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NUMERICAL METHODS FOR APPLIED SCIENCES

Academic Year 2025/2026 - Teacher: Giovanni RUSSO

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

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°


First part: 3 CFU - 21 hours (prof. A. Coco)

Second part: 3 CFU - 21 hours (prof. G. Russo)

Total: 42 hours

 

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.

Required Prerequisites

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

Attendance of Lessons

Highly recommended.


Detailed Course Content

The course includes the analysis of numerical algorithms for accurately solving or approximating linear algebra problems, such as linear systems and eigenvalue problems.

It also aims to provide solid implementation skills by developing small software programs for the various numerical algorithms studied, with applications to real-world problems.


Contents of the second part


Introduction to iterative methods for solving linear systems.

Jacobi, Gauss-Seidel, and SOR methods, both pointwise and block versions. Convergence conditions, stopping criteria, and computational complexity.


Mathematical models.

Detailed description of the mathematical modeling process. Real-world applications to problems solvable with multigrid methods: electromagnetism, gravity, elastic deformation, diffusion processes. Finite difference discretization of the heat equation and Poisson's equation, and corresponding solution methods in one spatial dimension (Thomas algorithm).


Multigrid methods.

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


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.

Learning Assessment

Learning Assessment Procedures

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.


Examples of frequently asked questions and / or exercises

Examples of questions 

Convergence properties of Jacobi method 

Smoothing factor of weighted-Jacobi. 

Relation between interpolation and restriction operators in matrix form. 

Finite difference approximation of the diffusion equation

Thomas algorithm


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