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1 Geology, Geophysics and Environment
1 Research Bulletin The Lower Silesian University of Entrepreneurship and Technology Studies in Technical Sciences
1 Zeszyty Naukowe Dolnośląskiej Wyższej Szkoły Przedsiębiorczości i Techniki. Studia z Nauk Technicznych

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Numeryczne otrzymywanie rozkładu Fermiego-Diraca z definicji zespołu kanonicznego

Jastrzębski G.

Zeszyty Naukowe Dolnośląskiej Wyższej Szkoły Przedsiębiorczości i Techniki. Studia z Nauk Technicznych

2014

Nr 3

99--109

Praca przedstawia kilka prób wyliczenia rozkładu Fermiego-Diraca na podstawie definicji zespołu kanonicznego. Rozpatruje prosty model kwantowy, do którego rozwiązania zastosowano metody kombinatoryczne, metodę Monte-Carlo i jej modyfikację, zwaną metodą Metropolisa. Poglądowe wyniki mają dużą wartość dydaktyczną.

This paper presents several attempts to calculate the Fermi-Dirac distribution based on the canonical ensemble definition. The calculation is based on a simple quantum model, for which combinatorial algorithms, the Monte-Carlo method and its modification called the Metropolis algorithm were used. The illustrative results are of great value in teaching.

Inversion of seismic tomography data by Metropolis and simulated annealing algorithm

Dwornik M.

Geology, Geophysics and Environment

2012

Vol. 38, no. 4

463--464

Inversion of seismic tomography is non-uniqueness and bad-conditioned problem. Reconstruction of velocity field is a process of minimization error function between estimated and received travel times. Classical, deterministic method, like matrix decomposition or conjugate gradient, is known for finish calculation in local minimums. Other problems with deterministic methods were application of constraints to the solution. Stochastic algorithms are methods that can be helpful in solving inverse problem in seismic tomography. This paper presents application of the following two stochastic algorithms to reconstruct velocity field: Metropolis algorithm (MA) and simulated algorithm (SA). The Metropolis algorithm is an iterative method and it was first described by Metropolis et al. (1953). This method uses Boltzmann distribution to calculate probability of replacing current solution by worse one, which is modification of current. Level of acceptance is given by value of a temperature. The simulated annealing was first described by Kirkpatrick et al. (1983) and it is a modification of Metropolis algorithm. This algorithm decreased temperature during iterative process. Both algorithms were modified by adding two dimensional median filtration in a place of modification of current velocity field. This filtration was applied with some small probability. Estimation of travel times of primary seismic waves was performed using two ray-based methods: a straight line and a shortest path method (Moser 1991, Pięta & Dwornik 2009). The first method was very fast but nonrealistic in heterogeneous geological medium. The second method had over one hundred times longer calculation time, but provided real ray trajectories. The algorithms were tested in series by ten independent numerical simulations for each parameter of configuration to minimize random effects of stochastic methods. Both algorithms were initialized in two ways: by random velocity fields and by velocity field obtained by SIRT algorithm (Lo & Inderwiesen 1994). Application of median filtration and initializing by SIRT solution decreased calculation time and improved quality of inversion.