Inverse Problems module (MA51006)
About the module
This is a Level 5 course that offers a robust understanding of the inverse problems theoretical framework and methods suitable for medical and financial applications. The aim is to achieve comprehensive knowledge in the theoretical fundaments and general methodology for inverse problems in various heterogeneous media, including medical applications and finance. This module may optionally be taken in combination with other modules at this level by Level 5 students on the MMath in Mathematics or MSci in Mathematics and Physics degrees. If you have questions about this module or the possible combinations, please contact your Advisor of Studies.
Students taking this module must usually have achieved a pass mark in the module MA32001, or equivalent.
Examples of Inverse Problems
Examples from medical applications and finance.
Inverse Methodology Preliminary Foundation
Necessary Basic Definitions and Theorems in Measure Theory and Function Spaces
General Regularisation Theory
Tikhonov′s regularization method. Landweber Iteration. The Discrepancy Principle of Morozov. Conjugate gradient method.
Galerkin General formulation. The Least Squares Method. The Dual Least Squares Method.
The Truncated Singular Value Decomposition Method
Stable inversion via the Mollification Method
Inverse Problems in General Heterogeneous Media and Medical Applications
Backward heat conduction problem. Inverse problems in reaction-diffusion equations.
Inverse problems in finance
Formulation of forward model: Black-Scholes and Dupire's Formula. Inverse Problem formulation of market volatility. Reconstruction of time- and price-dependent volatilities.
Delivery and Assessment
The module is delivered in the form of lectures and assessed via coursework (100%) consisting of tests and homeworks.
This module is a Scottish Higher Education Level 5 or SCQF level 11 module and is rated as 15 SCOTCAT credits or 7.5 ECTS credits.