# NUMERICAL METHODS FOR APPLIED SCIENCES

**Academic Year 2023/2024**- Teacher:

**ARMANDO COCO**

## Expected Learning Outcomes

The learning outcomes of the course are as follows:

Demonstrate knowledge and understanding of the analysis of a range of numerical algorithms.

Analyse and choose appropriate techniques for solving numerical problems.

Use of specialist software for computational problems.

Synthesize concepts from linear algebra, analysis and numerical analysis and apply these to find numerical solutions to problems.

Understand the process of mathematical model-building in a range of application areas.

Demonstrate skills in designing and solving a model based on a real-world problem.

**Knowledge and understanding:** one of the objectives of this course is the knowledge and understanding of numerical algorithms for solving linear systems and finding the eigenvalues of a matrix. The student will be able to know advanced methods such as multigrid methods.

**Apply knowledge and understanding:** thanks to the use of specialized software for computational problems, students will be able to apply knowledge in practical cases. The computer implementation will allow them to verify the properties of numerical methods with their own hands and to have a more in-depth learning of theoretical and practical concepts.

**Expressing judgments:** the student will be able to analyze and choose the appropriate techniques to solve numerical problems, such as the choice between a direct method and an iterative method, recognizing which ones are more efficient and accurate based on the real application. They will be able to synthesize concepts from linear algebra, numerical analysis and mathematical analysis and apply them to find numerical solutions to real problems.

**Communication skills:** the development of a project will allow the student to improve communication skills both towards a specialist and general public.

**Learning skills:** interactive participation in lectures as well as group exercises will allow the student to improve their learning skills and understanding of mathematical models in different areas of application.

## Course Structure

**Credit Value**: 6**Scientific field**: MAT/08 - Numerical analysis**Taught classes**: 150 hours in total: 108 hours of individual study, 42 hours of lectures**Term / Semester**: 2°

Face-to-face lectures and individual and group works.

If the teaching is given in a mixed or remote mode, the necessary changes with respect to what was previously stated may be introduced, in order to comply with the program envisaged and reported in the syllabus.

**Information for students with disabilities and / or SLD**

To guarantee equal opportunities and in compliance with the laws in force, interested students can ask for a personal interview in order to plan any compensatory and / or dispensatory measures, based on the didactic objectives and specific needs. It is also possible to contact the referent teacher CInAP (Center for Active and Participated Integration - Services for Disabilities and / or SLD) of our Department, prof. Filippo Stanco.

## Required Prerequisites

Good knowledge of vector spaces and matrix properties, direct (LU factorization) and iterative (Jacobi and Gauss-Seidel) methods to solve linear systems.

## Attendance of Lessons

Highly recommended.

## Detailed Course Content

The course covers the design and analysis of numerical algorithms to solve or accurately approximate problems from linear algebra, such as linear systems and eigenvalue problems.

It also aims at providing solid implementation skills by developing small software programs of the different numerical algorithms, with applications to real-world problems.

** **

**Direct methods to solve linear systems.**

Introduction to projectors. Definition of complementary projectors and properties. Orthogonal projector and characterization theorem. QR factorization for rectangular matrices: full and reduced factorizations. Gram-Schmidt algorithm: classical and modified and their stability properties. Householder triangularization.

**Iterative methods to solve linear systems.**

Definition of the class of Krylov methods. Some examples: GMRES, Gradient descent, Conjugate gradient.

**Multigrid methods.**

Discretization of elliptic equations. Smoothing properties of some relaxations schemes: Jacobi, weighted-Jacobi and Gauss-Seidel. Two-Grid Correction Scheme: restriction and interpolation operator. Example of multigrid iterations: V-cycle, W-cycle, F-cycle. Convergence factor and computational cost.

**Eigenvalue problems.**

Hessenberg reduction, QR without shift and with Wilkinson shift, Divide-and-Conquer algorithm.

**Mathematical modelling.**

Fundamentals and detailed description of the modelling and simulation process for real-life applications. Real-world applications to problems that can be solved by multigrid methods: electro-magnetism, gravity, elastic deformation, diffusion processes.

**Financial Mathematics.**

## Textbook Information

1. Trefethen, L. N., & Bau III, D. (1997). *Numerical linear algebra* (Vol. 50). Society for Industrial and Applied Mathematics.

2. Briggs, W. L., Henson, V. E., & McCormick, S. F. (2000). *A multigrid tutorial*. Society for Industrial and Applied Mathematics.

3. Tung, K. K., & Tung, K. K. (2007). *Topics in mathematical modeling*. Princeton, NJ: Princeton University Press.

4. Campolieti, G., & Makarov, R. N. (2018). *Financial mathematics: a comprehensive treatment*. Chapman and Hall/CRC.

Author | Title | Publisher | Year | ISBN |
---|---|---|---|---|

Trefethen, L. N., & Bau III, D. | Numerical linear algebra | Society for Industrial and Applied Mathematics | 1997 | 0898713617 |

Briggs, W. L., Henson, V. E., & McCormick, S. F. | A multigrid tutorial | Society for Industrial and Applied Mathematics | 2000 | 0898714621 |

Tung, K. K., & Tung, K. K. | Topics in mathematical modeling | Princeton, NJ: Princeton University Press | 2007 | 0691116423 |

Campolieti, G., & Makarov, R. N. | Financial mathematics: a comprehensive treatment | Chapman and Hall/CRC | 2018 | 1439892423 |

## Course Planning

Subjects | Text References | |
---|---|---|

1 | QR factorization | 1 |

2 | Gram-Schmidt algorithm | 1 |

3 | triangolarizzazione di Householder | 1 |

4 | GMRES | 1 |

5 | Gradient descent | 1 |

6 | Conjugate gradient | 1 |

7 | smoothing factor | 2 |

8 | Two-Grid Correction Scheme | 2 |

9 | V-cycle, W-cycle, F-cycle | 2 |

10 | Hessenberg reduction | 1 |

11 | QR without shift and with Wilkinson shift | 1 |

12 | mathematical models | 3 |

13 | real-life applications | 3 |

14 | European put and call options | 4 |

15 | Trading strategy and arbitrage opportunity | 4 |

16 | Risk-neutral probability measure | 4 |

## Learning Assessment

### Learning Assessment Procedures

**Project**,

**Oral Exam**, marked out of 30 (worth 50% of the final mark)

**Criteria for assigning marks**: both for the project and for the oral exam, the following will be taken into account: clarity of presentation, completeness of knowledge, ability to connect different topics. The student must demonstrate that they have acquired sufficient knowledge of the main topics covered during the course and that they are able to carry out at least the simplest of the assigned exercises.

**Not approved**: the student has not acquired the basic concepts and is not able to carry out the exercises.

**18-23**: the student demonstrates minimal mastery of the basic concepts, their skills in exposition and connection of contents are modest, they are able to solve simple exercises.

**24-27**: the student demonstrates good mastery of the course contents, their presentation and content connection skills are good, they solve the exercises with few errors.

**28-30 cum laude**: the student has acquired all the contents of the course and is able to explain them fully and connect them with a critical spirit; they solve the exercises completely and without mistakes.

### Examples of frequently asked questions and / or exercises

Main differences between QR factorization and Householder triangularization. Characterization theorem of orthogonal projectors and demonstration. Example of problems where the modified Gram-Schmidt outperform the classical one. Convergence properties of the Conjugate Gradient method. Smoothing factor of weighted-Jacobi. Relation between interpolation and restriction operators in matrix form. Secular equation and graphical representation of the Divide-and-Conquer algorithm. Description of the modelling process stages. Example of arbitrage opportunity in financial mathematics.