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* log^{2} ε random bits. The number of arithmetic operations is of the order ε^{−2} + *d* log^{2} ε. Hence, the cost of our algorithm increases only mildly with the dimension *d*, we obtain the upper bound *C* · (ε^{−2} + *d* log^{2} ε) for the complexity. In particular, the problem is tractable for coin tossing algorithms.

# 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

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**Published Online**: 2008-05-09

**Published in Print**: 2004-12-01

**Citation Information: **Monte Carlo Methods and Applications mcma, Volume 10, Issue 3-4, Pages 491–498, ISSN (Online) 1569-3961, ISSN (Print) 0929-9629, DOI: https://doi.org/10.1515/mcma.2004.10.3-4.491.

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*Foundations of Computational Mathematics*, 2018

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