1Blurb:: Multifidelity uncertainty quantification using stochastic collocation 2 3Description:: 4As described in \ref method-stoch_collocation, stochastic collocation 5is a general framework for approximate representation of random response 6functions in terms of finite-dimensional interpolation bases, using 7interpolation polynomials that may be either local or global 8and either value-based or gradient-enhanced. 9 10In the multifidelity case, we decompose this interpolant expansion 11into several constituent expansions, one per model form or solution 12control level. In a bi-fidelity case with low-fidelity (LF) and 13high-fidelity (HF) models and an additive discrepancy approach, we have: 14 15\f[R = \sum_{i=1}^{N_p^{LF}} r^{LF}_i L_i(\xi) 16 + \sum_{i=1}^{N_p^{HF}} \delta_i L_i(\xi) \f] 17 18where \f$\delta_i\f$ is a coefficient for the discrepancy expansion. 19 20The same specification options are available as described in 21\ref method-stoch_collocation with one key difference: the 22coefficient estimation inputs change from a scalar input for a single 23expansion to a <i>sequence</i> specification for a low-fidelity expansion 24followed by multiple discrepancy expansions. 25 26To obtain the coefficients \f$r_i\f$ and \f$\delta_i\f$ for each of 27the expansions, the following options are provided: 28 29<ol> 30<li> multidimensional integration by a tensor-product of Gaussian 31 quadrature rules (specified with \c quadrature_order_sequence, and, 32 optionally, \c dimension_preference). 33<li> multidimensional integration by the Smolyak sparse grid method 34 (specified with \c sparse_grid_level_sequence and, optionally, 35 \c dimension_preference) 36</ol> 37 38It is important to note that, while \c quadrature_order_sequence and \c 39sparse_grid_level_sequence are 40array inputs, only one scalar from these arrays is active at a time 41for a particular expansion estimation. In order to specify anisotropy 42in resolution across the random variable set, a \c dimension_preference 43specification can be used to augment these scalar specifications. 44 45Multifidelity UQ using SC requires that the model selected for 46iteration by the method specification is a multifidelity surrogate 47model (see \ref model-surrogate-hierarchical), which defines an 48\c ordered_model_sequence (see \ref model-surrogate-hierarchical). 49Two types of hierarchies are supported: (i) a hierarchy of model forms 50composed from more than one model within the \c ordered_model_sequence, 51or (ii) a hierarchy of discretization levels comprised from a single 52model within the \c ordered_model_sequence which in turn specifies a 53\c solution_level_control (see 54\ref model-single-solution_level_cost-solution_level_control). 55 56In both cases, an expansion will first be formed for the low fidelity 57model or coarse discretization, using the first value within the 58coefficient estimation sequence, along with any specified refinement 59strategy. Second, expansions are formed for one or more model 60discrepancies (the difference between response results if \c additive 61\c correction or the ratio of results if \c multiplicative \c 62correction), using all subsequent values in the coefficient estimation 63sequence (if the sequence does not provide a new value, then the 64previous value is reused) along with any specified refinement 65strategy. The number of discrepancy expansions is determined by the 66number of model forms or discretization levels in the hierarchy. 67 68After formation and refinement of the constituent expansions, each of 69the expansions is combined (added or multiplied) into an expansion 70that approximates the high fidelity model, from which the final set of 71statistics are generated. 72 73<b> Additional Resources </b> 74 75%Dakota provides access to multifidelity SC methods through the 76NonDMultilevelStochCollocation class. Refer to the Stochastic Expansion 77Methods chapter of the Theory Manual \cite TheoMan for additional 78information on the Multifidelity SC algorithm. 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-se_moments (expansion moments only) 87- \ref hdf5_results-pdf 88- \ref hdf5_results-level_mappings 89 90In addition, the execution group has the attribute \c equiv_hf_evals, which 91records the equivalent number of high-fidelity evaluations. 92 93Topics:: 94 95Examples:: 96\verbatim 97method, 98 multifidelity_stoch_collocation 99 model_pointer = 'HIERARCH' 100 sparse_grid_level_sequence = 4 3 2 101 102model, 103 id_model = 'HIERARCH' 104 surrogate hierarchical 105 ordered_model_fidelities = 'LF' 'MF' 'HF' 106 correction additive zeroth_order 107\endverbatim 108 109Theory:: 110 111Faq:: 112See_Also:: method-adaptive_sampling, method-gpais, method-local_reliability, method-global_reliability, method-sampling, method-importance_sampling, method-stoch_collocation 113