Differential Equations Solutions 97

Differential Equations Solutions 97 - the dimension d is...

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Chapter 18 Solutions: Case Study: Multidimensional Integration: Partition and Conquer CHALLENGE 18.1. A sample program is given on the website. Method 2 gives somewhat better results, since it averages the function values themselves rather than just using them to decide whether a point is inside or outside the region. Three digit accuracy is achieved for 100000 points in Method 1 and for 1000 and 100000 points for Method 2. The convergence rate for Method 1 is consistent with 1 / n , since the product of the error with n is approximately constant for large n , but for Method 2, the results are somewhat more variable. MATLAB’s function quad uses 13 function evaluations to get three digit accuracy. Clearly, for low dimensional integration of smooth functions, Monte Carlo methods are not the methods of choice! Their value becomes apparent only when
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Unformatted text preview: the dimension d is large so that methods like quad would be forced to use a lot of function evaluations. CHALLENGE 18.2. See challenge2.m on the website. CHALLENGE 18.3. A sample program is available on the website. Importance sampling produces better estimates at lower cost: see the answer to Challenge 4 for detailed results. CHALLENGE 18.4. The results are shown in Figure 18.1. The pseudo-random points from MATLAB’s rand are designed to have good statistical properties, but they leave large gaps in space. The quasi-random points are both more predictable and more evenly distributed. They tend to lie on diagonal lines, with longer strings as the coordinate number increases. Other algorithms for generating quasi-random points avoid this defect. 107...
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