hierarchical

Hierarchical surrogates use an ordered hierarchy of model fidelities and/or resolutions to trade accuracy versus cost.

Specification

• Alias: None

• Arguments: None

Child Keywords:

Required/Optional	Description of Group	Dakota Keyword	Dakota Keyword Description
Required		ordered_model_fidelities	Specification of an hierarchy of model fidelities, ordered from low to high.
Optional		correction	Correction approaches for surrogate models

Description

Multifidelity modeling

involves the use of lower-fidelity physics-based models as surrogates for a high-fidelity “truth” model. These low-fidelity models can involve variations in model form, resolution, or both. Model form variations for reduced fidelity may involve omitted physics or solution of approximated or averaged physics. Resolution variations typically involve coarsened space-time discretizations (e.g., h- derefinement), reduced element order (e.g., p- derefinement), or loosened convergence tolerances. Unlike local, global, and multipoint surrogates, these approximations are not data-driven (no high-fidelity data required for construction) and are stand-alone simulation models in their own right. As an example from computational fluid dynamics (CFD), both model form and resolution are varied when an inviscid, incompressible Euler model on a coarse discretization is used as a low-fidelity surrogate for a high-fidelity large eddy simulation model on a fine discretization.

The required ordered_model_fidelities specification points to a sequence of model specifications of varying fidelity, ordered from lowest to highest fidelity. The highest fidelity model provides the “truth” model, and each of the lower fidelity alternatives provides different levels of approximation at different levels of cost. This specification defines the sequence of model forms, and each model specification identified in this ordered listing may identify a set of resolution controls. Either or both of these sequences may be specified, and this is referred to as a “multilevel hierarchy” in the case of a hierarchy of resolutions (one entry in ordered_model_fidelities that includes active resolution control), a “multifidelity hierarchy” in the case of a hierarchy of model forms (multiple entries in ordered_model_fidelities without active resolution control), or a “multilevel-multifidelity hierarchy” in the case of two-dimensional hierarchy including both model forms and resolutions (multiple entries in ordered_model_fidelities with one or more including active resolution control). Note that the multilevel-multifidelity case can be “ragged” in the sense that not all models need to provide the same number of (or any) resolution controls.

The correction specification identifies what type ( additive, multiplicative, combined) and order ( zeroth_order, first_order, second_order) of correction technique will be applied to the low fidelity results in order to match high fidelity results (value and potentially gradient and Hessian) at one or more points. As described below, this is essential in the optimization context.

Use cases

In multifidelity surrogate-based optimization (SBO), the search algorithm relies primarily on the lower fidelity models, which are corrected for consistency with higher fidelity models. The higher fidelity models are used primarily for verifying candidate steps based on solution of low fidelity approximate subproblems and for updating low fidelity corrections. In the hierarchical SBO case (as compared to SBO with data fits), the correction specification is required, since the omission of a correction technique would effectively eliminate the purpose of the high fidelity model (to use a low fidelity model without corrections, then a single model can be used rather than a hierarchical model). Refer to model-surrogate-global for additional information on available correction approaches.

In multifidelity uncertainty quantification (UQ), response differences are tracked for purposes of decomposing variance across model/resolution levels or for constructing separate discrepancy emulators. In this context, correction specifications are still valid for defining discrepancy emulation details but they are optional with the most common cases used as defaults.

Examples

Theory: