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.

Pass or fail.

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.

Introductory module:

• Principles of modelling and derivation of mathematical models
• Analysis of ordinary differential equations (ODEs)
• Linear partial differential equations (PDEs)
• Prominent results from functional analysis and their application to ODEs and PDEs
• Numerical methods for computing of solutions of PDEs

Functional analysis:

• Completeness for normed spaces
• Hilbert spaces, compact and diagonalisable operators
• Theory of topological vector spaces
• Test functions, distributions and the Fourier transform
• Sobolev spaces and fundamental solutions of partial differential equations

Biosystems:

• Mathematical models for biological systems
• Analytical and numerical methods for simulation of system response
• Actuators and sensors for stimulation and measurements of biological systems
• Interaction of biological and measurement system

Modern physics:

• Monte Carlo techniques
• Splines and other expansion techniques
• Applications of expansions in spherical harmonics
• Numerical problems in general relativity and quantum physics
• Manifolds with geometric structures central to physics and engineering.

Within all specializations, the content may be adjusted to accommodate for the research area of each PhD candidate.

Cloud computing is an emerging paradigm of utilising large-scale computing services over the internet that will affect the computing needs of individuals and organisations. Over the past decade, many cloud computing platforms have been set up by companies such as Google, Yahoo!, Amazon, Microsoft, Salesforce, eBay and Facebook. Some of the platforms are open to the public via various pricing models. They operate at different levels and enable businesses to harness different amounts of computing power from the cloud.

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.

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:

• has a deep understanding of how cloud computing, large services and infrastructures play a crucial role in todays digitised society;
• has an interdisciplinary view on cloud computing due to its central role in the digitised society;
• understands the fundamental principles of distributed computing and particularly cloud computing;
• understands the importance of virtualisation in distributed computing and how this has enabled the development of cloud computing;
• understands the business models that underlie cloud computing;
• has an understanding of the architecture and concept of different cloud models: IaaS, PaaS, and SaaS;
• is knowledgeable in the various methods available to monitor and evaluate cloud infrastructure;
• has a deep knowledge of the common security issues in the field of cloud computing;
• has an understanding of the concept of threat intelligence in the field of cloud computing;
• understands the use of security policies as part of the overall security strategy of an organization;

Skills

On successful completion of the course, the student can:

• design highly distributed digital systems.
• create virtual machine images and deploy them on a cloud.
• design and develop scalable cloud-based applications by creating and configuring virtual machines in the cloud.
• analyse cloud infrastructures with regard to properties such as resilience, security, performance and manageability.
• identify cloud security weaknesses by recognising and discovering threats and vulnerabilities to cloud computing.
• implement cloud features to secure and harden the infrastructure.
• use tools to monitor and evaluate cloud infrastructure.
• use tools to analyse system logs to detect possible security or performance problems.

General competence

On successful completion of the course, the student:

• can discuss his/her area of expertise with a non-expert audience by combining insights across disciplines.
• can discuss and debate the impact of technological development on our society in the future

The course is part of a series of seminars in which the students actively participate together with members of relevant research groups. The students present papers and listen to paper presentations from other PhD students and staff. The students are also expected to actively critique and challenge fellow participants. The students are provided with a sound foundation in research skills and are naturally integrated into the local research community and its research discourse.

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

• Three individual oral presentations of assigned publications.
• Participate as prepared discussant for three presentations by other group members.
• Independently find and study publications relevant to the research discourse.
• 80% attendance at seminars.

Oral exam, 30 minutes per student.

The oral exam cannot be appealed.

None.