# Master's programme in Applied and Computational Mathematics

Students from the master’s programme in Applied and Computational Mathematics will become skilled applied mathematicians, well-prepared for advanced industrial positions or continuing graduate studies. The programme offers four tracks: Computational Mathematics, Financial Mathematics, Optimisation and Systems Theory, and Mathematics of Data Science.

## Application dates for studies starting in Autumn 2020

**16 October 2019: **Application opens

**15 January 2020: **Application deadline

**3 February 2020:** Deadline for supporting academic documents (all applicants) and documentation of fee exempt status (if required) or receipt of application fee (if required)

**3 April 2020:** Notification of selection results

**August 2020: **Arrival and start of studies

## Applied and Computational Mathematics at KTH

The programme consists of foundation courses that are mandatory for all students, and once the individual specialisation track is chosen, there are relevant mandatory courses within that area as well. The programme offers four tracks: Computational Mathematics, Financial Mathematics, Optimisation and Systems Theory, and Mathematics of Data Science.

Regardless of which track students attend, the final term consist of a degree project that may be carried out in an academic or industrial environment in Sweden or abroad. Students are welcome to discuss project ideas with the staff of the Department of Mathematics, but are also encouraged to seek other contacts, in the academic world and in industry, to identify suitable projects. The result of the degree project is provided as a written report and as a presentation at a seminar.

### Computational Mathematics track

The field of computer simulations is of great importance for high-tech industry and scientific/engineering research, for example virtual processing, climate studies, fluid dynamics and advanced materials. Thus, computational science and engineering is an enabling technology for scientific discovery and engineering design. It involves mathematical modelling, numerical analysis, computer science, high-performance computing and visualisation. The remarkable development of large scale computing in the last decades has turned computational science and engineering into the "third pillar" of science, complementing theory and experiment.

The Computational Mathematics track is mainly concerned with the mathematical foundations of computational science and engineering. However, in this track we will also discuss issues of high-performance computing. Given the interdisciplinarity, the final curriculum may vary greatly depending on your interests. The Computational Mathematics track contains courses providing knowledge of design, analysis and application of numerical methods for mathematical modelling, usable in computer simulations catering to both research and prototyping.

### Financial mathematics track

Financial mathematics is applied mathematics used to analyse and solve problems related to financial markets. Any informed market participant would exploit an opportunity to make a profit without any risk of loss. This fact is the basis of the theory of arbitrage-free pricing of derivative instruments. Arbitrage opportunities exist but are rare. Typically both potential losses and gains need to be considered. Hedging and diversification aim at reducing risk. Speculative actions on financial markets aim at making profits. Market participants have different views of the future market prices and combine their views with current market prices to take actions that aim at managing risk while creating opportunities for profits. Portfolio theory and quantitative risk management present theory and methods that form the theoretical basis of market participants’ decision making.

Financial mathematics has received lots of attention from academics and practitioners over recent decades and the level of mathematical sophistication has risen substantially. However, a mathematical model is at best a simplification of the real world phenomenon that is being modelled, and mathematical sophistication can never replace common sense and knowledge of the limitations of mathematical modelling.

### Optimisation and Systems Theory track

Optimisation and Systems Theory is a discipline in applied mathematics primarily devoted to methods of optimisation, including mathematical programming and optimal control, and systems theoretic aspects of control and signal processing. The discipline is also closely related to mathematical economics and applied problems in operations research, systems engineering and control engineering. The master’s education in Optimisation and Systems Theory provides knowledge and competence to handle various optimisation problems, both linear and nonlinear, to build up and analyse mathematical models for various engineering systems, and to design optimal algorithms, feedback control, and filters and estimators for such systems.

Optimisation and Systems Theory has wide applications in both industry and research. Examples of applications include aerospace industry, engineering industry, radiation therapy, robotics, telecommunications, and vehicles. Furthermore, many new areas in biology, medicine, energy and environment, and information and communications technology require an understanding of both optimisation and system integration.

### Mathematics of Data Science track

Statistics is the science of learning from data. Classical statistics is trying to understand data by determining a plausible model for data, and testing whether the data fits the model. Modern learning is concerned with computational statistics and automated methods for extracting information from data. The technological progress and the increased availability of information contributes to the emergence of massive and complex data sets. A variety of scientific fields are contributing to the analysis of such data at the interface of mathematics, statistics, optimization and computational methods for learning. Optimal decision making under uncertainty based in such circumstances require modelling and discovering relevant features in data, optimization of decision policies and model parameters, dimension reduction and large scale computations. Data science based on applied mathematics has the potential for transformative impact on natural sciences, business and social sciences.

This is a two year programme (120 ECTS credits) given in English. Graduates are awarded the degree of Master of Science. The programme is given mainly at KTH Campus in Stockholm by the School of Engineering Sciences (at KTH).

## Topics covered

Optimisation, mathematical systems theory, systems engineering, modelling and simulation, numerical methods and applications, parallel and high-performance computations, big data, machine learning, arbitrage pricing, portfolio theory and risk management.

## Career

Advanced mathematics and computer simulations are present within several important fields, their use having increased dramatically by the rapid development in computer software and hardware. Financial mathematics, medicine and biology are prevalent areas, but students will be able to bring the usage of mathematics and simulations into a multitude of applications.

The graduates of this programme are in high demand on the labour market as well as in academia. Alumni work in large and smaller companies like Ericsson, ABB, Comsol, SAAB, RaySearch Labs, Modelon, If, Citibank, Brainlab, ÅF, Atlas Copco, Elekta, Process Systems Enterprise, Goldman Sachs, and many others. Another alternative is an academic carrier in which the programme’s alumni continue with their doctoral studies at KTH, other Swedish universities, or other leading European and US universities.

## After graduation

Technology manager, deep learning software engineer, team lead, professor, PhD student, credit risk analyst, CEO, head of marketing and sales, technical director, development engineer.

## Students

Find out what students from the programme think about their time at KTH.

"During my first year at KTH, I joined a student association called KTH Formula. We designed and built an electric car that we took to an international competition in the UK. It was very fun and rewarding."

**Dina Faraj, Sweden/Iraq**

## Sustainable development

Graduates from KTH have the knowledge and tools for moving society in a more sustainable direction, as sustainable development is an integral part of all programmes. The particular strength of mathematics is its high degree of abstraction, with one and the same mathematical model used to describe very different features in many different areas of application. This versatility leads to the effect that once you are able to quantify phenomena, you will be able to investigate these phenomena independently of their source, for example in science, engineering, society and the economy. Many of the UN goals of sustainable development are accordingly linked to Applied Mathematics, to name just a few: Good health and well-being, Affordable and clean energy, Decent work and economic growth, Industry, innovation and infrastructure, Sustainable cities and communities, Climate action, Life below water, Reduced inequality and others. The master’s programme in Applied and Computational Mathematics provides the student with the knowledge and tools applicable for their successful treatment. You will see examples of how to do this in different courses. It is not uncommon for the final master’s degree project to be devoted to questions related to sustainable development and its various goals. The examples of sustainable development goals addressed by the programme are:

Examples of master’s degree projects relating to Climate Action are: Efficient computational methods for climate models (collaboration with SMHI); Consequences of climate change for the electric power supply (in collaboration with SWECO); Polynomial chaos expansion for climate economy assessment (in collaboration with Karlsruhe Institute of Technology)

Examples of master’s degree projects relating to Good Health and Well-Being are: Optimal construction of medical equipment for cancer treatment (in collaboration with RaySearch Labs); Simulation of suturing for surgeon training (in collaboration with SenseGraphics); Proton arc therapy optimisation (in collaboration with RaySearch Labs);

Examples of master’s degree projects relating to Industry, Innovation and Infrastructure are: Optimal traffic planning for autonomous vehicles (in collaboration with Volvo Construction Equipment); Optimal energy management for parallel hybrid electric vehicles (in collaboration with Scania); Optimal driving decision based on energy and time costs (in collaboration with Volvo).

### Webinar on sustainable development

Watch a recording of the webinar offered by the School of Engineering Sciences in December 2019. The webinar focused on the schools' master’s programmes and how they relate to sustainable development.

## Faculty and research

The programme is run by the Department of Mathematics. The Department of Mathematics at KTH hosts some of the strongest Swedish research groups in mathematics. It comprises four units: Mathematics, Mathematical Statistics, Optimization and Systems Theory, and Numerical Analysis. Jointly, these units perform research in a broad spectrum of mathematical disciplines, ranging from pure to applied mathematics. Some of the current larger research centres hosted at the department are:

- Random matrices, sponsored by the Wallenberg foundation
- Image processing, sponsored by SSF
- PDE, sponsored by the ERC/VR/Gustafsson's foundation
- MathDataLab, sponsored by Brummer & Partners

Research carried out at the Division of Optimization and Systems Theory includes various topics in mathematical systems theory (Xiaoming Hu, Per Enqvist and Johan Karlsson), with particular emphasis on stochastic systems, filtering, identification and robust and nonlinear control; mathematical programming (Anders Forsgren and Per Enqvist), with large-scale nonlinear programming, structural optimization; and a wide range of applications. Examples of applications include radiation therapy (Forsgren), robotics (Hu), and telecommunications (Karlsson).

The research in the Division of Numerical Analysis includes numerical methods for stochastic and deterministic differential equations (Anders Szepessy (member of the Royal Swedish Academy of Sciences), Mattias Sandberg), computational modeling in systems biology (Michael Hanke), numerical methods for micro and complex flow (Anna-Karin Tornberg (member of the Royal Swedish Academy of Sciences and the Royal Swedish Academy of Engineering Sciences), Katarina Gustavsson), multiscale methods (Olof Runborg, Patrick Henning), finite element methods for multiphase flow (Sara Zahedi). The researchers are working actively in many interdisciplinary cooperative ventures, e.g., the Swedish e-Science Research Centre (SeRC), the Linné FLOW Centre, and with Karolinska Institutet. Students will also have access to Sweden’s fastest supercomputers via the PDC Centre for High-Performance Computing.

The Division of Mathematical Statistics hosts active groups in probability theory, stochastic analysis and statistical inference with applications to extreme value theory (Boualem Djehiche, Henrik Hult, Pierre Nyquist), finance and risk management (Boualem Djehiche, Henrik Hult), stochastic optimal control (Boualem Djehiche, Thomas Önskog), statistical learning (Henrik Hult, Timo Koski, Tatjana Pavlenko), Monte Carlo methods (Henrik Hult, Jimmy Olsson), computational statistics (Timo Koski, Jimmy Olsson, Tatjana Pavlenko), and statistical methodologies for high-dimensional models with applications (Tatjana Pavlenko).

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