Continuous dependence estimates for the ergodic problem of Bellman equation with an application to the rate of convergence for the homogenization problem

This paper is devoted to establish continuous dependence estimates for the ergodic problem for Bellman operators (namely, estimates of (v_1-v_2) where v_1 and v_2 solve two equations with different coefficients). We shall obtain an estimate of ||v_1-v_2||_\infty with an explicit dependence on the L^\infty-distance between the coefficients and an explicit characterization of the constants and also, under some regularity conditions, an estimate of ||v_1-v_2||_{C^2(\R^n)}. Afterwards, the former result will be crucial in the estimate of the rate of convergence for the homogenization of Bellman equations. In some regular cases, we shall obtain the same rate of convergence established in the monographs [11,26] for regular linear problems

Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients

Elliptic partial differential equations with diffusion coefficients of lognormal form, that is $a=exp(b)$, where $b$ is a Gaussian random field, are considered. We study the $\ell^p$ summability properties of the Hermite polynomial expansion of the solution in terms of the countably many scalar parameters appearing in a given representation of $b$. These summability results have direct consequences on the approximation rates of best $n$-term truncated Hermite expansions. Our results significantly improve on the state of the art estimates available for this problem. In particular, they take into account the support properties of the basis functions involved in the representation of $b$, in addition to the size of these functions. One interesting conclusion from our analysis is that in certain relevant cases, the Karhunen-Lo\`eve representation of $b$ may not be the best choice concerning the resulting sparsity and approximability of the Hermite expansion