Dakota Reference Manual: multilevel_polynomial_chaos

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multilevel_polynomial_chaos

Multilevel uncertainty quantification using polynomial chaos expansions

## Specification

Alias: none

Argument(s): none

Child Keywords:

| Required/Optional | Description of Group | Dakota Keyword | Dakota Keyword Description
--- | --- | --- | --- | ---
| Optional | max_iterations | Number of iterations allowed for optimizers and adaptive UQ methods
| Optional | allocation_control | Sample allocation approach for multilevel expansions
| 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 | discrepancy_emulation | Formulation for emulation of model discrepancies.
| Required (Choose One) | Coefficient Computation Approach (Group 1) | expansion_order_sequence | Sequence of expansion orders used in a multi-stage expansion
orthogonal_least_interpolation | Build a polynomial chaos expansion from simulation samples using orthogonal least interpolation.
| Optional (Choose One) | Basis Polynomial Family (Group 2) | 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 | normalized | The normalized specification requests output of PCE coefficients that correspond to normalized orthogonal basis polynomials
| Optional | export_expansion_file | Export the coefficients and multi-index of a Polynomial Chaos Expansion (PCE) to a file
| 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 3) | 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 polynomial_chaos, the polynomial chaos expansion (PCE) is a general framework for the approximate representation of random response functions in terms of series expansions in standardized random variables:

where is a deterministic coefficient, is a multidimensional orthogonal polynomial and is a vector of standardized random variables.

In the multilevel and multifidelity cases, we decompose this 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, we have:

where is a coefficient for the discrepancy expansion.

For the case of regression-based PCE (least squares, compressed sensing, or orthogonal least interpolation), an optimal sample allocation procedure can be applied for the resolution of each level within a multilevel sampling procedure as in multilevel_sampling. The core difference is that a Monte Carlo estimator of the statistics is replaced with a PCE-based estimator of the statistics, requiring approximation of the variance of these estimators.

Initial prototypes for multilevel PCE can be explored using `dakota/share/dakota/test/dakota_uq_diffusion_mlpce.in`, and will be stabilized in future releases.

Additional Resources

Dakota provides access to multilevel PCE methods through the NonDMultilevelPolynomialChaos class. Refer to the Stochastic Expansion Methods chapter of the Theory Manual[15] for additional information on the Multilevel PCE 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,
        multilevel_polynomial_chaos
          model_pointer = 'HIERARCH'
          pilot_samples = 10
          expansion_order_sequence = 2
          collocation_ratio = .9
          seed = 1237
          orthogonal_matching_pursuit
          convergence_tolerance = .01
        output silent

    model,
        id_model = 'HIERARCH'
        surrogate hierarchical
          ordered_model_fidelities = 'SIM1'
          correction additive zeroth_order

    model,
        id_model = 'SIM1'
        simulation
          solution_level_control = 'mesh_size'
          solution_level_cost = 1. 8. 64. 512. 4096.

## See Also

These keywords may also be of interest:

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