multilevel_function_train

Multilevel uncertainty quantification using function train expansions

Specification

• Alias: None

• Arguments: 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	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:

\[Q^H \approx \hat{Q}_{r_L}^L + \hat{\Delta}_{r_{ML}}^{ML} + \hat{\Delta}_{r_{HM}}^{HM}\]

where \(\Delta^{ij}\) represents a discrepancy expansion computed from \(Q^i - Q^j\) and reduced rank representations of these discrepancies may be targeted ( \(r_{HM} < r_{ML} < r_L\) ).

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.