Master in Applied and Computational Mathematics

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

Applied and Computational Mathematics at KTH

The program consists of foundation courses that are mandatory for all students, and once the individual specialization track is chosen, there are relevant mandatory courses within that area as well. The program 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 consists 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 contracts, 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 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, 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 modeling, usable in computer simulations catering to both research and prototyping.

Financial mathematics track

Financial mathematics is applied mathematics used to analyze 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 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. The master’s education in Optimisation 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 the 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 optimization 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 requires modeling 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 a transformative impact on natural sciences, business and social sciences.

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

Career

Advanced mathematics and computer simulations are present within several important fields, their use has 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 program are in high demand on the labor 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 program’s alumni continue with their doctoral studies at KTH, other Swedish universities, or other leading European and US universities.

Students

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

Dina Faraj, Sweden/Iraq: "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 international competition in the UK. It was very fun and rewarding."

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 programs. 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 program 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 program are:

• Climate Action
• Good Health and Well-Being
• Industry, Innovation, and Infrastructure

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 optimization (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)

Courses

The two-year master's program in Applied and Computational Mathematics consists of three terms of courses and one final term dedicated to the master's degree project. Each term consists of approximately 30 ECTS credits. Depending on which track you choose, you will study different courses. The courses presented on this page apply to studies starting in autumn 2020.

Year 1

Mandatory courses for all tracks

• Theory and Methodology of Science with Applications Computational Science) (AK2040) 7.5 credits
• Applied Numerical Methods (SF2520) 7.5 credits
• Probability Theory (SF2940) 7.5 credits

Conditionally elective courses for all tracks

• Applied Linear Optimization (SF2812) 7.5 credits
• Mathematical Systems Theory (SF2832) 7.5 credits
• Systems Engineering (SF2863) 7.5 credits

Recommended courses for all tracks

• Visualization (DD2257) 7.5 credits
• Methods in High-Performance Computing (DD2356) 7.5 credits
• Advanced Computation in Fluid Mechanics (DD2365) 7.5 credits
• Machine Learning (DD2421) 7.5 credits
• Machine Learning, Advanced Course (DD2434) 7.5 credits
• Mathematical Modelling of Biological Systems (DD2435) 9.0 credits
• Optimization (SF1811) 6.0 credits
• Computational Methods for Stochastic Differential Equations and Machine Learning (SF2525) 7.5 credits
• Numerical algorithms for data-intensive science (SF2526) 7.5 credits
• Program Construction in C++ for Scientific Computing (SF2565) 7.5 credits
• Computational Fluid Dynamics (SG2212) 7.5 credits
• Applied Computational Fluid Dynamics (SG2224) 5.0 credits

Computational Mathematics track

Conditionally elective courses

• Advanced Computation in Fluid Mechanics (DD2365) 7.5 credits
• Numerical Solutions of Differential Equations (SF2521) 7.5 credits
• Computational Methods for Stochastic Differential Equations and Machine Learning (SF2525) 7.5 credits
• Numerical algorithms for data-intensive science (SF2526) 7.5 credits
• Project Course in Scientific Computing (SF2567) 7.5 credits
• Parallel Computations for Large- Scale Problems (SF2568) 7.5 credits

Financial Mathematics track

Mandatory courses

• Financial Mathematics, Basic Course (SF2701) 7.5 credits

Conditionally elective courses

• Regression Analysis (SF2930) 7.5 credits
• Time Series Analysis (SF2943) 7.5 credits
• Martingales and Stochastic Integrals (SF2971) 7.5 credits

Optimization and Systems Theory track

Conditionally elective courses

• Applied Linear Optimization (SF2812) 7.5 credits
• Applied Nonlinear Optimization (SF2822) 7.5 credits
• Mathematical Systems Theory (SF2832) 7.5 credits
• Geometric Control Theory (SF2842) 7.5 credits
• Systems Engineering (SF2863) 7.5 credits
• Applied Systems Engineering (SF2866) 7.5 credits

Mathematics of Data Science track

Mandatory courses

• Computer Intensive Methods in Mathematical Statistics (SF2955) 7.5 credits

Conditionally elective courses

• Algorithms and Complexity (DD2352) 7.5 credits
• Computational Methods for Stochastic Differential Equations and Machine Learning (SF2525) 7.5 credits
• Numerical algorithms for data-intensive science (SF2526) 7.5 credits
• Parallel Computations for Large- Scale Problems (SF2568) 7.5 credits
• Regression Analysis (SF2930) 7.5 credits
• Time Series Analysis (SF2943) 7.5 credits

Year 2

Conditionally elective courses for all tracks

• Mathematical Systems Theory (SF2832) 7.5 credits
• Systems Engineering (SF2863) 7.5 credits

Recommended courses for all tracks

• Visualization (DD2257) 7.5 credits
• Machine Learning (DD2421) 7.5 credits
• Machine Learning, Advanced Course (DD2434) 7.5 credits
• Mathematical Modelling of Biological Systems (DD2435) 9.0 credits
• Optimization (SF1811) 6.0 credits
• Program Construction in C++ for Scientific Computing (SF2565) 7.5 credits

Computational Mathematics track

Mandatory courses

• Matrix Computations for Large-scale Systems (SF2524) 7.5 credits

Conditionally elective courses

• The Finite Element Method (SF2561) 7.5 credits
• Program Construction in C++ for Scientific Computing (SF2565) 7.5 credits
• Project Course in Scientific Computing (SF2567) 7.5 credits

Financial Mathematics track

Mandatory courses

• Portfolio Theory and Risk Management (SF2942) 7.5 credits

Conditionally elective courses

• Financial Derivatives (SF2975) 7.5 credits
• Risk Management (SF2980) 7.5 credits

Optimization and Systems Theory track

Conditionally elective courses

• Mathematical Systems Theory (SF2832) 7.5 credits
• Optimal Control Theory (SF2852) 7.5 credits
• Systems Engineering (SF2863) 7.5 credits
• Applied Systems Engineering (SF2866) 7.5 credits

Mathematics of Data Science track

Mandatory courses

• Modern Methods of Statistical Learning (SF2935) 7.5 credits

Conditionally elective courses

• Topological Data Analysis (SF2956) 7.5 credits
• Statistical Machine Learning (SF2957) 7.5 credits

Admission requirements

To be eligible for the program, you must have been awarded a bachelor's degree, be proficient in English and meet the program-specific requirements.

Bachelor's degree

A bachelor's degree, equivalent to a Swedish bachelor's degree, or equivalent academic qualifications from an internationally recognized university, is required. Students who are following longer technical programs, and have completed courses equivalent to a bachelor's degree, will be considered on a case-by-case basis.

English proficiency

English language proficiency equivalent to (the Swedish upper secondary school) English course B/6 is required. The requirement can be satisfied through a result equal to, or higher than, those stated in the following internationally recognized English tests:

• TOEFL Paper-based: Score of 4.5 (scale 1-6) in written test, a total score of 575.
  TOEFL ITP is not accepted.
• TOEFL iBT internet-based: Score of 20 (scale 0-30) in written test, a total score of 90
• IELTS Academic: A minimum overall mark of 6.5, with no section lower than 5.5
• Cambridge ESOL: Cambridge English: Advanced (CAE) Certificate in Advanced English or Cambridge English: Proficiency (CPE) (Certificate of Proficiency in English)
• Michigan English Language Assessment Battery (MELAB): Minimum score of 90
• The University of Michigan, ECPE (Examination for the Certificate of Proficiency in English)
• Pearson PTE Academic: Score of 62 (writing 61)

Specific requirements for the master's program in Applied and Computational Mathematics

A bachelor's degree corresponding to 180 ECTS credits, or equivalent, with at least 45 ECTS credits in Mathematics.

The students are required to have documented knowledge corresponding to basic university courses in analysis in one and several variables, linear algebra, numerical analysis, ordinary and partial differential equations, and integral transforms, mathematical statistics and basics of programming in a higher programming language.

Application documents

1. Certificates and diplomas from previous university studies
2. Transcript of completed courses and grades included in your degree
3. Proof of English proficiency
4. A copy of your passport including personal data and photograph, or other identification documents

Specific documents for the master's program in Applied and Computational Mathematics

• Letter of motivation
• Summary sheet *

*In order for your application to be considered complete, you need to fill out the online summary sheet. If you do not include a summary sheet, this will negatively affect your evaluation score. Please be sure to fill out all of the required information before you submit the form.