
Basic Properties of Multigrid Methods
Minarovičová, Anna Marie ; Papež, Jan (advisor) ; Pultarová, Ivana (referee)
Multigrid methods are among the most effective iterative methods for the numerical solution of partial differential equations (PDEs). In the thesis, we consider Poisson's equation as the model problem and present its discretization by the finite difference method. Discretization of PDEs gives typically large algebraic systems of linear equations. Various iterative methods can struggle to find an enough accurate approximation within the allocated time. In particular, relaxation methods such as Jacobi or GaussSeidel effectively reduce oscillating parts of the error but are inefficient in reducing smooth error components. Multigrid methods combine relaxation methods with correction on a coarser grid to overcome this deficiency. The problem discretized on a coarser grid is smaller and easier to solve. Typically, a recursive error correction is considered using a hierarchy of grids until the coarsest problem is small enough to get a solution quickly by a direct solver. The purpose of this thesis is to discuss the main principles and thoughts behind the multigrid methods, alongside some practical examples and numerical experiments.


Programs and Algorithms of Numerical Mathematics 21 : Jablonec nad Nisou, June 1924, 2022 : Proceedings of Seminar
Chleboun, J. ; Kůs, Pavel ; Papež, Jan ; Rozložník, Miroslav ; Segeth, Karel ; Šístek, Jakub
These proceedings contain peerreviewed papers that are based on the invited lectures, short communications, and posters presented at the 21st seminar Programs and Algorithms of Numerical Mathematics (PANM) held in Merkur Hotel, Jablonec nad Nisou, Czech Republic, June 1924, 2022.\nThe seminar was organized by the Institute of Mathematics of the Czech Academy of Sciences under the auspices of EUMATHSIN.CZ, Czech Network for Mathematics in Industry, and with the nancial support provided by the RSJ Foundation. It continued the previous seminars on mathematical software and numerical methods held (biennially, with only one exception) in Alšovice, Bratříkov, Janov nad Nisou, Kořenov, Lázně Libverda, Dolní Maxov, Prague, and Hejnice in the period 19832020. The objective of this series of seminars is to provide a forum for presenting and discussing advanced topics in numerical analysis, computer implementation of numerical algorithms, new approaches to mathematical modeling, and single or multiprocessor applications of computational methods.


Computation of roots of polynomials using comrade matrices
Novák, Martin ; Tichý, Petr (advisor) ; Papež, Jan (referee)
The bachelor thesis describes the relationship between the roots of the polynomial and the eigenvalues of the companion matrix, which is formed from the coefficients of the given polynomial. For numerical computing, it can be better to express the polynomial in basis of some orthogonal polynomials. After that, the coefficients can be used to form the comrade matrix. A similar relationship between roots of the polynomial and eigenvalues of the comrade matrix holds. We show that comrade matrices are nonderogatory matrices. The thesis contains numerical experiments programmed in MATLAB. 1


Deflated Conjugate Gradient Method
Piskalla, Adam ; Papež, Jan (advisor) ; Tichý, Petr (referee)
Conjugate gradient method is one of the basic iterative methods for solving systems of linear algebraic equations with a symmetric positive definite matrix. We present two different derivations of the method and show some its properties. In situations where the method converges slowly or almost stagnates, techniques that transform the original system are usually used to speed up the convergence. Among them there is a precon ditioning, for which we briefly present the basic idea and algorithm of preconditioned conjugate gradients. We then focus in more detail on the socalled deflation. We present the context in which it has been described in the literature, and comment on various approaches to the derivation of the deflated CG algorithm. We explain the principle of deflation and derive thoroughly the algorithm, describing steps that are not explic itly stated or discussed in detail in the literature. On simple numerical experiments we illustrate the effect of the deflation on the convergence rate. 1


Estimation of the algebraic error and stopping criteria in numerical solution of partial differential equations
Papež, Jan
Title: Estimation of the algebraic error and stopping criteria in numerical solution of partial differential equations Author: Jan Papež Department: Department of Numerical Mathematics Supervisor of the master thesis: Zdeněk Strakoš Abstract: After introduction of the model problem and its properties we describe the Conjugate Gradient Method (CG). We present the estimates of the energy norm of the error and a heuristic for the adaptive refinement of the estimate. The difference in the local behaviour of the discretization and the algebraic error is illustrated by numerical experiments using the given model problem. A posteriori estimates for the discretization and the total error that take into account the inexact solution of the algebraic system are then discussed. In order to get a useful perspective, we briefly recall the multigrid method. Then the Cascadic Conjugate Gradient Method of Deuflhard (CCG) is presented. Using the estimates for the error presented in the preceding parts of the thesis, the new stopping criteria for CCG are proposed. The CCG method with the new stopping criteria is then tested. Keywords: numerical PDE, discretization error, algebraic error, error es timates, locality of the error, adaptivity


Estimation of the algebraic error and stopping criteria in numerical solution of partial differential equations
Papež, Jan
Title: Estimation of the algebraic error and stopping criteria in numerical solution of partial differential equations Author: Jan Papež Department: Department of Numerical Mathematics Supervisor of the master thesis: Zdeněk Strakoš Abstract: After introduction of the model problem and its properties we describe the Conjugate Gradient Method (CG). We present the estimates of the energy norm of the error and a heuristic for the adaptive refinement of the estimate. The difference in the local behaviour of the discretization and the algebraic error is illustrated by numerical experiments using the given model problem. A posteriori estimates for the discretization and the total error that take into account the inexact solution of the algebraic system are then discussed. In order to get a useful perspective, we briefly recall the multigrid method. Then the Cascadic Conjugate Gradient Method of Deuflhard (CCG) is presented. Using the estimates for the error presented in the preceding parts of the thesis, the new stopping criteria for CCG are proposed. The CCG method with the new stopping criteria is then tested. Keywords: numerical PDE, discretization error, algebraic error, error es timates, locality of the error, adaptivity


Odhady algebraické chyby a zastavovací kritéria v numerickém řešení parciálních diferenciálních rovnic
Papež, Jan ; Strakoš, Zdeněk (advisor) ; Vlasák, Miloslav (referee)
Title: Estimation of the algebraic error and stopping criteria in numerical solution of partial differential equations Author: Jan Papež Department: Department of Numerical Mathematics Supervisor of the master thesis: Zdeněk Strakoš Abstract: After introduction of the model problem and its properties we describe the Conjugate Gradient Method (CG). We present the estimates of the energy norm of the error and a heuristic for the adaptive refinement of the estimate. The difference in the local behaviour of the discretization and the algebraic error is illustrated by numerical experiments using the given model problem. A posteriori estimates for the discretization and the total error that take into account the inexact solution of the algebraic system are then discussed. In order to get a useful perspective, we briefly recall the multigrid method. Then the Cascadic Conjugate Gradient Method of Deuflhard (CCG) is presented. Using the estimates for the error presented in the preceding parts of the thesis, the new stopping criteria for CCG are proposed. The CCG method with the new stopping criteria is then tested. Keywords: numerical PDE, discretization error, algebraic error, error es timates, locality of the error, adaptivity

 

Algebraic Error in Matrix Computations in the Context of Numerical Solution of Partial Differential Equations
Papež, Jan ; Strakoš, Zdeněk (advisor) ; Ramage, Alison (referee) ; Vejchodský, Tomáš (referee)
Title: Algebraic Error in Matrix Computations in the Context of Numerical Solution of Partial Differential Equations Author: Jan Papež Department: Department of Numerical Mathematics Supervisor: prof. Ing. Zdeněk Strakoš, DrSc., Department of Numerical Mathe matics Abstract: Solution of algebraic problems is an inseparable and usually the most timeconsuming part of numerical solution of PDEs. Algebraic computations are, in general, not exact, and in many cases it is even principally desirable not to perform them to a high accuracy. This has consequences that have to be taken into account in numerical analysis. This thesis investigates in this line some closely related issues. It focuses, in particular, on spatial distribution of the errors of different origin across the solution domain, backward error interpretation of the algebraic error in the context of function approximations, incorporation of algebraic errors to a posteriori error analysis, influence of algebraic errors to adaptivity, and construction of stopping criteria for (preconditioned) iterative algebraic solvers. Progress in these issues requires, in our opinion, understanding the interconnections between the phases of the overall solution process, such as discretization and algebraic computations. Keywords: Numerical solution of partial...

 