# 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

 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

#### 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.

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.

#### 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

Charles University, Faculty of Mathematics and Physics
Ke Karlovu 3, 121 16 Praha 2, Czech Republic
VAT ID: CZ00216208