1Blurb:: 2Correction approaches for surrogate models 3Description:: 4 5Some of the surrogate model types support the use of correction 6factors that improve the local accuracy of the surrogate models. 7 8The \c correction specification specifies that the approximation will 9be corrected to match truth data, either matching truth values in the 10case of \c zeroth_order matching, matching truth values and gradients 11in the case of \c first_order matching, or matching truth values, 12gradients, and Hessians in the case of \c second_order matching. For 13\c additive and \c multiplicative corrections, the correction is local 14in that the truth data is matched at a single point, typically the 15center of the approximation region. The \c additive correction adds a 16scalar offset (\c zeroth_order), a linear function (\c first_order), 17or a quadratic function (\c second_order) to the approximation to 18match the truth data at the point, and the \c multiplicative 19correction multiplies the approximation by a scalar (\c zeroth_order), 20a linear function (\c first_order), or a quadratic function (\c 21second_order) to match the truth data at the point. The \c additive 22\c first_order case is due to \cite Lew00 23and the \c multiplicative \c first_order case is commonly known as 24beta correction \cite Haftka1991. For the \c combined 25correction, the use of both additive and multiplicative corrections 26allows the satisfaction of an additional matching condition, typically 27the truth function values at the previous correction point (e.g., the 28center of the previous trust region). The \c combined correction is 29then a multipoint correction, as opposed to the local \c additive and 30\c multiplicative corrections. Each of these correction capabilities 31is described in detail in \cite Eld04. 32 33 34The 35correction factors force the surrogate models to match the true 36function values and possibly true function derivatives at the center 37point of each trust region. 38Currently, Dakota supports either zeroth-, 39first-, or second-order accurate correction methods, each of which can 40be applied using either an additive, multiplicative, or combined 41correction function. For each of these correction approaches, the 42correction is applied to the surrogate model and the corrected model 43is then interfaced with whatever algorithm is being employed. The 44default behavior is that no correction factor is applied. 45 46The simplest correction approaches are those that enforce consistency 47in function values between the surrogate and original models at a 48single point in parameter space through use of a simple scalar offset 49or scaling applied to the surrogate model. First-order corrections 50such as the first-order multiplicative correction (also known as beta 51correction \cite Cha93) and the first-order additive 52correction \cite Lew00 also enforce consistency in the gradients and 53provide a much more substantial correction capability that is 54sufficient for ensuring provable convergence in SBO algorithms. 55SBO convergence rates can be further 56accelerated through the use of second-order corrections which also 57enforce consistency in the Hessians \cite Eld04, where the 58second-order information may involve analytic, finite-difference, or 59quasi-Newton Hessians. 60 61Correcting surrogate models with additive corrections involves 62<!-- cannot reuse anchor in a DUPLICATE: \anchor eq-correct_val_add --> 63\f{equation} 64\hat{f_{hi_{\alpha}}}({\bf x}) = f_{lo}({\bf x}) + \alpha({\bf x}) 65\f} 66where multifidelity notation has been adopted for clarity. For 67multiplicative approaches, corrections take the form 68<!-- cannot reuse anchor in a DUPLICATE: \anchor eq-correct_val_mult --> 69\f{equation} 70\hat{f_{hi_{\beta}}}({\bf x}) = f_{lo}({\bf x}) \beta({\bf x}) 71\f} 72where, for local corrections, \f$\alpha({\bf x})\f$ and \f$\beta({\bf x})\f$ 73are first or second-order Taylor series approximations to the exact 74correction functions: 75<!-- cannot reuse anchor in a DUPLICATE: \anchor eq-taylor_a --> 76<!-- cannot reuse anchor in a DUPLICATE: \anchor eq-taylor_b --> 77\f{eqnarray} 78\alpha({\bf x}) & = & A({\bf x_c}) + \nabla A({\bf x_c})^T 79({\bf x} - {\bf x_c}) + \frac{1}{2} ({\bf x} - {\bf x_c})^T 80\nabla^2 A({\bf x_c}) ({\bf x} - {\bf x_c}) \\ 81\beta({\bf x}) & = & B({\bf x_c}) + \nabla B({\bf x_c})^T 82({\bf x} - {\bf x_c}) + \frac{1}{2} ({\bf x} - {\bf x_c})^T \nabla^2 83B({\bf x_c}) ({\bf x} - {\bf x_c}) 84\f} 85where the exact correction functions are 86<!-- cannot reuse anchor in a DUPLICATE: \anchor eq-exact_B --> 87<!-- cannot reuse anchor in a DUPLICATE: \anchor eq-exact_A --> 88\f{eqnarray} 89A({\bf x}) & = & f_{hi}({\bf x}) - f_{lo}({\bf x}) \\ 90B({\bf x}) & = & \frac{f_{hi}({\bf x})}{f_{lo}({\bf x})} 91\f} 92Refer to \cite Eld04 for additional details on the derivations. 93 94A combination of additive and multiplicative corrections can provide 95for additional flexibility in minimizing the impact of the correction 96away from the trust region center. In other words, both additive and 97multiplicative corrections can satisfy local consistency, but through 98the combination, global accuracy can be addressed as well. This 99involves a convex combination of the additive and multiplicative 100corrections: 101\f[ \hat{f_{hi_{\gamma}}}({\bf x}) = \gamma \hat{f_{hi_{\alpha}}}({\bf x}) + 102(1 - \gamma) \hat{f_{hi_{\beta}}}({\bf x}) \f] 103where \f$\gamma\f$ is calculated to satisfy an additional matching 104condition, such as matching values at the previous design iterate. 105 106It should be noted that in both first order correction methods, the 107function \f$\hat{f}(x)\f$ matches the function value and gradients of 108\f$f_{t}(x)\f$ at \f$x=x_{c}\f$. This property is necessary in proving that 109the first order-corrected SBO algorithms are provably convergent to a 110local minimum of \f$f_{t}(x)\f$. However, the first order correction 111methods are significantly more expensive than the zeroth order 112correction methods, since the first order methods require computing 113both \f$\nabla f_{t}(x_{c})\f$ and \f$\nabla f_{s}(x_{c})\f$. When the SBO 114strategy is used with either of the zeroth order correction methods, 115or with no correction method, convergence is not guaranteed to a local 116minimum of \f$f_{t}(x)\f$. That is, the SBO strategy becomes a heuristic 117optimization algorithm. From a mathematical point of view this is 118undesirable, but as a practical matter, the heuristic variants of SBO 119are often effective in finding local minima. 120 121<b> Usage guidelines </b> 122 123\li Both the \c additive zeroth_order and 124 \c multiplicative zeroth_order correction methods are 125 "free" since they use values of \f$f_{t}(x_{c})\f$ that are normally 126 computed by the SBO strategy. 127 128\li The use of either the \c additive first_order method or 129 the \c multiplicative first_order method does not necessarily 130 improve the rate of convergence of the SBO algorithm. 131 132\li When using the first order correction methods, the 133 gradient-related response keywords must be modified to allow either analytic or 134 numerical gradients to be computed. This provides the gradient data 135 needed to compute the correction function. 136 137\li For many computationally expensive engineering optimization 138 problems, gradients often are too expensive to obtain or are 139 discontinuous (or may not exist at all). In such cases the heuristic 140 SBO algorithm has been an effective approach at identifying optimal 141 designs \cite Giu02. 142 143Topics:: 144Examples:: 145Theory:: 146Faq:: 147See_Also:: 148