Information-based complexity
SIAM Journal on Computing
Sampling and integration of near log-concave functions
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
The real number model in numerical analysis
Journal of Complexity
The optimal error of Monte Carlo integration
Journal of Complexity
Journal of Complexity
Computing in Science and Engineering
Guest Editors' Introduction: Monte Carlo Methods
Computing in Science and Engineering
Rapidly Mixing Markov Chains with Applications in Computer Science and Physics
Computing in Science and Engineering
Explicit error bounds for lazy reversible Markov chain Monte Carlo
Journal of Complexity
Rigorous confidence bounds for MCMC under a geometric drift condition
Journal of Complexity
Optimal Monte Carlo integration with fixed relative precision
Journal of Complexity
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We study the integration of functions with respect to an unknown density. Information is available as oracle calls to the integrand and to the non-normalized density function. We are interested in analyzing the integration error of optimal algorithms (or the complexity of the problem) with emphasis on the variability of the weight function. For a corresponding large class of problem instances we show that the complexity grows linearly in the variability, and the simple Monte Carlo method provides an almost optimal algorithm. Under additional geometric restrictions (mainly log-concavity) for the density functions, we establish that a suitable adaptive local Metropolis algorithm is almost optimal and outperforms any non-adaptive algorithm.