# 5 Degree Plans – Numerical and Computational Mathematics

Coordinated by: Department of Numerical Mathematics
Study branch coordinator: doc. Mgr. Petr Knobloch, Dr.

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. They can specialise in industrial mathematics, numerical analysis, or matrix computations.

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 algebra (matrix calculus, vector spaces).
Foundations of functional analysis (Banach and Hilbert spaces, duals, bounded operators, compact operators, basics of the theory of distributions), theory of ordinary differential equations (basic properties of the solution and maximal solutions, systems of linear equations, stability) and partial differential equations (quasilinear equations of first order, Laplace equation, heat equation and 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

 Code Subject Credits Winter Summer NMMA405 Partial Differential Equations 1 6 3/1 C+Ex — NMMA406 Partial Differential Equations 2 6 — 3/1 C+Ex NMNV401 Functional Analysis 5 2/2 C+Ex — NMNV402 Nonlinear Functional Analysis 5 — 2/2 C+Ex NMNV403 Numerical Software 1 5 2/2 C+Ex — NMNV404 Numerical Software 2 5 — 2/2 C+Ex NMNV405 Finite Element Method 1 5 2/2 C+Ex — NMNV407 Matrix Iterative Methods 1 6 4/0 Ex — NMNV501 Solution of Nonlinear Algebraic Equations 5 2/2 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 28 credits from elective courses. The selection of elective courses should take into account the planned choice of the third subject area for the final exam. The subject area for which the course is recommended is shown in parentheses (3A, 3B or 3C). The course NMNV451 Seminar in Numerical Mathematics can be taken repeatedly. We recommend enrolling in it for each semester of study.

 Code Subject Credits Winter Summer NMNV436 Finite Element Method 2 (3B) 5 — 2/2 C+Ex NMNV438 Matrix Iterative Methods 2 (3C) 5 — 2/2 C+Ex NMNV451 Seminar in Numerical Mathematics 2 0/2 C 0/2 C NMNV531 Inverse Problems and Regularization 5 2/2 C+Ex — NMNV532 Parallel Matrix Computations (3C) 5 — 2/2 C+Ex NMNV533 Sparse Matrices in Direct Methods (3C) 5 2/2 C+Ex — NMNV534 Numerical Optimization Methods 5 — 2/2 C+Ex NMNV535 Nonlinear Differential Equations (3B) 3 2/0 Ex — NMNV536 Numerical Solution of Evolutionary Equations (3A) 3 — 2/0 Ex NMNV537 Mathematical Methods in Fluid Mechanics 1 (3A) 3 2/0 Ex — NMNV538 Mathematical Methods in Fluid Mechanics 2 (3A) 3 — 2/0 Ex NMNV539 Numerical Solution of ODE (3B) 5 2/2 C+Ex — NMNV540 Fundamentals of Discontinuous Galerkin Method (3B) 3 — 2/0 Ex NMNV541 Shape and Material Optimisation 1 (3A) 3 2/0 Ex — NMNV542 Shape and Material Optimisation 2 (3A) 3 — 2/0 Ex NMNV543 Approximation Theory 4 2/1 C+Ex —

#### 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 28 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 two common subject areas ''1. Mathematical and Functional Analysis" and ''2. Numerical Methods" and a choice of one of three subject areas ''3A. Industrial Mathematics", ''3B. Numerical Analysis", or ''3C. Matrix Computations". One question is asked from each of the subject areas 1 and 2 and one question is asked from the subject area selected among 3A, 3B, or 3C.

#### 1. Mathematical and functional analysis

Partial differential equations, spectral analysis of linear operators, monotone and potential operators, solution of variational problems

#### 2. Numerical methods

Finite element method, basic matrix iterative methods, methods for the solution of systems of nonlinear algebraic equations, basics of the implementation of numerical methods

#### 3. Choice of one of the following topics:

3A. Industrial Mathematics
Mathematical methods in fluid mechanics, methods of material optimization, methods of solution of evolutionary equations

3B. Numerical Analysis
Nonlinear differential equations, numerical methods for ordinary differential equations, numerical solution of convection-diffusion problems

3C. Matrix Computations
Methods of Krylov subspaces, projections and problem of moments, connection between spectral information and convergence, direct methods for sparse matrices

#### 1st year

 Code Subject Credits Winter Summer NMMA405 Partial Differential Equations 1 6 3/1 C+Ex — NMNV407 Matrix Iterative Methods 1 6 4/0 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 — NMNV451 Seminar in Numerical Mathematics 2 0/2 C — NMMA406 Partial Differential Equations 2 6 — 3/1 C+Ex NSZZ023 Diploma Thesis I 6 — 0/4 C NMNV402 Nonlinear Functional Analysis 5 — 2/2 C+Ex NMNV404 Numerical Software 2 5 — 2/2 C+Ex NMNV451 Seminar in Numerical Mathematics 2 — 0/2 C Optional and Elective Courses 7

#### 2nd year

 Code Subject Credits Winter Summer NSZZ024 Diploma Thesis II 9 0/6 C — NMNV501 Solution of Nonlinear Algebraic Equations 5 2/2 C+Ex — 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 27