Monte Carlo Method (1)
Monte Carlo Method (1)
What is a monte carlo simulation? Monte carlo simulation is a numerical method based on the extensive use of random numbers for solving different problems Remark Monte Carlo simulation can be used to study not only stochastic problems(e.g, diffusion) but also non stochastic ones(Monte Carlo integration)
What is a Monte Carlo simulation? Monte Carlo simulation is a numerical method based on the extensive use of random numbers for solving different problems. Remark: Monte Carlo simulation can be used to study not only stochastic problems (e.g., diffusion) but also non stochastic ones (Monte Carlo integration)
A short history Buffon's problem of needle throwing: A needle of length L is thrown at random onto a plane with straight parallel lines which are separated by a distance d(d>l) What is the probability p that the needle will intersect one of these li Ines Georges Louis Leclerc comte de buffon(1707-1788): Eminent French naturalist Solution P=2L/(d Reference: G. Comte de buffon, Essai darithmetique morale, Supplement a I Histoire naturelle. VoL 41777 Remark Laplace suggested that the method described in Buffon's problem can be used as a stochastic method to calculate the value of t
A short history Buffon’s problem of needle throwing: A needle of length L is thrown at random onto a plane with straight parallel lines which are separated by a distance d (d > L). What is the probability P that the needle will intersect one of these lines? Georges Louis Leclerc comte de Buffon (1707 - 1788): Eminent French naturalist. Solution: P = 2L/(pd) Reference: G. Comte de Buffon, Essai d ’arithmétique morale, Supplément à l ’Histoire Naturelle, Vol. 4, 1777. Remark: Laplace suggested that the method described in Buffon’s problem can be used as a stochastic method to calculate the value of p
Condition for intersection h=Lsin(0)/2>X Geometric implication ALSIna P=S/(d/2/2) S1=L2 d/2 L/2 兀/20
q h x d Condition for intersection: h=Lsin(q)/2 > x Geometric implication: P = SI /(d/2 p/2) SI = L/2 x d/2 L/2 p/2 q x=Lsin(q)/2 I
Anecdote In 1901, Lazzerini (ltalian mathematician) performed a simulation by spinning round and dropping a needle 3407 times. He estimated T to be 3. 1415929(accurate to the seventh number after the point!
Anecdote: In 1901, Lazzerini (Italian mathematician) performed a simulation by spinning round and dropping a needle 3407 times. He estimated p to be 3.1415929 (accurate to the seventh number after the point!)