Monte Carlo Methods & Integration for Circles IT Assignment Help

Assignment Task

Task
R6.1 Use a Monte Carlo method to find the area of a circle, the volume of a sphere, and so on, for higher-dimensional “circles” and “spheres”. Make a statistical estimate of the accuracy of your result, and compare with known results. R6.2 Do either (a) or (b):a) From computer problems 9.3 in Sauer: In a biased random walk, the probability of going up one unit is 0 < p xss=removed xss=removed xss=removed> i. [?2, 5]
ii. [?5, 3]
iii. [?8, 3]

Additional
Attempts at these exercises can earn additional marks, but will not count towards the grade of 1–5 for the exercises. Completing all of these exercises does not mean that 4 marks will be obtained—the marks depend on the quality of the answers. It is possible to earn all 4 marks without completing all of these additional exercises.

• A6.1 Do the other part (i.e., (a) or (b)) for R6.2.
• A6.2 Implement and test a random number generator.
• A6.3 Generate a distribution of random numbers other than uniform or normal.
• What kind of problem would this distribution be useful for?
• A6.4 Find or implement code for high-dimensional optimisation. Test, and discuss.
• A6.5 Implement Monte Carlo integration for circles, spheres, etc., using C or some other compiled language. Compare the performance of your code with your Matlab version. Investigate the effect of using single precision instead of double precision. A6.6 Write a popular science article describing a problem and the use of Monte Carlo methods to solve it. Programming hints Note that rand() and randn() can return a vector or matrix of random numbers. Instead of getting one random number at a time in a loop, you can get them all at once!
• • rand(N) and randn(N) will return N × N matrices of random numbers.
• • rand(M,N) and randn(M,N) will return M × N matrices of random numbers.
• R6.1 You need to demonstrate a sufficient understanding of the question. This means that you will need to do for multiple dimensions. Going from a 1-sphere (a sphere in 1 dimension) to a 10-sphere is sufficient (That is: 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10). If you are using rand() in MATLAB or similar, one way to do this is to sample points inside a box. This method is finding the ratio of In/N, where In is how many random points lie inside the sphere, and N is how many points used in the simulation. It is important to note that this is a ratio of the “box” it covers; this means you will need to multiply the ratio by the volume of the box.
   

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