Code

COMPM078

Year

4

Prerequisites

Term

2

Taught By

Simon Arridge (100%)

Aims

To introduce the concepts of optimisation, and appropriate mathematical and numerical tools. Applications in image processing and image reconstruction.

Learning Outcomes

To understand the principles of optimisation and to acquire skills in mathematical methods and programming techniques.

Introduction

Example problems;
Data Fitting Concepts;
Variational and Iterative Concepts.

Mathematical Tools

Linear Algebra: Solving Systems of Linear Equations; Over and Under Determined Problems; Eigen-Analysis and SVD; Preconditioning
Variational Methods: Calculus of Variation; Multivariate Derivatives; Frechet and Gateaux Derivatives
Regulariation: Tikhonov and Generalised Tikhonov Non-Quadratic Regularisation, Non-Convex Regularisation.

Numerical Tools

Non-Gradient Methods: Simplex Method; Powell's Method
Descent Methods: Steepest Descent; Conjugate Gradients; Line Search
Newton Methods: Gauss Newton and Full Newton; TrustRegion and Globalisation; Quasi-Newton; Inexact Newton

Unconstrained Optimisation

Least-Squares Problems: Linear Least Squares; Non-linear Lesat Squares
Non-Quadratic Problems: Poisson Likelihood; Kullback-Leibler Divergence

Regularisation Parameter Selection

Discrepancy Principles; The L-Curve Method; Generalised Cross-Validation

Constrained Optimisation

Equality Constraints: Lagranian Penalties
Inequality Constraints: Positivity Constraints; Upper and Lower Bounds; Active Sets
Primal-Dual Methods: Primal-Dual Interior Point Methods

Bayesian Approach

Bayesian Priors and Penalty Functions, Maximum Likelihood and Maximum A Posterior: Best Linear Unbiased Estimation; Expectation-Minimisation
Posterior Sampling: Confidence-Limits; Monte Carlo Markov Chain; Applications

Image Deblurring: Deconvolution; Anisotropic Denoising
Linear Image Reconstruction: Tomographic Reconstruction; Reconstruction from Incomplete Data
Non-Linear Parameter Estimation: General Concepts; Direct and Adjoint Differentiation

Other Approaches

Simulated Annealing; Genetic Algorithms

Lecture presentations with associated class coursework and laboratory sessions

The course has the following assessment components:

• Written Examination (2.5 hours, 75%)
• Coursework Section (2 pieces, 25%)

To pass this course, students must:

• Obtain an overall pass mark of 50% for all sections combined

The examination rubric is:
Choice of 3 questions from 5. All questions carry equal marks

C.R. Vogel, Computational Methods for Inverse Problems (SIAM 2002)

J.E. Dennis and R.B Schnabel, Numerical Methods for Unconstrained Optimisation and Nonlinear Equations (SIAM 1996)

J. Nocedal and S.J Wright, Numercal Optimisation (Springer 1999)

S.Boyd and L.Vandenberghe, Convex Optimisation, (Cambridge University Press, 2004)