Modeling, simulation, Monte Carlo methods
Prof. dr hab. Elżbieta Richter-Wąs
Follow the course/slides from
B. Chopard et al., coursera lectures, University of Geneva
S. Paltani, Statistical Course for Astrophysicits, University of Geneva
• Modeling and simulation of natural processes
introduction and general concepts
• Monte Carlo methods
Examples of natural processes
2
What is a model?
3
What is a model?
4
Computational Science
5
Computational Science
6
Why a model? What is a good model?
7
Level of reality
8
Several models/ different language of description
9
…to a virtual model of reality
10
Example of modeling method
11
From a model to a simulation
12
From a model to a simulation
13
From a model to a simulation: illustration
14
Space and time
15
Time evolution
16
Time evolution
17
Time evolution
18
Modeling space: Eurelian approach
19
Modeling space: Lagrangian approach
20
Modeling space
21
Beyond the physical space: complex networks
22
Example
23
Complex networks
24
Monte Carlo methods
25
A Monte Carlo computer simulation
26
A more difficult problem
27
Historical note
28
Markov-chain Monte Carlo (MCMC)
29
Sampling the diffusion equation in 1D
30
Diffusion equation in 1D
31
More general case: Master equation
32
Detailed balance
33
Metropolis Rule
34
Metropolis Rule in practice
35
Metropolis Rule in practice
36
Glauber Rule
37
Kinetic/Dynamic Mote Carlo
38
Monte Carlo simulation
39
Monte Carlo simulation
40
Gillespie’s algorithm
41
More on Monte Carlo methods
42
Monte Carlo methods
43
Numerical integration
44
Error estimation
45
Optimisation problems
46
Numerical simulations
47
Numerical simulations
48
Example: probability estimation
49
Example: error estimation
50
Example: numerical integration
51
Random numer generators
52
Random numer generators
53
Random numer generators
54
Random numer generators
55
Transformation method
56
Transformation method
57
Transformation method
58
Rejection method
59
Rejection method
60
Rejection method
61
Distributions
62
Quasi-random numbers
63
Quasi-random numbers
64
Quasi-random numbers
65
Quasi-random numbers
66
Quasi-random numbers
67
BONUS slides: For more
mathematically oriented students
68 Follow the course/slides from
M. Chrzaszcz , Lectures on Monte Carlo methods, ETH Zurich
Monte Carlo methods: more maths
69
Applications of MC methods
70
Euler number determination
71
Let’s test the √N
72
Let’s test the √N
73
Let’s test the √N
74
Mathematical foundations of MC methods
75
Mathematical foundations of MC methods
76
Mathematical foundations of MC methods
77
Mathematical foundations of MC methods
78
Mathematical foundations of MC methods
79
Law of large numbers
80
Wrap up
81
Monte Carlo and integration
82
Uncertainty from MC methods
83
Buffon needle – p number calculus
84
Buffon needle – simples Carlo method
85
Head and tails Monte Carlo
86
Buffon needle
87
Crude Monte Carlo method of integration
88
Crude vs „hits or miss”
89
Generalization to multi-dimension case. Crude method
90
Generalization to multi-dimension case. „Hits or miss”
91
Crude MC vs „Hits or miss”
92
Crude MC vs „Hits or miss”
93
Crude MC vs „Hits or miss”
94
Classical methods of variance reduction
95
Stratified sampling
96
Stratified sampling – mathematical details
97
Stratified sampling in practice
98
Stratified sampling for the Buffon needle
99
Importance sampling
100
Importance sampling - Example
101
Control variates
102
Antithetic variates
103
Wrap up
104
Classical methods of variance reduction
105
Disadvantages of classical variance reduction methods
106
Schematic of running this kind of methods
107
VEGAS algorithm
108
VEGAS algorithm
109
VEGAS algorithm – futher improvements
110
VEGAS algorithm – 2D case
111
VEGAS algorithm – an example
112
FOAM algorithm
113
FOAM algorithm
114
FOAM algorithm
115
Monte Carlo vs numerical methods
116
Wrap up
117
Random and pseudorandom numbers
118
Random numbers = history remark
119
Pseudorandom numbers
120
General schematic
121
Linear generators
122
Linear generators
123
Linear generators
124
RANLUX generator
125
Wrap up
126
Literature
127
!["Iz istorii riewolucionnoj mysli w Rossii : izbrannyje trudy", B. P. Koźmin, Moskwa 1961 : [recenzja]](data:image/gif;base64,R0lGODlhAQABAIAAAP///wAAACH5BAEAAAAALAAAAAABAAEAAAICRAEAOw==)