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Differential Equations Solutions 94

# Differential Equations Solutions 94 - R n 1 by adding an...

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104 Chapter 17. Solutions: Case Study: Monte-Carlo Minimization and Counting 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Late stage TSP for 100 cities, log cooling schedule Figure 17.8. Tour produced for TSP by simulated annealing, logarithmic cooling schedule. V ol ( K ). The connection to simulated annealing comes in a couple of ways. For one thing, the random walk can done using a Metropolis algorithm with a different rejection rate (i.e. a different temperature) for each i . A more recent idea is to recognize that the volume is the integral of the characteristic function of the set K so we can try to approach this integral by integrating a sequence of other, easier functions instead. In particular we can embed the problem in
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Unformatted text preview: R n +1 by adding an extra coeﬃcient x to the points in R n and then choose functions f < f 1 < f 2 < ...f m where f m is the characteristic function of K but the others look like exp( − x /T ) in the extra coeﬃcient, x . Another example of use of the idea of simulated annealing is the KRS algo-rithm used in the next challenge. Those who have become fascinated by this subject might want to try to identify the “temperature” in this case in order to understand why KRS is a form of simulated annealing. CHALLENGE 17.4. (a) Here are some explicit counts, some done by hand and some by latticecount.m by Thomas DuBois....
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