Gaussian process emulators for computer experiments with inequality constraints

Abstract

Physical phenomena are observed in many fields (sciences and engineering) and are often studied by time-consuming computer codes. These codes are analyzed with statistical models, often called emulators. In many situations, the physical system (computer model output) may be known to satisfy inequality constraints with respect to some or all input variables. Our aim is to build a model capable of incorporating both data interpolation and inequality constraints into a Gaussian process emulator. By using a functional decomposition, we propose to approximate the original Gaussian process by a finite-dimensional Gaussian process such that all conditional simulations satisfy the inequality constraints in the whole domain. The mean, mode (maximum a posteriori) and prediction intervals (uncertainty quantification) of the conditional Gaussian process are calculated. To investigate the performance of the proposed model, some conditional simulations with inequality constraints such as boundary, monotonicity or convexity conditions are given. 1. Introduction. In the engineering activity, runs of a computer code can be expensive and time-consuming. One solution is to use a statistical surrogate for conditioning computer model outputs at some input locations (design points). Gaussian process (GP) emulator is one of the most popular choices [23]. The reason comes from the property of the GP that uncertainty quantification can be calculated. Furthermore, it has several nice properties. For example, the conditional GP at observation data (linear equality constraints) is still a GP [5]. Additionally, some inequality constraints (such as monotonicity and convexity) of output computer responses are related to partial derivatives. In such cases, the partial derivatives of the GP are also GPs. Incorporating an infinite number of linear inequality constraints into a GP emulator, the problem becomes more difficult. The reason is that the resulting conditional process is not a GP. In the literature of interpolation with inequality constraints, we find two types of meth-ods. The first one is deterministic and based on splines, which have the advantage that the inequality constraints are satisfied in the whole input domain (see e.g. [16], [24] and [25]). The second one is based on the simulation of the conditional GP by using the subdivision of the input set (see e.g. [1], [6] and [11]). In that case, the inequality constraints are satisfied in a finite number of input locations. Notice that the advantage of such method is that un-certainty quantification can be calculated. In previous work, some methodologies have been based on the knowledge of the derivatives of the GP at some input locations ([11], [21] and [26]). For monotonicity constraints with noisy data, a Bayesian approach was developed in [21]. In [11] the problem is to build a GP emulator by using the prior monotonicity