PENG9570 Applied Mathematical Modelling and Analysis Course description

Students taking the course must have a thorough knowledge of advanced calculus, including ordinary and partial differential equations. The student should also be familiar with linear algebra and Fourier and Laplace transform theory. In terms of programming, the candidate should have some experience in implementing numerical methods, including schemes for solving partial differential equations.

The candidate should also have a certain knowledge of mathematical analysis, modern physics or physiology – depending on specialization.

The course will be offered once a year, provided 3 or more students sign up for the course. If less than 3 students sign up for a course, the course will be cancelled for that year.

Both the presentation of the case in Module 3 of the course and the tool summary document in the practical training part the course will form basis of assessment.

Both exams must be passed in order to pass the course.

The oral presentation cannot be appealed.

None.

Students who complete the course are expected to have the following learning outcomes, defined in terms of knowledge, skills and general competence:

Knowledge

On successful completion of the course, the student:

• knows how mathematical models can be derived from facts and first principles.
• has a repertoire of methods to solve and/or analyse both ordinary differential equation (ODE) systems and certain partial differential equations (PDEs).
• is able to apply analytical and/or numerical solution methods for PDEs to models of heat transfer, wave propagation and diffusion-convection and discuss the relevance of these models to real-world phenomena.
• is able to construct and develop relevant models and discuss the validity of the models.

Skills

On successful completion of the course, the student can:

• can determine steady states of ODE systems and use linear approximation to elucidate the stability properties of these states.
• can solve and/or analyse selected PDE models.
• is able to implement and use some numerical methods for solving relevant PDEs.
• can devise the solution of certain composite quantitative problems.
• can disseminate results and findings in an accessible manner – both orally and in writing.

General competence

• is aware of the usefulness and limitations of mathematical modelling as well as of pitfalls frequently encountered in modelling and simulation.
• is able to discuss properties of a system using the equations of the mathematical model that describes the system.
• can explain and use numerical methods, know their strengths and weaknesses and interpret results of numerical simulations.

All aids are permitted.

The teaching is organised as sessions where the subject material is presented, and as sessions where the students solve problems using analytical and/or numerical methods. Between these sessions, the students should work individually with literature studies and problem solving.

In the last, specialised part, the students are required to complete and present a rather extensive individual project involving theoretical and practical/implementational aspects.

The following required coursework must be approved before the student can take the exam:

• Completion of an extensive individual project in the specialised module.

An individual, oral examination. The examination will address both general topics from within the course and the specific project developed by the student.

The oral examination cannot be appealed.

Knowledge

On successful completion of the course, the student:

• has an overview of the different elements that comprise the architecture of today’s internet.
• has a good understanding about the approaches for conducting internet measurements and the latest advances in this field.
• be familiar of a broad set of tools that can help analyzing Internet measurments. Of a particular relevance here are tools that originate in other disciplines like Machine Learning and Statisitcal Physics. This will not only expand the available toolset but also increases the potential for interdisciplinory collaboration going forward.

Skills

On successful completion of the course, the student can:

• plan and carry out state-of-the-art measurement tasks
• can formulate research questions on the robustness and performance of operational networks, and design measurements for evaluating these questions.
• will have a general practical understanding of how different parts of the internet's architecture interplay to offer a performant end-to-end service.

General competence

On successful completion of the course, the student can:

• participate in debates and present aspects of his/her expertise in a way that promotes such discussions.
• drive innovation

Module 1 will take the form of lectures. Module 2 will take the form of lab and homework assignments. Module 3 will take the form of seminars. In module 3, the student will present a case to the other students. We will also invite guest lecturers from research groups that focuses on machine learning and network science to introduce the students to potential tools and analysis methods.

Practical training

The students will participate in lab experiments to explore how once can measure various aspects of internet's robustness and performance. The students will write a summary of one of the tools that were introduced in the lab and discuss its benefits and limitations.

None.