Dakota Reference Manual: multilevel_function_train

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multilevel_function_train

Multilevel uncertainty quantification using function train 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.
| Optional | rounding_tolerance | An accuracy tolerance that is used to guide rounding during rank adaptation.
| Optional | arithmetic_tolerance | A secondary rounding tolerance used for post-processing
| Optional | regression_type | Type of solver for forming function train approximations by regression
| Optional | max_solver_iterations | Maximum iterations in determining polynomial coefficients
| Optional | max_cross_iterations | Maximum number of iterations for cross-approximation during a rank adaptation.
| Optional | solver_tolerance | Convergence tolerance for the optimizer used during the regression solve.
| Optional | response_scaling | Perform bounds-scaling on response values prior to surrogate emulation
| Optional | tensor_grid | Use sub-sampled tensor-product quadrature points to build a polynomial chaos expansion.
| Optional | collocation_points_sequence | Sequence of collocation point counts used in a multi-stage expansion
| Optional | collocation_ratio | Set the number of points used to build a PCE via regression to be proportional to the number of terms in the expansion.
| Optional | start_order_sequence | Sequence of start orders used in a multi-stage expansion
| Optional | adapt_order | Activate adaptive procedure for determining the best basis order
| Optional | kick_order | increment used when adapting the basis order in function train methods
| Optional | max_order | Maximum polynomial order of each univariate function within the functional tensor train.
| Optional | max_cv_order_candidates | Limit the number of cross-validation candidates for basis order
| Optional | start_rank_sequence | Sequence of start ranks used in a multi-stage expansion
| Optional | adapt_rank | Activate adaptive procedure for determining best rank representation
| Optional | kick_rank | The increment in rank employed during each iteration of the rank adaptation.
| Optional | max_rank | Limits the maximum rank that is explored during a rank adaptation.
| Optional | max_cv_rank_candidates | Limit the number of cross-validation candidates for rank
| 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 1) | 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 the function_train method and the function_train model, the function train (FT) approximation is a polynomial expansion that exploits low rank structure within the mapping from input random variables to output quantities of interest (QoI). For multilevel and multifidelity function train approximations, we decompose this expansion into several constituent expansions, one per model form or solution control level, where independent function train approximations are constructed for the low-fidelity/coarse resolution model and one or more levels of model discrepancy.

In a three-model case with low-fidelity (L), medium-fidelity (M), and high-fidelity (H) models and an additive discrepancy approach, we can denote this as:

where represents a discrepancy expansion computed from and reduced rank representations of these discrepancies may be targeted ( ).

In multilevel approaches, sample allocation for the constituent expansions is performed as described in allocation_control.

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.

## See Also

These keywords may also be of interest:

- function_train
- function_train