1. Introduction
Approximation problems (also known as regression problems) arise quite often in industrial design, and solutions of such problems are conventionally referred to as surrogate models [1]. The most common application of surrogate modeling in engineering is in connection to engineering optimization [2]. Indeed, on the one hand, design optimization plays a central role in the industrial design process; on the other hand, a single optimization step typically requires the optimizer to create or refresh a model of the response function whose optimum is sought, to be able to come up with a reasonable next design candidate. The surrogate models used in optimization range from simple local linear regression employed in the basic gradient-based optimization [3] to complex global models employed in the so-called Surrogate-Based Optimization (SBO) [4]. Aside from optimization, surrogate modeling is used in dimension reduction [5,6], sensitivity analysis [7–10], and for visualization of response functions. Mathematically, the approximation problem can generally be described as follows. We assume that we are given a finite sample of pairs (xn, yn)N n=1 (the “training data”), where xn ∈ Rdin , yn ∈ Rdout . These pairs represent sampled inputs and outputs of an unknown response function y = f(x). Our goal is to construct a function (a surrogate model) f : Rdin → Rdout which should be as close as possible to the true function f.
