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 continued graduate studies. The programme contains five tracks: Computational Mathematics, Financial Mathematics, Mathematical Statistics, Optimization and Systems Theory, and Statistical Learning and Data Analytics.
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 required courses within that area as well. 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 at 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, advanced materials, etc. Thus, computational science and engineering is an enabling technology for scientific discovery and engineering design. It involves mathematical modeling, numerical analysis, computer science, high-performance computing and visualization. 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, your 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 modeling, 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 the last 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 modeled, and mathematical sophistication can never replace common sense and knowledge of the limitations of mathematical modeling.
Optimization and Systems Theory track
Optimization and Systems Theory is a discipline in applied mathematics primarily devoted to methods of optimization, 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. Master’s education in Optimization and Systems Theory provides knowledge and competence to handle various optimization problems, both linear and nonlinear, to build up and analyze mathematical models for various engineering systems, and to design optimal algorithms, feedback control, and filters and estimators for such systems.
Optimization 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 understanding of both optimization and system integration.
Statistical Learning and Data Analytics 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.
As a result of technological progress and the emergence of massive data sets, a variety of scientific fields and their approaches to data analysis are converging at the interface of statistics and machine learning. This new field of data analytics focuses on modeling and knowledge extraction for predictive purposes. In Statistical Learning and Data Analytics, focus is on discovering new features in the data and on confirming or falsifying existing hypotheses. Predictive data analytics applies statistical models for predictive forecasting or classification. Data analytics, when it includes at its core mathematical statistics and computational learning, has the potential for transformative impact on science, business, and social sciences.
Optimization, mathematical systems theory, systems engineering, modelling och simulation, numerical methods and applications, parallel and high-performance computations, big data, machine learning, arbitrage pricing, portfolio theory and risk management.
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 where the programme’s alumni continue with doctoral studies at KTH, other Swedish universities, or other leading European and US universities.
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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 centra 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 performed 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 at 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 cooperations, 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), Monte Carlo methods (Henrik Hult, Jimmy Olsson), and computational statistics (Timo Koski, Jimmy Olsson, Tatjana Pavlenko).
Changes in the programme may occur.