1Blurb:: Multilevel methods for sampling-based UQ 2 3Description:: A nascent sampling method that utilizes both multifidelity and 4multilevel relationships within a hierarchical surrogate model in order to 5improve convergence behavior in sampling methods. 6 7In the case of a multilevel relationship, multilevel Monte Carlo 8methods are used to compute an optimal sample allocation per level, 9and in the case of a multifidelity relationship, control variate Monte 10Carlo methods are used to compute an optimal sample allocation per 11fidelity. These two approaches can also be combined, resulting in the 12three approaches below. 13 14<b> Multilevel Monte Carlo </b> 15 16The Monte Carlo estimator for the mean is defined as 17\f[ \mathbb{E}[Q] \equiv \hat{Q}^{MC} = \frac{1}{N} \sum_{i=1}^N Q^{(i)} \f] 18 19In a multilevel method with \f$L\f$ levels, we replace this estimator 20with a telescoping sum: 21\f[ \mathbb{E}[Q] \equiv \hat{Q}^{ML} 22 = \sum_{l=0}^L \frac{1}{N_l} \sum_{i=1}^{N_l} (Q_l^{(i)} - Q_{l-1}^{(i)}) 23 \equiv \sum_{l=0}^L \hat{Y}^{MC}_l \f] 24 25This decomposition forms discrepancies for each level greater than 0, 26seeking reduction in the variance of the discrepancy \f$Y\f$ relative 27to the variance of the original response \f$Q\f$. The number of samples 28allocated for each level (\f$N_l\f$) is based on a total cost minimization 29procedure that incorporates the relative cost and observed variance for each 30of the \f$Y_\ell\f$. 31 32<b> Control Variate Monte Carlo </b> 33 34In the case of two model fidelities (low fidelity denoted as LF and 35high fidelity denoted as HF), we employ a control variate approach: 36 37\f[ \hat{Q}_{HF}^{CV} = \hat{Q}_{HF}^{MC} 38- \beta (\hat{Q}_{LF}^{MC} - \mathbb{E}[Q_{LF}]) \f] 39 40As opposed to the traditional control variate approach, we do not know 41\f$\mathbb{E}[Q_{LF}]\f$ precisely, but rather estimate it more 42accurately than \f$\hat{Q}_{LF}^{MC}\f$ based on a sampling increment 43applied to the LF model. This sampling increment is based again on 44a total cost minimization procedure that incorporates the relative LF 45and HF costs and the observed Pearson correlation coefficient 46\f$\rho_{LH}\f$ between \f$Q_{LF}\f$ and \f$Q_{HF}\f$. The 47coefficient \f$\beta\f$ is then determined from the observed LF-HF 48covariance and LF variance. 49 50<b> Multilevel Control Variate Monte Carlo </b> 51 52If both multifidelity and multilevel structure 53are included within the hierarchical model specification, then a control 54variate can be applied across fidelities for each level within an 55outer multilevel approach. 56 57On each level a control variate is active for the discrepancy \f$Y_{\ell}\f$ 58based on 59\f[ Y_{\ell}^\star = Y_{\ell} + \alpha_\ell \left( \hat{Y}^{\mathrm{LF}}_\ell - \mathbb{E}\left[ Y^{\mathrm{LF}}_\ell \right] \right), \f] 60where \f$Y^{\mathrm{LF}}_\ell = \gamma_\ell Q^{\mathrm{LF}}_\ell - Q^{\mathrm{HF}}_\ell\f$. 61 62The optimal parameter \f$\gamma_\ell\f$ is computed from the correlation properties 63between model forms and discretization levels (see the theory manual for further details) 64and the optimal allocation \f$N_\ell\f$ (per level) is finally obtained from it. 65 66<b> Default Behavior </b> 67 68The multilevel sampling method employs Monte Carlo sampling be 69default, but this default can be overridden to use Latin hypercube 70sampling using \c sample_type \c lhs. 71 72<b> Expected Output </b> 73 74The multilevel sampling method reports estimates of the first four 75moments and a summary of the evaluations performed for each model 76fidelity and discretization level. The method does not support any 77level mappings (response, probability, reliability, generalized 78reliability) at this time. 79 80<b> Expected HDF5 Output </b> 81 82If Dakota was built with HDF5 support and run with the 83\ref environment-results_output-hdf5 keyword, this method 84writes the following results to HDF5: 85 86- \ref hdf5_results-sampling_moments (moments only, not confidence intervals) 87 88In addition, the execution group has the attribute \c equiv_hf_evals, which 89records the equivalent number of high-fidelity evaluations. 90 91<b> Usage Tips </b> 92 93The multilevel sampling method must be used in combination with a 94hierarchical model specification. When exploiting multiple 95discretization levels, it is necessary to identify the variable string 96identifier that controls these levels using \c solution_level_control. 97Associated relative costs also need to be supplied using \c 98solution_level_cost. 99 100<b> Additional Discussion </b> 101 102Also see multilevel regression in \ref method-polynomial_chaos. 103 104 105Topics:: 106 107Examples:: 108The following method block 109\verbatim 110method, 111 model_pointer = 'HIERARCH' 112 multilevel_sampling 113 pilot_samples = 20 seed = 1237 114 max_iterations = 10 115 convergence_tolerance = .001 116\endverbatim 117 118results in multilevel Monte Carlo when the HIERARCH model 119specification contains a single model fidelity with multiple discretization 120levels, in control variate Monte Carlo when the HIERARCH model 121specification has multiple ordered model fidelities each with a single 122discretization level, and multilevel control variate Monte Carlo when 123the HIERARCH model specification contains multiple model fidelities 124each with multiple discretization levels. 125 126An example of the former (single model fidelity with multiple discretization 127levels) follows: 128\verbatim 129model, 130 id_model = 'HIERARCH' 131 surrogate hierarchical 132 ordered_model_fidelities = 'SIM1' 133 correction additive zeroth_order 134 135model, 136 id_model = 'SIM1' 137 simulation 138 solution_level_control = 'N_x' 139 solution_level_cost = 630. 1260. 2100. 4200. 140\endverbatim 141Refer to \c dakota/share/dakota/test/dakota_uq_heat_eq_{mlmc,cvmc,mlcvmc}.in for 142additional examples. 143 144Theory:: 145Faq:: 146See_Also:: method-polynomial_chaos 147