1Blurb:: Variance of mean estimator within multilevel polynomial chaos 2 3Description:: 4Multilevel Monte Carlo performs optimal resource allocation based on a 5known estimator variance for the mean statistic: 6 7\f[ Var[\hat{Q}] = \frac{\sigma^2_Q}{N} \f] 8 9Replacing the simple ensemble average estimator in Monte Carlo with a 10polynomial chaos estimator results in a different and unknown 11relationship between the estimator variance and the number of samples. 12In one approach to multilevel PCE, we can employ a parameterized 13estimator variance: 14 15\f[ Var[\hat{Q}] = \frac{\sigma^2_Q}{\gamma N^\kappa} \f] 16 17for free parameters \f$\gamma\f$ and \f$\kappa\f$, with default values 18that may be overridden as part of this specification block. 19 20This approach is supported for regression-based PCE approaches 21(over-determined least squares, under-determined compressed 22sensing, or othogonal least interpolation). 23 24In practice, it can be challenging to estimate a smooth convergence 25rate for estimator variance in the presence of abrupt transitions in 26the quality of sparse recoveries. As a result, sample allocation by 27greedy refinement is generally preferred. 28 29Topics:: 30 31Examples:: 32 33This example starts with sparse recovery for a second-order candidate 34expansion at each level. As the number of samples is adapted for 35each level, as dictated by the convergence of the estimator variance, 36the candidate expansion order is incremented as needed 37in order to synchronize with the specified collocation ratio. 38 39\verbatim 40method, 41 model_pointer = 'HIERARCH' 42 multilevel_polynomial_chaos 43 orthogonal_matching_pursuit 44 expansion_order_sequence = 2 45 pilot_samples = 10 46 collocation_ratio = .9 47 allocation_control 48 estimator_variance estimator_rate = 2.5 49 seed = 1237 50 convergence_tolerance = .01 51\endverbatim 52 53Theory:: 54 55Faq:: 56See_Also:: 57