Computational Mathematics
Coordinated by: Department of Numerical Mathematics
Study branch coordinator: doc. Mgr. Petr Knobloch, Dr., DSc.
This programme focuses on design, analysis, algorithmization, and implementation of methods for computer processing of mathematical models. It represents a transition from theoretical mathematics to practically useful results. An emphasis is placed on the creative use of information technology and production of programming applications. An integral part of the programme is the verification of employed methods. The students will study modern methods for solving partial differential equations, the finite element method, linear and non-linear functional analysis, and methods for matrix calculation. They will choose the elective courses according to the topic of their master's thesis.
The graduate will have attained the knowledge needed for numerical solution of practical problems from discretization through numerical analysis up to implementation and verification. He/she will be able to choose an appropriate numerical method for a given problem, conduct its numerical analysis, and implement its computation including analysis of numerical error. The graduate will be able to critically examine, assess, and tune the whole process of the numerical solution, and can assess the agreement between the numerical results and reality. He/she will be able to carry out an analytical approach to the solution of a general problem based on thorough and rigorous reasoning. The graduate will be qualified for doctoral studies and for employment in industry, basic or applied research, or government institutions.
Assumed knowledge
It is assumed that an incoming student of this branch has sufficient knowledge of the following topics and fields:
- –Differential calculus for functions of one and several real variables.
Integral calculus for functions of one variable. Measure theory, Lebesgue
measure and Lebesgue integral. Basics of linear algebra (matrix calculus, vector
spaces).
- –Foundations of functional analysis (Banach and Hilbert spaces, duals, bounded operators, compact operators), theory of ordinary differential equations (basic properties of the solutions and maximal solutions, systems of linear equations, stability) and theory of partial differential equations (quasilinear equations of first order, Laplace equation, heat equation, wave equation).
- –Foundations of numerical mathematics (numerical quadrature, basics of the numerical solution of ordinary differential equations, finite difference method for partial differential equations) and of analysis of matrix computations (Schur theorem, orthogonal transformations, matrix decompositions, basic iterative methods).
- –Foundations of functional analysis (Banach and Hilbert spaces, duals, bounded operators, compact operators), theory of ordinary differential equations (basic properties of the solutions and maximal solutions, systems of linear equations, stability) and theory of partial differential equations (quasilinear equations of first order, Laplace equation, heat equation, wave equation).
Should an incoming student not meet these entry requirements, the coordinator of the study programme may assign a method of acquiring the necessary knowledge and abilities, which may for example mean taking selected bachelor's courses, taking a reading course with an instructor, or following tutored independent study.
5.1 Obligatory Courses
| Code | Subject | Credits | Winter | Summer | |
| NMMA405 | Partial Differential Equations 1 | 6 | 3/1 C+Ex | — | |
| NMNV401 | Functional Analysis | 5 | 2/2 C+Ex | — | |
| NMNV403 | Numerical Software 1 | 5 | 2/2 C+Ex | — | |
| NMNV405 | Finite Element Method 1 | 5 | 2/2 C+Ex | — | |
| NMNV406 | Nonlinear differential equations | 5 | — | 2/2 C+Ex | |
| NMNV411 | Algorithms for matrix iterative methods | 5 | 2/2 C+Ex | — | |
| NMNV412 | Analysis of matrix iterative methods — principles and interconnections | 6 | — | 4/0 Ex | |
| NMNV503 | Numerical Optimization Methods 1 | 6 | 3/1 C+Ex | — | |
| NSZZ023 | Diploma Thesis I | 6 | — | 0/4 C | |
| NSZZ024 | Diploma Thesis II | 9 | 0/6 C | — | |
| NSZZ025 | Diploma Thesis III | 15 | — | 0/10 C | |
5.2 Elective Courses
It is required to earn at least 30 credits from elective courses.
| Code | Subject | Credits | Winter | Summer | |
| NMMA406 | Partial Differential Equations 2 | 6 | — | 3/1 C+Ex | |
| NMNV404 | Numerical Software 2 | 5 | — | 2/2 C+Ex | |
| NMNV436 | Finite Element Method 2 | 5 | — | 2/2 C+Ex | |
| NMNV461 | Techniques for a posteriori error estimation | 3 | 2/0 Ex | — | |
| NMNV464 | A Posteriori Numerical Analysis Based on the Method of Equilibrated Fluxes | 3 | — | 2/0 Ex | |
| NMNV531 | Inverse Problems and Regularization | 5 | 2/2 C+Ex | — | |
| NMNV532 | Parallel Matrix Computations | 5 | — | 2/2 C+Ex | |
| NMNV533 | Sparse Matrices in Numerical Mathematics | 5 | 2/2 C+Ex | — | |
| NMNV537 | Mathematical Methods in Fluid Mechanics 1 | 3 | 2/0 Ex | — | |
| NMNV538 | Mathematical Methods in Fluid Mechanics 2 | 3 | — | 2/0 Ex | |
| NMNV539 | Numerical Solution of ODE | 5 | 2/2 C+Ex | — | |
| NMNV540 | Fundamentals of Discontinuous Galerkin Method | 3 | — | 2/0 Ex | |
| NMNV543 | Approximation of functions 1 | 5 | 2/2 C+Ex | — | |
| NMNV544 | Numerical Optimization Methods 2 | 5 | — | 2/2 C+Ex | |
5.3 Recommended Optional Courses
| Code | Subject | Credits | Winter | Summer | |
| NMMO401 | Continuum Mechanics | 6 | 2/2 C+Ex | — | |
| NMMO403 | Computer Solutions of Continuum Physics Problems | 5 | — | 2/2 C+Ex | |
| NMMO461 | Seminar in Continuum Mechanics | 2 | 0/2 C | 0/2 C | |
| NMMO535 | Mathematical Methods in Mechanics of Solids | 3 | 2/0 Ex | — | |
| NMMO536 | Mathematical Methods in Mechanics of Compressible Fluids | 3 | — | 2/0 Ex | |
| NMMO537 | Saddle Point Problems and Their Solution | 5 | — | 2/2 C+Ex | |
| NMMO539 | Mathematical Methods in Mechanics of Non-Newtonian Fluids | 3 | 2/0 Ex | — | |
| NMNV361 | Fractals and Chaotic Dynamics | 3 | 2/0 Ex | — | |
| NMNV451 | Seminar in Numerical Mathematics | 2 | 0/2 C | 0/2 C | |
| NMNV466 | Domain Decomposition Methods | 3 | — | 2/0 Ex | |
| NMNV462 | Numerical Modelling of Electrical Engineering Problems | 3 | — | 2/0 Ex | |
| NMNV468 | Numerical Linear Algebra for data science and informatics | 5 | — | 2/2 C+Ex | |
| NMNV541 | Shape and Material Optimisation 1 | 3 | 2/0 Ex | — | |
| NMNV542 | Shape and Material Optimisation 2 | 3 | — | 2/0 Ex | |
| NMNV561 | Bifurcation Analysis of Dynamical Systems 1 | 3 | 2/0 Ex | — | |
| NMNV562 | Bifurcation Analysis of Dynamical Systems 2 | 3 | — | 2/0 Ex | |
| NMNV565 | High-Performance Computing for Computational Science | 5 | 2/2 C+Ex | — | |
| NMNV568 | Approximation of functions 2 | 3 | — | 2/0 Ex | |
| NMNV569 | Numerical Computations with Verification | 5 | — | 2/2 C+Ex | |
| NMNV571 | Multilevel Methods | 3 | 2/0 Ex | — | |
| NMNV623 | Contemporary Problems in Numerical Mathematics | 3 | 0/3 C | 0/3 C | |
| NMST442 | Matrix Computations in Statistics | 5 | — | 2/2 C+Ex | |
5.4 State Final Exam
Requirements for taking the final exam
- – Earning at least 120 credits during the course of the study.
- – Completion of all obligatory courses prescribed by the study plan.
- – Earning at least 30 credits by completion of elective courses.
- – Submission of a completed master's thesis by the submission deadline.
- – Completion of all obligatory courses prescribed by the study plan.
Oral part of the state final exam
The oral part of the final exam consists of three questions from topics described below. The contents of these topics are covered by obligatory courses.
Requirements for the oral part of the final exam
1. Partial differential equations
Linear elliptic, parabolic and hyperbolic equations, nonlinear differential equations in divergence form, Sobolev spaces, variational formulation, existence and properties of solutions, monotone and potential operators.
2. Finite element method
Finite element spaces and their approximative properties, Galerkin approximation of linear elliptic problems, error estimates, solution of nonlinear differential equations in divergence form.
3. Numerical linear algebra
Basic direct and iterative matrix methods, Krylov methods, projections and problem of moments, connection between spectral information and convergence.
Numerical quadrature, error estimates, adaptivity. Numerical methods for ordinary differential equations, estimates of local error, adaptive choice of time step.
5. Numerical optimization methods
Methods for solution of nonlinear algebraic equations and their systems, methods for minimization of functionals without constraints, local and global convergence.
5.5 Recommended Course of Study
1st year
| Code | Subject | Credits | Winter | Summer | |
| NMMA405 | Partial Differential Equations 1 | 6 | 3/1 C+Ex | — | |
| NMNV401 | Functional Analysis | 5 | 2/2 C+Ex | — | |
| NMNV403 | Numerical Software 1 | 5 | 2/2 C+Ex | — | |
| NMNV405 | Finite Element Method 1 | 5 | 2/2 C+Ex | — | |
| NMNV411 | Algorithms for matrix iterative methods | 5 | 2/2 C+Ex | — | |
| NMNV451 | Seminar in Numerical Mathematics | 2 | 0/2 C | — | |
| NMNV406 | Nonlinear differential equations | 5 | — | 2/2 C+Ex | |
| NMNV412 | Analysis of matrix iterative methods — principles and interconnections | 6 | — | 4/0 Ex | |
| NSZZ023 | Diploma Thesis I | 6 | — | 0/4 C | |
| NMNV451 | Seminar in Numerical Mathematics | 2 | — | 0/2 C | |
| Optional and Elective Courses | 13 | ||||
2nd year
| Code | Subject | Credits | Winter | Summer | |
| NMNV503 | Numerical Optimization Methods 1 | 6 | 3/1 C+Ex | — | |
| NSZZ024 | Diploma Thesis II | 9 | 0/6 C | — | |
| NMNV451 | Seminar in Numerical Mathematics | 2 | 0/2 C | — | |
| NSZZ025 | Diploma Thesis III | 15 | — | 0/10 C | |
| NMNV451 | Seminar in Numerical Mathematics | 2 | — | 0/2 C | |
| Optional and Elective Courses | 26 | ||||
