Stanislaw Ulam and Monte Carlo Method By Chris

Stanislaw Ulam and Monte Carlo Method By Chris Towles

Stanislaw Ulam � After spending a lot of time trying to estimate them by pure combinatorial calculations, I wondered whether a more practical method than “abstract thinking”.

Stanislaw Ulam – cont. � “Later … [in 1946, I] described the idea to John von Neumann, and we began to plan actual calculations. ” � Ulam is primarily known for designing the hydrogen bomb with Edward Teller in 1951. � April 13, 1909 – May 13, 1984

Stanislaw Ulam � Ulam’s contribution was to recognize the potential for the newly invented electronic computer to automate such sampling. � He was Polish born mathematician who worked for John von Neumann on the United States’ Manhattan Project. � Adventures of a mathematician (Bio) � April 13, 1909 – May 13, 1984 Ulam's ID badge photo from Los Alamos

Monte Carlo Method � Any technique of statistical sampling employed to approximate solutions to quantitative problems. � For simulating physical and mathematical systems � Really useful for solving a set of problems where the time required to find the exact solution grows exponential. � Monte Carlo Method coined by physicists working on Los Alamos Nuclear projects in the 1940’s.

Solving an Old Problem � We’ve studied how others came to figure out PI. To show this idea I actually wrote an some simple code to demonstrate how it could be used to find pi.

Monte Carlo solution for PI

Results for PI � � � In. Circle: 78488 Points: 10^5 Pi = 3. 13952 dif =0. 00207 In. Circle: 785726 Points: 10^6 Pi = 3. 142904 dif =0. 00131 In. Circle: 7854613 Points: 10^7 Pi = 3. 1418452 dif =0. 00025 In. Circle: 78535596 Points: 10^8 Pi = 3. 14142384 dif =0. 000168 All four solutions ran consecutively in under 30 seconds.

A MC solution for ‘Go’ (1993) “Let me quickly point out the obvious. The above idea is certainly simple, and I believe that any good idea can be stated simply. However, not every simple idea is a good one, and at first sight the above one may sound ridiculous. The main portion of this article is dedicated to convince the reader that at least this is not necessarily a bad Idea. ”

Resource Links � http: //www. geocities. com/College. Park/Quad /2435/history. html � http: //math. fullerton. edu/mathews/n 2003/M onte. Carlo. Pi. Mod. html � http: //en. wikipedia. org/wiki/Monte_Carlo_m ethod � http: //en. wikipedia. org/wiki/Stanislaw_Ulam � http: //www. riskglossary. com/link/monte_carl o_method. htm � ***http: //www. ideanest. com/vegos/Monte. Ca rlo. Go. pdf