . . . . 107

6.2.1 The forward modeling operator . . . . . . . . . . . . . . . . 107

6.2.2 Formulation of the inversion . . . . . . . . . . . . . . . . . . 108

6.3 Numerical optimization through DIRECT algorithm . . . . . . . . . 111

6.3.1 The DIRECT algorithm . . . . . . . . . . . . . . . . . . . . 111

6.3.2 Discussion of the constraints . . . . . . . . . . . . . . . . . . 112

6.4 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.4.1 Numerical Experiment 1: S/R = 0:5 . . . . . . . . . . . . . 121

6.4.2 Numerical Experiment 2: S/R unfixed . . . . . . . . . . . . 123

7 Summary and future work 125

About this Dissertation

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School	Laney Graduate School
Department	Math and Computer Science
Degree	PhD
Submission	Dissertation
Language	English
Research Field	Education, Mathematics
Mot-clé	Optimal Design
Committee Chair / Thesis Advisor	Haber, Eldad, UBC Nagy, James, Emory University
Committee Members	Veneziani, Alessandro, Emory University Benzi, Michele, Emory University

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