# 2023/24 Taught Postgraduate Module Catalogue

## COMP5930M Scientific Computation

### 15 creditsClass Size: 150

Module manager: Dr Toni Lassila
Email: T.Lassila@leeds.ac.uk

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2023/24

This module is not approved as an Elective

### Module summary

This module covers a range of numerical algorithms for computational problems that can be formulated as continuous nonlinear equations and large linear equation systems. Starting from standard algorithms, such as the Newton-Raphson method, we cover the various properties and difficulties faced by such numerical algorithms (robustness, complexity, generality) and ways to tackle these problems reliably. By introducing the solution of nonlinear partial differential equations through numerical discretisation techniques, we demonstrate how state-of-the-art numerical methods can be designed to achieve maximum efficiency and allow the solution of large, sparse systems of linear and nonlinear systems of equations.

### Objectives

On completion of this module, students should be able to:
- understand the role of computational methods in scientific computing and the importance of robustness, computational efficiency, stability and accuracy of numerical methods
- demonstrate awareness of the state-of-the-art in scientific computing algorithms for the solution of nonlinear and linear problems and identify the best methods for a given problem
- use sparsity and sparse data structures to develop and analyse efficient implementation of numerical methods
- perform discretisation to systems of differential equations to approximate their solutions numerically
- understand the practical issues associated with code implementation of numerical methods and program basic implementations of numerical methods themselves
- develop awareness of typical applications for numerical analysis software in engineering, computer science and computational mathematics
- analyse the computational complexity of a numerical method for a given system of equations

Learning outcomes
1. Formulate and solve systems of nonlinear equations to solve challenging real-world problems arising from engineering and computational science.
2. Analyse a given nonlinear equation and choose and implement the best numerical method for its solution.
3. Implement algorithmic solutions to solve computational differential equation problems based on mathematical theory.
4. Analyse computational linear algebra problems to identify and implement the most efficient and scalable solution algorithm to apply for large problems.
5. Develop a broad understanding of the theory of numerical analysis and its applications to the solution of nonlinear and differential equations.

### Syllabus

Solving one nonlinear equation: Newton's method, bisection method, secant method, Dekker’s method; Issues: divergence of Newton's method, finding an initial guess, local convergence; Solving systems of equations: Jacobian computation, Gradient descent method, Newton with line-search, continuation methods; Partial differential equations: discretisation in space and time; time-stepping algorithms, sparse Jacobian computation; Linear solvers for large systems: Sparsity: computational complexity, pivoting, reordering; Direct methods: LU factorisation; Iterative linear solvers: Gauss-Seidel, Jacobi, conjugate gradient; Inexact Newton-Krylov method.

### Teaching methods

 Delivery type Number Length hours Student hours Lectures 20 1.00 20.00 Tutorial 10 1.00 10.00 Independent online learning hours 20.00 Private study hours 100.00 Total Contact hours 30.00 Total hours (100hr per 10 credits) 150.00

### Private study

Private study consists of 3 hours of review of the lecture/tutorial materials per week (total 30 hours), 20 hours of review for the final exam, and 50 hours for each of the two coursework assessments.

### Opportunities for Formative Feedback

Attendance and formative coursework.

### Methods of assessment

Coursework
 Assessment type Notes % of formal assessment In-course Assessment Coursework 1 20.00 In-course Assessment Coursework 2 20.00 Total percentage (Assessment Coursework) 40.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated.

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 2 hr 00 mins 60.00 Total percentage (Assessment Exams) 60.00

The exam will be a computer-based exam. The module will be reassessed by computer-based examination.