.. _method-multifidelity_stoch_collocation:

"""""""""""""""""""""""""""""""
multifidelity_stoch_collocation
"""""""""""""""""""""""""""""""


Multifidelity uncertainty quantification using stochastic collocation


.. toctree::
   :hidden:
   :maxdepth: 1

   method-multifidelity_stoch_collocation-p_refinement
   method-multifidelity_stoch_collocation-h_refinement
   method-multifidelity_stoch_collocation-max_refinement_iterations
   method-multifidelity_stoch_collocation-convergence_tolerance
   method-multifidelity_stoch_collocation-metric_scale
   method-multifidelity_stoch_collocation-statistics_mode
   method-multifidelity_stoch_collocation-allocation_control
   method-multifidelity_stoch_collocation-discrepancy_emulation
   method-multifidelity_stoch_collocation-quadrature_order_sequence
   method-multifidelity_stoch_collocation-sparse_grid_level_sequence
   method-multifidelity_stoch_collocation-piecewise
   method-multifidelity_stoch_collocation-askey
   method-multifidelity_stoch_collocation-wiener
   method-multifidelity_stoch_collocation-use_derivatives
   method-multifidelity_stoch_collocation-samples_on_emulator
   method-multifidelity_stoch_collocation-sample_type
   method-multifidelity_stoch_collocation-rng
   method-multifidelity_stoch_collocation-probability_refinement
   method-multifidelity_stoch_collocation-final_moments
   method-multifidelity_stoch_collocation-response_levels
   method-multifidelity_stoch_collocation-probability_levels
   method-multifidelity_stoch_collocation-reliability_levels
   method-multifidelity_stoch_collocation-gen_reliability_levels
   method-multifidelity_stoch_collocation-distribution
   method-multifidelity_stoch_collocation-variance_based_decomp
   method-multifidelity_stoch_collocation-diagonal_covariance
   method-multifidelity_stoch_collocation-full_covariance
   method-multifidelity_stoch_collocation-import_approx_points_file
   method-multifidelity_stoch_collocation-export_approx_points_file
   method-multifidelity_stoch_collocation-seed_sequence
   method-multifidelity_stoch_collocation-fixed_seed
   method-multifidelity_stoch_collocation-model_pointer


**Specification**

- *Alias:* None

- *Arguments:* None


**Child Keywords:**

+-------------------------+--------------------+--------------------------------+-----------------------------------------------+
| Required/Optional       | Description of     | Dakota Keyword                 | Dakota Keyword Description                    |
|                         | Group              |                                |                                               |
+=========================+====================+================================+===============================================+
| Optional (Choose One)   | Automated          | `p_refinement`__               | Automatic polynomial order refinement         |
|                         | Refinement Type    +--------------------------------+-----------------------------------------------+
|                         |                    | `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 | `quadrature_order_sequence`__  | Sequence of quadrature orders used in a       |
|                         | Type               |                                | multi-stage expansion                         |
|                         |                    +--------------------------------+-----------------------------------------------+
|                         |                    | `sparse_grid_level_sequence`__ | Sequence of sparse grid levels used in a      |
|                         |                    |                                | multi-stage expansion                         |
+-------------------------+--------------------+--------------------------------+-----------------------------------------------+
| Optional (Choose One)   | Basis Polynomial   | `piecewise`__                  | Use piecewise local basis functions           |
|                         | Family             +--------------------------------+-----------------------------------------------+
|                         |                    | `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    | `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                                        |
+----------------------------------------------+--------------------------------+-----------------------------------------------+

.. __: method-multifidelity_stoch_collocation-p_refinement.html
__ method-multifidelity_stoch_collocation-h_refinement.html
__ method-multifidelity_stoch_collocation-max_refinement_iterations.html
__ method-multifidelity_stoch_collocation-convergence_tolerance.html
__ method-multifidelity_stoch_collocation-metric_scale.html
__ method-multifidelity_stoch_collocation-statistics_mode.html
__ method-multifidelity_stoch_collocation-allocation_control.html
__ method-multifidelity_stoch_collocation-discrepancy_emulation.html
__ method-multifidelity_stoch_collocation-quadrature_order_sequence.html
__ method-multifidelity_stoch_collocation-sparse_grid_level_sequence.html
__ method-multifidelity_stoch_collocation-piecewise.html
__ method-multifidelity_stoch_collocation-askey.html
__ method-multifidelity_stoch_collocation-wiener.html
__ method-multifidelity_stoch_collocation-use_derivatives.html
__ method-multifidelity_stoch_collocation-samples_on_emulator.html
__ method-multifidelity_stoch_collocation-sample_type.html
__ method-multifidelity_stoch_collocation-rng.html
__ method-multifidelity_stoch_collocation-probability_refinement.html
__ method-multifidelity_stoch_collocation-final_moments.html
__ method-multifidelity_stoch_collocation-response_levels.html
__ method-multifidelity_stoch_collocation-probability_levels.html
__ method-multifidelity_stoch_collocation-reliability_levels.html
__ method-multifidelity_stoch_collocation-gen_reliability_levels.html
__ method-multifidelity_stoch_collocation-distribution.html
__ method-multifidelity_stoch_collocation-variance_based_decomp.html
__ method-multifidelity_stoch_collocation-diagonal_covariance.html
__ method-multifidelity_stoch_collocation-full_covariance.html
__ method-multifidelity_stoch_collocation-import_approx_points_file.html
__ method-multifidelity_stoch_collocation-export_approx_points_file.html
__ method-multifidelity_stoch_collocation-seed_sequence.html
__ method-multifidelity_stoch_collocation-fixed_seed.html
__ method-multifidelity_stoch_collocation-model_pointer.html



**Description**


As described in :dakkw:`method-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:


.. math:: R = \sum_{i=1}^{N_p^{LF}} r^{LF}_i L_i(\xi)
   + \sum_{i=1}^{N_p^{HF}} \delta_i L_i(\xi) 

where :math:`\delta_i`  is a coefficient for the discrepancy expansion.

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

To obtain the coefficients :math:`r_i`  and :math:`\delta_i`  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 an ensemble surrogate model
(see :dakkw:`model-surrogate-ensemble`), which defines an an ordered
sequence of model fidelities or resolutions.  Two types of hierarchies
are supported: (i) a hierarchy of model forms composed from more than
one model within the ``ordered_model_fidelities`` specification, or
(ii) a hierarchy of discretization levels comprised from a single
model (either from a ``truth_model_pointer`` specification or a single
entry within an ``ordered_model_fidelities`` specification) which in
turn specifies a ``solution_level_control`` (see
:dakkw:`model-single-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 
:ref:`theory:uq:expansion` section within the theory portion of the Users Guide
for additional information on the Multifidelity SC algorithm.

*Expected HDF5 Output*

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


- :ref:`hdf5_results-se_moments` (expansion moments only)
- :ref:`hdf5_results-pdf`
- :ref:`hdf5_results-level_mappings`

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



**Examples**



.. code-block::

    method,
     multifidelity_stoch_collocation
       model_pointer = 'HIERARCH'
       sparse_grid_level_sequence = 4 3 2
    
    model,
     id_model = 'HIERARCH'
     surrogate ensemble
       ordered_model_fidelities = 'LF' 'MF' 'HF'
       correction additive zeroth_order




