Monte Carlo Methods and Applications
Managing Editor: Sabelfeld, Karl K.
Editorial Board: Binder, Kurt / Bouleau, Nicolas / Chorin, Alexandre J. / Dimov, Ivan / Dubus, Alain / Egorov, Alexander D. / Ermakov, Sergei M. / Halton, John H. / Heinrich, Stefan / Kalos, Malvin H. / Lepingle, D. / Makarov, Roman / Mascagni, Michael / Mathe, Peter / Niederreiter, Harald / Platen, Eckhard / Sawford, Brian R. / Schmid, Wolfgang Ch. / Schoenmakers, John / Simonov, Nikolai A. / Sobol, Ilya M. / Spanier, Jerry / Talay, Denis
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Coin Tossing Algorithms for Integral Equations and Tractability
Integral equations with Lipschitz kernels and right-hand sides are intractable for deterministic methods, the complexity increases exponentially in the dimension d. This is true even if we only want to compute a single function value of the solution. For this latter problem we study coin tossing algorithms (or restricted Monte Carlo methods), where only random bits are allowed. We construct a restricted Monte Carlo method with error ε that uses roughly ε−2 function values and only d log2 ε random bits. The number of arithmetic operations is of the order ε−2 + d log2 ε. Hence, the cost of our algorithm increases only mildly with the dimension d, we obtain the upper bound C · (ε−2 + d log2 ε) for the complexity. In particular, the problem is tractable for coin tossing algorithms.
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