correction
Correction approaches for surrogate models
Specification
Alias: None
Arguments: None
Default: no surrogate correction
Child Keywords:
Required/Optional 
Description of Group 
Dakota Keyword 
Dakota Keyword Description 

Required (Choose One) 
Correction Order 
Specify that truth values must be matched. 

Specify that truth values and gradients must be matched. 

Specify that truth values, gradients and Hessians must be matched. 

Required (Choose One) 
Correction Type 
Additive correction factor for local surrogate accuracy 

Multiplicative correction factor for local surrogate accuracy. 

Multipoint correction for a hierarchical surrogate 
Description
Some of the surrogate model types support the use of correction factors that improve the local accuracy of the surrogate models.
The correction
specification specifies that the approximation will
be corrected to match truth data, either matching truth values in the
case of zeroth_order
matching, matching truth values and gradients
in the case of first_order
matching, or matching truth values,
gradients, and Hessians in the case of second_order
matching. For
additive
and multiplicative
corrections, the correction is local
in that the truth data is matched at a single point, typically the
center of the approximation region. The additive
correction adds a
scalar offset ( zeroth_order
), a linear function ( ,first_order
),
or a quadratic function ( second_order
) to the approximation to
match the truth data at the point, and the multiplicative
correction multiplies the approximation by a scalar ( zeroth_order
),
a linear function ( first_order
), or a quadratic function (
second_order
) to match the truth data at the point. The additive
first_order
case is due to [LN00]
and the multiplicative
first_order
case is commonly known as
beta correction [Haf91]. For the combined
correction, the use of both additive and multiplicative corrections
allows the satisfaction of an additional matching condition, typically
the truth function values at the previous correction point (e.g., the
center of the previous trust region). The combined
correction is
then a multipoint correction, as opposed to the local additive
and
multiplicative
corrections. Each of these correction capabilities
is described in detail in [EGC04].
The correction factors force the surrogate models to match the true function values and possibly true function derivatives at the center point of each trust region. Currently, Dakota supports either zeroth, first, or secondorder accurate correction methods, each of which can be applied using either an additive, multiplicative, or combined correction function. For each of these correction approaches, the correction is applied to the surrogate model and the corrected model is then interfaced with whatever algorithm is being employed. The default behavior is that no correction factor is applied.
The simplest correction approaches are those that enforce consistency in function values between the surrogate and original models at a single point in parameter space through use of a simple scalar offset or scaling applied to the surrogate model. Firstorder corrections such as the firstorder multiplicative correction (also known as beta correction [CHGK93]) and the firstorder additive correction [LN00] also enforce consistency in the gradients and provide a much more substantial correction capability that is sufficient for ensuring provable convergence in SBO algorithms. SBO convergence rates can be further accelerated through the use of secondorder corrections which also enforce consistency in the Hessians [EGC04], where the secondorder information may involve analytic, finitedifference, or quasiNewton Hessians.
Correcting surrogate models with additive corrections involves
f{equation} hat{f_{hi_{alpha}}}({bf x}) = f_{lo}({bf x}) + alpha({bf x}) f} where multifidelity notation has been adopted for clarity. For multiplicative approaches, corrections take the form
f{equation} hat{f_{hi_{beta}}}({bf x}) = f_{lo}({bf x}) beta({bf x}) f} where, for local corrections, \(\alpha({\bf x})\) and \(\beta({\bf x})\) are first or secondorder Taylor series approximations to the exact correction functions:
f{eqnarray} alpha({bf x}) & = & A({bf x_c}) + nabla A({bf x_c})^T ({bf x}  {bf x_c}) + frac{1}{2} ({bf x}  {bf x_c})^T nabla^2 A({bf x_c}) ({bf x}  {bf x_c}) \ beta({bf x}) & = & B({bf x_c}) + nabla B({bf x_c})^T ({bf x}  {bf x_c}) + frac{1}{2} ({bf x}  {bf x_c})^T nabla^2 B({bf x_c}) ({bf x}  {bf x_c}) f} where the exact correction functions are
f{eqnarray} A({bf x}) & = & f_{hi}({bf x})  f_{lo}({bf x}) \ B({bf x}) & = & frac{f_{hi}({bf x})}{f_{lo}({bf x})} f} Refer to [EGC04] for additional details on the derivations.
A combination of additive and multiplicative corrections can provide for additional flexibility in minimizing the impact of the correction away from the trust region center. In other words, both additive and multiplicative corrections can satisfy local consistency, but through the combination, global accuracy can be addressed as well. This involves a convex combination of the additive and multiplicative corrections:
where \(\gamma\) is calculated to satisfy an additional matching condition, such as matching values at the previous design iterate.
It should be noted that in both first order correction methods, the function \(\hat{f}(x)\) matches the function value and gradients of \(f_{t}(x)\) at \(x=x_{c}\) . This property is necessary in proving that the first ordercorrected SBO algorithms are provably convergent to a local minimum of \(f_{t}(x)\) . However, the first order correction methods are significantly more expensive than the zeroth order correction methods, since the first order methods require computing both \(\nabla f_{t}(x_{c})\) and \(\nabla f_{s}(x_{c})\) . When the SBO strategy is used with either of the zeroth order correction methods, or with no correction method, convergence is not guaranteed to a local minimum of \(f_{t}(x)\) . That is, the SBO strategy becomes a heuristic optimization algorithm. From a mathematical point of view this is undesirable, but as a practical matter, the heuristic variants of SBO are often effective in finding local minima.
Usage guidelines
Both the
additive
zeroth_order andmultiplicative
zeroth_order correction methods are “free” since they use values of \(f_{t}(x_{c})\) that are normally computed by the SBO strategy.The use of either the
additive
first_order method or themultiplicative
first_order method does not necessarily improve the rate of convergence of the SBO algorithm.When using the first order correction methods, the gradientrelated response keywords must be modified to allow either analytic or numerical gradients to be computed. This provides the gradient data needed to compute the correction function.
For many computationally expensive engineering optimization problems, gradients often are too expensive to obtain or are discontinuous (or may not exist at all). In such cases the heuristic SBO algorithm has been an effective approach at identifying optimal designs [Giu02].