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