Quadrature Points via Heat Kernel Repulsion

We discuss the classical problem of how to pick $N$ weighted points on a $d-$dimensional manifold so as to obtain a reasonable quadrature rule $$\frac{1}{|M|}\int_{M}{f(x) dx} \simeq \frac{1}{N} \sum_{n=1}^{N}{a_i f(x_i)}.$$ This problem, naturally, has a long history; the purpose of our paper is to propose selecting points and weights so as to minimize the energy functional \sum_{i,j =1}^{N}{ a_i a_j \exp\left(-\frac{d(x_i,x_j)^2}{4t}\right) } \rightarrow \min, \quad \mbox{where}~t \sim N^{-2/d}, $d(x,y)$ is the geodesic distance and $d$ is the dimension of the manifold. This yields point sets that are theoretically guaranteed, via spectral theoretic properties of the Laplacian $-\Delta$, to have good properties. One nice aspect is that the energy functional is universal and independent of the underlying manifold; we show several numerical examples

Parametric estimation of complex mixed models based on meta-model approach

Complex biological processes are usually experimented along time among a collection of individuals. Longitudinal data are then available and the statistical challenge is to better understand the underlying biological mechanisms. The standard statistical approach is mixed-effects model, with regression functions that are now highly-developed to describe precisely the biological processes (solutions of multi-dimensional ordinary differential equations or of partial differential equation). When there is no analytical solution, a classical estimation approach relies on the coupling of a stochastic version of the EM algorithm (SAEM) with a MCMC algorithm. This procedure needs many evaluations of the regression function which is clearly prohibitive when a time-consuming solver is used for computing it. In this work a meta-model relying on a Gaussian process emulator is proposed to replace this regression function. The new source of uncertainty due to this approximation can be incorporated in the model which leads to what is called a mixed meta-model. A control on the distance between the maximum likelihood estimates in this mixed meta-model and the maximum likelihood estimates obtained with the exact mixed model is guaranteed. Eventually, numerical simulations are performed to illustrate the efficiency of this approach