Dakota Reference Manual: multifidelity_stoch_collocation

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multifidelity_stoch_collocation

Multifidelity uncertainty quantification using stochastic collocation

## Specification

Alias: none

Argument(s): none

Child Keywords:

| Required/Optional | Description of Group | Dakota Keyword | Dakota Keyword Description
--- | --- | --- | --- | ---
| Optional (Choose One) | Automated Refinement Type (Group 1) | p_refinement | Automatic polynomial order refinement
h_refinement | Employ h-refinement to refine the grid
| Optional | max_refinement_iterations | Maximum number of expansion refinement iterations
| Optional | convergence_tolerance | Stopping criterion based on objective function or statistics convergence
| Optional | metric_scale | define scaling of statistical metrics when adapting UQ surrogates
| Optional | statistics_mode | type of statistical metric roll-up for multifidelity UQ methods
| Optional | allocation_control | Sample allocation approach for multifidelity expansions
| Optional | discrepancy_emulation | Formulation for emulation of model discrepancies.
| Required (Choose One) | Interpolation Grid Type (Group 2) | quadrature_order_sequence | Sequence of quadrature orders used in a multi-stage expansion
sparse_grid_level_sequence | Sequence of sparse grid levels used in a multi-stage expansion
| Optional (Choose One) | Basis Polynomial Family (Group 3) | piecewise | Use piecewise local basis functions
askey | Select the standardized random variables (and associated basis polynomials) from the Askey family that best match the user-specified random variables.
wiener | Use standard normal random variables (along with Hermite orthogonal basis polynomials) when transforming to a standardized probability space.
| Optional | use_derivatives | Use derivative data to construct surrogate models
| Optional | samples_on_emulator | Number of samples at which to evaluate an emulator (surrogate)
| Optional | sample_type | Selection of sampling strategy
| Optional | rng | Selection of a random number generator
| Optional | probability_refinement | Allow refinement of probability and generalized reliability results using importance sampling
| Optional | final_moments | Output moments of the specified type and include them within the set of final statistics.
| Optional | response_levels | Values at which to estimate desired statistics for each response
| Optional | probability_levels | Specify probability levels at which to estimate the corresponding response value
| Optional | reliability_levels | Specify reliability levels at which the response values will be estimated
| Optional | gen_reliability_levels | Specify generalized relability levels at which to estimate the corresponding response value
| Optional | distribution | Selection of cumulative or complementary cumulative functions
| Optional | variance_based_decomp | Activates global sensitivity analysis based on decomposition of response variance into main, interaction, and total effects
| Optional (Choose One) | Covariance Type (Group 4) | diagonal_covariance | Display only the diagonal terms of the covariance matrix
full_covariance | Display the full covariance matrix
| Optional | import_approx_points_file | Filename for points at which to evaluate the PCE/SC surrogate
| Optional | export_approx_points_file | Output file for surrogate model value evaluations
| Optional | seed_sequence | Sequence of seed values for multi-stage random sampling
| Optional | fixed_seed | Reuses the same seed value for multiple random sampling sets
| Optional | model_pointer | Identifier for model block to be used by a method

## Description

As described in stoch_collocation, stochastic collocation is a general framework for approximate representation of random response functions in terms of finite-dimensional interpolation bases, using interpolation polynomials that may be either local or global and either value-based or gradient-enhanced.

In the multifidelity case, we decompose this interpolant expansion into several constituent expansions, one per model form or solution control level. In a bi-fidelity case with low-fidelity (LF) and high-fidelity (HF) models and an additive discrepancy approach, we have:

where is a coefficient for the discrepancy expansion.

The same specification options are available as described in stoch_collocation with one key difference: the coefficient estimation inputs change from a scalar input for a single expansion to a sequence specification for a low-fidelity expansion followed by multiple discrepancy expansions.

To obtain the coefficients and for each of the expansions, the following options are provided:

1. multidimensional integration by a tensor-product of Gaussian quadrature rules (specified with `quadrature_order_sequence`, and, optionally, `dimension_preference`).
2. multidimensional integration by the Smolyak sparse grid method (specified with `sparse_grid_level_sequence` and, optionally, `dimension_preference`)

It is important to note that, while `quadrature_order_sequence` and `sparse_grid_level_sequence` are array inputs, only one scalar from these arrays is active at a time for a particular expansion estimation. In order to specify anisotropy in resolution across the random variable set, a `dimension_preference` specification can be used to augment these scalar specifications.

Multifidelity UQ using SC requires that the model selected for iteration by the method specification is a multifidelity surrogate model (see hierarchical), which defines an `ordered_model_sequence` (see hierarchical). Two types of hierarchies are supported: (i) a hierarchy of model forms composed from more than one model within the `ordered_model_sequence`, or (ii) a hierarchy of discretization levels comprised from a single model within the `ordered_model_sequence` which in turn specifies a `solution_level_control` (see solution_level_control).

In both cases, an expansion will first be formed for the low fidelity model or coarse discretization, using the first value within the coefficient estimation sequence, along with any specified refinement strategy. Second, expansions are formed for one or more model discrepancies (the difference between response results if `additive` `correction` or the ratio of results if `multiplicative` `correction`), using all subsequent values in the coefficient estimation sequence (if the sequence does not provide a new value, then the previous value is reused) along with any specified refinement strategy. The number of discrepancy expansions is determined by the number of model forms or discretization levels in the hierarchy.

After formation and refinement of the constituent expansions, each of the expansions is combined (added or multiplied) into an expansion that approximates the high fidelity model, from which the final set of statistics are generated.

Additional Resources

Dakota provides access to multifidelity SC methods through the NonDMultilevelStochCollocation class. Refer to the Stochastic Expansion Methods chapter of the Theory Manual[15] for additional information on the Multifidelity SC algorithm.

Expected HDF5 Output

If Dakota was built with HDF5 support and run with the hdf5 keyword, this method writes the following results to HDF5:

- Integration and Expansion Moments (expansion moments only)
- Probability Density
- Level Mappings

In addition, the execution group has the attribute `equiv_hf_evals`, which records the equivalent number of high-fidelity evaluations.

## Examples

    method,
        multifidelity_stoch_collocation
          model_pointer = 'HIERARCH'
          sparse_grid_level_sequence = 4 3 2

    model,
        id_model = 'HIERARCH'
        surrogate hierarchical
          ordered_model_fidelities = 'LF' 'MF' 'HF'
          correction additive zeroth_order

## See Also

These keywords may also be of interest:

- adaptive_sampling
- gpais
- local_reliability
- global_reliability
- sampling
- importance_sampling
- stoch_collocation