Showing papers by "Luc Pronzato published in 2020"

Bayesian quadrature, energy minimization and space-filling design

Abstract: A standard objective in computer experiments is to approximate the behavior of an unknown function on a compact domain from a few evaluations inside the domain. When little is known about the function, space-filling design is advisable: typically, points of evaluation spread out across the available space are obtained by minimizing a geometrical (for instance, covering radius) or a discrepancy criterion measuring distance to uniformity. The paper investigates connections between design for integration (quadrature design), construction of the (continuous) BLUE for the location model, space-filling design, and minimization of energy (kernel discrepancy) for signed measures. Integrally strictly positive definite kernels define strictly convex energy functionals, with an equivalence between the notions of potential and directional derivative, showing the strong relation between discrepancy minimization and more traditional design of optimal experiments. In particular, kernel herding algorithms, which are special instances of vertex-direction methods used in optimal design, can be applied to the construction of point sequences with suitable space-filling properties.

Incremental construction of nested designs based on two-level fractional factorial designs

Abstract: The incremental construction of nested designs having good spreading properties over the d-dimensional hypercube is considered, for values of d such that the 2 d vertices of the hypercube are too numerous to be all inspected. A greedy algorithm is used, with guaranteed efficiency bounds in terms of packing and covering radii, using a 2 d−m fractional-factorial design as candidate set for the sequential selection of design points. The packing and covering properties of fractional-factorial designs are investigated and a review of the related literature is provided. An algorithm for the construction of fractional-factorial designs with maximum packing radius is proposed. The spreading properties of the obtained incremental designs, and of their lower dimensional projections, are investigated. An example with d = 50 is used to illustrate that their projection in a space of dimension close to d has a much higher packing radius than projections of more classical designs based on Latin hypercubes or low discrepancy sequences.

Sequential online subsampling for thinning experimental designs

Abstract: We consider a design problem where experimental conditions (design points $X_i$) are presented in the form of a sequence of i.i.d.\ random variables, generated with an unknown probability measure $\mu$, and only a given proportion $\alpha\in(0,1)$ can be selected. The objective is to select good candidates $X_i$ on the fly and maximize a concave function $\Phi$ of the corresponding information matrix. The optimal solution corresponds to the construction of an optimal bounded design measure $\xi_\alpha^*\leq \mu/\alpha$, with the difficulty that $\mu$ is unknown and $\xi_\alpha^*$ must be constructed online. The construction proposed relies on the definition of a threshold $\tau$ on the directional derivative of $\Phi$ at the current information matrix, the value of $\tau$ being fixed by a certain quantile of the distribution of this directional derivative. Combination with recursive quantile estimation yields a nonlinear two-time-scale stochastic approximation method. It can be applied to very long design sequences since only the current information matrix and estimated quantile need to be stored. Convergence to an optimum design is proved. Various illustrative examples are presented.

Adaptive design criteria motivated by a plug-in percentile estimator

Abstract: Increasingly complex numerical models are involved in a variety of modern engineering applications, ranging from evaluation of environmental risks to optimisation of sophisticated industrial processes. Study of climat change is an extremely well-known example, while its current use in other domains like pharmaceutics (the so-called in vitro experiments), aeronautics or even cosmetics are less well known of the general public. These models allow the prediction of a number of variables of interest for a given configuration of a number of factors that potentially affect them. Complex models depend in general on a large number of such factors, and their execution time may range from a couple of hours to several days. In many cases, collectively falling in the domain of risk analysis, the interest is in identifying how often, under what conditions, or how strongly, a certain phenomenon may happen. In addition to the numerical model that predicts the variable of interest, it is then necessary to define a probabilis-tic structure in the set of its input factors, most often using a frequenciest approach. "How often" requires then the evaluation of the probability of occurence of the event of interest, while "how strongly" implies the determination of the set of the most extreme possible situations. In the former case we face a problem of estimation of an exceedance probability, while in latter is usually referred to as percentile estimation. For instance, in a study of the risk of flooding in a given coastal region, in the first case we want to estimate the probability α that a certain level of inundation η will not be exceeded, while in the second we are interest in the inundation level η that, with probability α, is not exceeded. In the context of the current planetary concern