arXiv subjects

Xavier Bay

Publications and source records attributed to Xavier Bay.

5 recordsLinked to original sources

arXiv2026

Optimal Linear Interpolation under Differential Information: application to the prediction of perfect flows

Approximation of functions satisfying partial differential equations (PDEs) is paramount for simulation of physical fluid flows and other problems in physics. Recently, physics-informed machine learning approaches have proven useful as a data-driven complement to numerical models for partial differential equations, bringing faster responses and allowing us to capitalize on past observations. However, their efficiency and convergence depend on the availability of vast training datasets. For sparse observations, Gaussian process regression or Kriging has emerged as a powerful interpolation model, offering principled estimates and uncertainty quantification. Several attempts have been made to condition Gaussian processes on linear PDEs via artificial or collocation observations and kernel design.These methods suffer from scalability issues in higher dimensions and limited generalizability. The aim of this study is to explore the extension of the Kriging predictor in the presence of linear PDE information at a finite number of collocation points. Two approaches are proposed: 1) A collocated co-Kriging with primary observations of the physical field and auxiliary differential observations; 2) A constrained Kriging optimization problem strongly satisfying linear PDE constraints at the points of prediction through a Lagrangian formulation. Numerical experiments are given for ordinary differential equations, 2D harmonic PDEs and an application to perfect flows around a cylinder. This work highlights a trade-off between the computational efficiency of the Lagrange multipliers approach and the strict interpolation of observations.

math.NA↗

arXiv2017

Karhunen-Lo\`eve decomposition of Gaussian measures on Banach spaces

The study of Gaussian measures on Banach spaces is of active interest both in pure and applied mathematics. In particular, the spectral theorem for self-adjoint compact operators on Hilbert spaces provides a canonical decomposition of Gaussian measures on Hilbert spaces, the so-called Karhunen-L{\`o} eve expansion. In this paper, we extend this result to Gaussian measures on Banach spaces in a very similar and constructive manner. In some sense, this can also be seen as a generalization of the spectral theorem for covariance operators associated to Gaussian measures on Banach spaces. In the special case of the standard Wiener measure, this decomposition matches with Paul L{\'e}vy's construction of Brownian motion.

math.PR↗

arXiv2016

Gaussian process emulators for computer experiments with inequality constraints

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

math.PR↗

arXiv2016

Generalization of the Kimeldorf-Wahba correspondence for constrained interpolation

In this paper, we extend the correspondence between Bayes' estimation and optimal interpolation in a Reproducing Kernel Hilbert Space (RKHS) to the case of linear inequality constraints such as boundedness, monotonicity or convexity. In the unconstrained interpolation case, the mean of the posterior distribution of a Gaussian Process (GP) given data interpolation is known to be the optimal interpolation function minimizing the norm in the RKHS associated to the GP. In the constrained case, we prove that the Maximum A Posteriori (MAP) or Mode of the posterior distribution is the optimal constrained interpolation function in the RKHS. So, the general correspondence is achieved with the MAP estimator and not the mean of the posterior distribution. A numerical example is given to illustrate this last result.

math.PR↗

arXiv2016

An analytic comparison of regularization methods for Gaussian Processes

Gaussian Processes (GPs) are a popular approach to predict the output of a parameterized experiment. They have many applications in the field of Computer Experiments, in particular to perform sensitivity analysis, adaptive design of experiments and global optimization. Nearly all of the applications of GPs require the inversion of a covariance matrix that, in practice, is often ill-conditioned. Regularization methodologies are then employed with consequences on the GPs that need to be better understood.The two principal methods to deal with ill-conditioned covariance matrices are i) pseudoinverse and ii) adding a positive constant to the diagonal (the so-called nugget regularization).The first part of this paper provides an algebraic comparison of PI and nugget regularizations. Redundant points, responsible for covariance matrix singularity, are defined. It is proven that pseudoinverse regularization, contrarily to nugget regularization, averages the output values and makes the variance zero at redundant points. However, pseudoinverse and nugget regularizations become equivalent as the nugget value vanishes. A measure for data-model discrepancy is proposed which serves for choosing a regularization technique.In the second part of the paper, a distribution-wise GP is introduced that interpolates Gaussian distributions instead of data points. Distribution-wise GP can be seen as an improved regularization method for GPs.

math.OC↗