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A new method for the solution of the Schrödinger equation
Kocák, Jakub ; Uhlík, Filip (advisor) ; Demel, Ondřej (referee)
Title: A new method for the solution of the Schrödinger equation Author: Jakub Kocák Department: Department of Physical and Macromolecular Chemistry Supervisor: doc. RNDr. Filip Uhlík, Ph.D. Abstract: In this thesis we study method for the solution of time-independent Schrö- dinger equation for ground state. The wave function, interpreted as probability density, is represented by samples. In each iteration we applied approximant of imaginary time propagator. Acting of the operator is implemented by Monte Carlo simulation. Part of the thesis is dedicated to methods of energy calculation from samples of wave function: method based on estimation of value of wave function, method of convolution with heat kernel, method of averaged energy weighed by wave function and exponential de- cay method. The method for the solution was used to find ground state and energy for 6-dimensional harmonic oscillator, anharmonic 3-dimensional octic oscillator and hydrogen atom. Keywords: imaginary time propagation, Monte Carlo method, variational principle, ground state 1
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Tests of independence for multivariate data
Kudlík, Michal ; Omelka, Marek (advisor) ; Hlávka, Zdeněk (referee)
Title: Tests of independence for multivariate data Author: Bc. Michal Kudlík Department: Department of Probability and Mathematical Statistics Supervisor: Ing. Marek Omelka, PhD., Department of Probability and Mathema- tical Statistics Abstract: This thesis is an overview of tests of independence for multidimensi- onal data. The report includes tests on independence of categorical and conti- nuous random variables, tests assuming normal distribution of data, asymptotic nonparametric tests and permutation tests with application of the Monte Carlo method. This thesis shows the suitability of tests with properly chosen real data and checks significance level and compares the strength of the selected tests by simulation study while using appropriate statistical software. Based on the simu- lation study the thesis discusses an appropriateness of the use of different tests for different situations. Keywords: independence, permutation and asymptotic tests of independence, Monte Carlo method, simulation study 1
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Quantum variational Monte Carlo method
Kocák, Jakub ; Uhlík, Filip (advisor) ; Srnec, Martin (referee)
Title: Quantum Variational Monte Carlo method Author: Jakub Kocák Department: Department of Physical and Macromolecular Chemistry Supervisor: RNDr. Filip Uhlík, Ph.D. Abstract: In this thesis, we study variational Monte Carlo method in quantum- mechanical systems. We analysed choice of trial wave function and afterwards we optimized selected function for helium singlet and triplet state. In the first chapter, we discuss basic notions of quantum mechanics and general properties of wave function and properties of ground state of the system. In the second chapter, we consider computational algorithms used to calculate integrals, to estimate errors and for optimization. In the third chapter, we present results of optimization and properties of optimized trial wave function. Keywords: variational principle, Monte Carlo method, ground state, helium 1
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