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.

Prerequisites

Students taking this module must usually have achieved a pass mark in the module MA32001, or equivalent.

Indicative Content

• 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 Methods

  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.

Credit Rating

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.