8 research outputs found
Regularity dependence of the rate of convergence of the learning curve for Gaussian process regression
This paper deals with the speed of convergence of the learning curve in a
Gaussian process regression framework. The learning curve describes the average
generalization error of the Gaussian process used for the regression. More
specifically, it is defined in this paper as the integral of the mean squared
error over the input parameter space with respect to the probability measure of
the input parameters. The main result is the proof of a theorem giving the mean
squared error in function of the number of observations for a large class of
kernels and for any dimension when the number of observations is large. From
this result, we can deduce the asymptotic behavior of the generalization error.
The presented proof generalizes previous ones that were limited to more
specific kernels or to small dimensions (one or two). The result can be used to
build an optimal strategy for resources allocation. This strategy is applied
successfully to a nuclear safety problem
Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients
A scalar, elliptic boundary-value problem in divergence form with stochastic diffusion coefficient a(x, ω) in a bounded domain D ⊂ ℝd is reformulated as a deterministic, infinite-dimensional, parametric problem by separation of deterministic (x ∈ D) and stochastic (ω ∈ Ω) variables in a(x, ω) via Karhúnen-Loève or Legendre expansions of the diffusion coefficient. Deterministic, approximate solvers are obtained by projection of this problem into a product probability space of finite dimension M and sparse discretizations of the resulting M-dimensional parametric problem. Both Galerkin and collocation approximations are considered. Under regularity assumptions on the fluctuation of a(x, ω) in the deterministic variable x, the convergence rate of the deterministic solution algorithm is analysed in terms of the number N of deterministic problems to be solved as both the chaos dimension M and the multiresolution level of the sparse discretization resp. the polynomial degree of the chaos expansion increase simultaneousl