Surrogate Models

A Taylor series expansionTaylor expansion of a function allows us to approximate a function $$f(\mathbf {x})$$ at any point $$\mathbf {x}$$ , based solely on information about the function at a single point $$\mathbf {x}^{i}$$ . Here, information about the function implies zero order, first order, second order and higher order information of the function at $$\mathbf {x}^{i}$$ . Surrogate modelling offers an alternative approach to Taylor series for constructing approximations of functions. Instead of constructing approximations based on ever higher and higher order information at a single point, surrogate modelling approximates functions using lower order information at numerous points in the domain of interest. The advantage of such an approach is that it is (i) computationally inexpensive to approximate zero and first order information of the function at additional points in the domain, and that (ii) lower order information can be computed in parallel on distributed computing platforms. Hence, the approximation functions can be exhaustively optimized, while the computationally demanding evaluations of the actual function can be distributed over multiple cores and computers. We consider surrogates constructed using only function values, function values and gradients, and only gradients.