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In recent years we have developed adaptive quasi-Monte Carlo (qMC) cubature algorithms using Sobol sequences and integration lattice sequences that meet the error tolerance prescribed by the user. These algorithms have been implemented in MATLAB http://gailgithub.github.io/GAIL_Dev/ and they are guaranteed for integrands whose Fourier series coeffcients decay not too erratically. This talk presents several applications of these adaptive qMC algorithms, including option pricing, multivariate normal probability, and Sobol indices. These examples illustrate how our algorithms need little a priori information. One does not need to know the decay rates of the Fourier coeffcients nor the weights defining the underlying function spaces containing the integrands. We also discuss how these adaptive qMC algorithms can work with other effciency enhancing methods such as control variates and importance sampling. (Joint work with F. J. Hickernell and Da Li.) |
