"Modelos de Apoio à Decisão" (Prof. C. Bana e Costa)
A supplement
by Miguel Casquilho (Chem. Eng., PhD, Prof., ret.), DEQ, IST
March, 2018
 1 PortugalNotes about Portugal → Architect Pardal Monteiro (1897–1957)
 2Operations Research

The Best of the 20.th Century.pdfTop 10 Algorithms
1946, Monte Carlo method; 1947, Simplex method; ...; 1962: Quicksort; ...

Stanislaw Marcin Ulam

 3(My) Computing over the Internet

Computation (since 1998) — Welcome to my webpages !
Prof. Victor M. Ponce (since 2000), 'Vlab', San Diego State Univ., CA (USA)

What is this (below) ? w3c

Simulation (≡ "Monte Carlo")
 4

ManyRandom.xlsx. Are these random numbers "good" ?
Why 0.5 and 0.288675....pdf ?

Play a game: The Kruskal count card trick

 5

RNG: random number generation
mid-square technique.xls, by John von Neuman
Linear congruential.xls generator

 6

Is simulation a panacea ? Yes, but beware... it may not be necessary:
Two Gaussian bars.xlsx . If you insist: Sum of two Gaussians (!)
(Why is all this done in Fortran ? Is it your fault ?)

(Shall I guess a digit that is in one of your notes ?)

 7

Sometimes, the results are surprising:
Add some bars (trick ? yes, they are uniform).
So, uniforms are friendly ? Try this (analytical). (The integral is, of course, 1.)
Throwing dice.
The "other" use of Monte Carlo is to integrate (numerically), namely for n ≥ 2.

 8

How to simulate ?
Inversion method for some common variables
Simulate a parabolic variable (inverting a cubic !)
(See also acceptance-rejection, Box-Muller, etc., as particular cases.)

 9

Examples:
Distance to points in circle
Distance to points in rectangle
Filling a conical hopper
Liquid in sphere

(In Excel)
Exponential, Gaussian, triangular:  Monte Carlo inversion.xlsx
Box-Muller.xlsx
Given V and d, find h, can_level.xlsx

Estimating σ from sums (only) from unequal size samples (CISTI 2018, Spain)
10

Simulate (Gaussian) correlated variables:
2D, area (such as area of a paving slab); or
3D, bricks (volume, with algorithm)
Cholesky decomposition

 

References

• Wikipedia: Stanislaw Ulam, Monte Carlo method

• Google random number generation. Wikipedia: Random number generation

• Sigman, Karl, 2007, Acceptance-rejection method.pdf (=), Columbia University. Examples for Gaussian and beta.

• Wikipedia: Box-Muller transform

• Wikipedia: Kruskal, Martin D. or at History of Mathematics, Univ. of St. Andrews, Scotland (UK).
(See also a Portuguese mathematician, Pedro Nunes.)

• Wikipedia: List of mathematical symbols

Acknowledgement: Prof. C. Bana e Costa and Prof. António Quintino

 
 
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Created: 2018-02-28 — Last modified: 2018-03-13