NUMERICAL METHODS FOR APPLIED SCIENCES

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.

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.ssa Patrizia Daniele.

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

Highly recommended.

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.

Contribution of teaching to the objectives of the 2030 Agenda for Sustainable Development.
The course aims to provide adequate knowledge of numerical methods for different mathematical models. The knowledge acquired will be usable in the numerous topics that make use of numerical methods for mathematical models, such as models for the exploitation of sustainable energy, infrastructure design, atmospheric models to reduce the effects of climate change, in accordance with points 7, 9, 11, 12 and 13 of the 2030 Agenda.

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.

1. Project, marked out of 30 (worth 50% of the final mark)

The student selects an exercise from a list containing at least 10 questions and carries it out independently over a period of 14 days.

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

The student is invited to discuss some aspects of the project in order to verify its authenticity and exclude possible collisions or plagiarism. Then, they are invited to present a topic chosen from those covered during the course (with the exception of topics related to the chosen project) and finally they are evaluated with two or three questions regarding the entire programme.

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.

The following criteria will normally be followed to assign the grade:

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.

Information for students with disabilities and / or SLD

Students with disabilities and/or SLD must contact the teacher, the CInAP representative of the DMI (Prof. Daniele) and CInAP well in advance of the exam date to communicate that they intend to take the exam using the appropriate compensatory measures (which will be indicated by CInAP).

PLEASE NOTE: The learning assessment can also be carried out electronically, should the conditions require it.

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.