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).

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

NMMA405Partial Differential Equations 1 63/1 C+Ex
NMNV401Functional Analysis 52/2 C+Ex
NMNV403Numerical Software 1 52/2 C+Ex
NMNV405Finite Element Method 1 52/2 C+Ex
NMNV406Nonlinear differential equations 52/2 C+Ex
NMNV411Algorithms for matrix iterative methods 52/2 C+Ex
NMNV412Analysis of matrix iterative methods — principles and interconnections 64/0 Ex
NMNV503Numerical Optimization Methods 1 63/1 C+Ex
NSZZ023Diploma Thesis I 60/4 C
NSZZ024Diploma Thesis II 90/6 C
NSZZ025Diploma Thesis III 150/10 C

5.2 Elective Courses

It is required to earn at least 30 credits from elective courses.

NMMA406Partial Differential Equations 2 63/1 C+Ex
NMNV404Numerical Software 2 52/2 C+Ex
NMNV436Finite Element Method 2 52/2 C+Ex
NMNV461Techniques for a posteriori error estimation 32/0 Ex
NMNV464A Posteriori Numerical Analysis Based on the Method of Equilibrated Fluxes 32/0 Ex
NMNV531Inverse Problems and Regularization 52/2 C+Ex
NMNV532Parallel Matrix Computations 52/2 C+Ex
NMNV533Sparse Matrices in Numerical Mathematics 52/2 C+Ex
NMNV537Mathematical Methods in Fluid Mechanics 1 32/0 Ex
NMNV538Mathematical Methods in Fluid Mechanics 2 32/0 Ex
NMNV539Numerical Solution of ODE 52/2 C+Ex
NMNV540Fundamentals of Discontinuous Galerkin Method 32/0 Ex
NMNV543Approximation of functions 1 52/2 C+Ex
NMNV544Numerical Optimization Methods 2 52/2 C+Ex

5.3 Recommended Optional Courses

NMMO401Continuum Mechanics 62/2 C+Ex
NMMO403Computer Solutions of Continuum Physics Problems 52/2 C+Ex
NMMO461Seminar in Continuum Mechanics 20/2 C0/2 C
NMMO535Mathematical Methods in Mechanics of Solids 32/0 Ex
NMMO536Mathematical Methods in Mechanics of Compressible Fluids 32/0 Ex
NMMO537Saddle Point Problems and Their Solution 52/2 C+Ex
NMMO539Mathematical Methods in Mechanics of Non-Newtonian Fluids 32/0 Ex
NMNV361Fractals and Chaotic Dynamics 32/0 Ex
NMNV451Seminar in Numerical Mathematics 20/2 C0/2 C
NMNV466Domain Decomposition Methods 32/0 Ex
NMNV462Numerical Modelling of Electrical Engineering Problems 32/0 Ex
NMNV468Numerical Linear Algebra for data science and informatics 52/2 C+Ex
NMNV541Shape and Material Optimisation 1 32/0 Ex
NMNV542Shape and Material Optimisation 2 32/0 Ex
NMNV561Bifurcation Analysis of Dynamical Systems 1 32/0 Ex
NMNV562Bifurcation Analysis of Dynamical Systems 2 32/0 Ex
NMNV565High-Performance Computing for Computational Science 52/2 C+Ex
NMNV568Approximation of functions 2 32/0 Ex
NMNV569Numerical Computations with Verification 52/2 C+Ex
NMNV571Multilevel Methods 32/0 Ex
NMNV623Contemporary Problems in Numerical Mathematics 30/3 C0/3 C
NMST442Matrix Computations in Statistics 52/2 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.

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.

4. Adaptive discretization methods
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

NMMA405Partial Differential Equations 1 63/1 C+Ex
NMNV401Functional Analysis 52/2 C+Ex
NMNV403Numerical Software 1 52/2 C+Ex
NMNV405Finite Element Method 1 52/2 C+Ex
NMNV411Algorithms for matrix iterative methods 52/2 C+Ex
NMNV451Seminar in Numerical Mathematics 20/2 C
NMNV406Nonlinear differential equations 52/2 C+Ex
NMNV412Analysis of matrix iterative methods — principles and interconnections 64/0 Ex
NSZZ023Diploma Thesis I 60/4 C
NMNV451Seminar in Numerical Mathematics 20/2 C
 Optional and Elective Courses 13  

2nd year

NMNV503Numerical Optimization Methods 1 63/1 C+Ex
NSZZ024Diploma Thesis II 90/6 C
NMNV451Seminar in Numerical Mathematics 20/2 C
NSZZ025Diploma Thesis III 150/10 C
NMNV451Seminar in Numerical Mathematics 20/2 C
 Optional and Elective Courses 26