Are there some approximate or randomised algorithms to numerically solve Poisson's Equation in Partial Differential Equations.(http://en.wikipedia.org/wiki/Poisson%27s_equation). The best algorithms I know of are Multigrid methods(http://en.wikipedia.org/wiki/Multigrid_methods), but they are deterministic and are O(n). Are there Randomised or approximate algos to solve this problem.
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$\begingroup$ I think your question should contain more details. For instance, what is $n$ in $O(n)$ ? $\endgroup$– Denis SerreCommented Dec 22, 2010 at 13:24
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$\begingroup$ There was an ICM talk about this. cs-www.cs.yale.edu/homes/spielman/icm2010.pdf $\endgroup$– dranxoCommented Dec 27, 2010 at 20:08
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3$\begingroup$ You ask for random or approximate algorithms. You then mention the multigrid method. This gives approximate solutions to PDEs. Haven't you answered your question already? In addition, you say "the best algorithms I know are ... but ... are O(n)". The use of "but" implies that O(n) (whatever n is) conflicts with "approximate or randomised". I can't see why and it suggests you've left something out. Maybe you particularly seek fast algorithms. What actually is your question? $\endgroup$– Dan PiponiCommented Jan 4, 2011 at 21:45
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2 Answers
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You can modify the method described here for the Laplace equation to work for Poisson.
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How about the fast multipole method? (Or does that count as multigrid?)
answered Dec 21, 2010 at 19:45
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$\begingroup$ @Steve, I guess FMM techniques use hierarchical techniques like MultiGrid techniques. Actually, I was looking for a randomised/approximate version. $\endgroup$ Commented Dec 23, 2010 at 4:15