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

Description::

Multifidelity modeling
<!-- (also known as variable fidelity or variable complexity 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 \c 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 \c
ordered_model_fidelities that includes active resolution control), a
"multifidelity hierarchy" in the case of a hierarchy of model forms
(multiple entries in \c 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 \c 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 \c correction specification identifies what type (\c additive, \c
multiplicative, \c combined) and order (\c zeroth_order, \c
first_order, \c 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.

<b> Use cases </b>

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 \c 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 \c single model can be used
rather than a \c hierarchical model). Refer to \ref
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.

Topics::
Examples::
Theory:


Faq::
See_Also::	model-surrogate-global, model-surrogate-local, model-surrogate-multipoint, method-multilevel_sampling, method-multilevel_polynomial_chaos, method-multifidelity_polynomial_chaos, method-multifidelity_stoch_collocation, method-multilevel_function_train, method-multifidelity_function_train, method-surrogate_based_local
