Gaussian Process Emulators for Computer Experiments with Inequality Constraints

Abstract

Physical phenomena are observed in many fields (science 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 a finite-dimensional approximation of Gaussian processes such that all conditional simulations satisfy the inequality constraints in the entire domain. The inequality mean and mode (i.e. mean and maximum a posteriori ) of the conditional Gaussian process are calculated and prediction intervals are quantified. To show the performance of the proposed model, some conditional simulations with inequality constraints such as boundedness, monotonicity or convexity conditions in one and two dimensions are given. A simulation study to investigate the efficiency of the method in terms of prediction and uncertainty quantification is included.Keywords Gaussian process emulator · inequality constraints · finite-dimensional approximation · uncertainty quantification · design and modeling of computer experiments 1 Introduction

Efficient Bayesian shape-restricted function estimation with constrained Gaussian process priors

“…Some representative examples include piecewise linear models [Neelon andDunson, 2004, Cai andDunson, 2007], Bernstein polynomials [Curtis and Ghosh, 2009], regression splines [Meyer et al, 2011], penalized spines [Brezger and Steiner, 2008], cumulative distribution functions [Bornkamp and Ickstadt, 2009], and restricted splines [Shively et al, 2011] used as the basis. Maatouk and Bay [2017] recently exploited a novel basis representation to equivalently represent various shape restrictions such as boundedness, monotonicity, convexity etc as non-negativity constraints on the basis coefficients. Although originally developed in the context of computer model emulation, the approach of Maatouk and Bay [2017] is broadly applicable to general shape constrained problems.…”

“…Maatouk and Bay [2017] recently exploited a novel basis representation to equivalently represent various shape restrictions such as boundedness, monotonicity, convexity etc as non-negativity constraints on the basis coefficients. Although originally developed in the context of computer model emulation, the approach of Maatouk and Bay [2017] is broadly applicable to general shape constrained problems. Zhou et al [2018] adapted their approach to handle a combination of shape constraints in a nuclear physics application to model the electric form factor of a proton.…”

“…Zhou et al [2018] adapted their approach to handle a combination of shape constraints in a nuclear physics application to model the electric form factor of a proton. The main idea of Maatouk and Bay [2017] is to expand a Gaussian process using a first or second order exact Taylor expansion, with the remainder term approximated using linear combinations of compactly supported triangular basis functions. A key observation is that the resulting approximation has the unique advantage of enforcing linear inequality constraints on the function space through an equivalent linear constraint on the basis coefficients.…”

Gaussian processes for computer experiments

“…The contributions which are presented in Sections 2, 3 and 4 correspond to the references [14,35], [49] and [12,13].…”

“…The difficulty of the problem comes from the fact that, when incorporating an infinite number of inequality constraints into a GP, the resulting process is not a GP in general. In this section, we show that the model developed in [35] is capable of incorporating an infinite number of inequality constraints into a GP model…”