A Dynamically Adaptive Sparse Grid Method for Quasi-Optimal Interpolation of Multidimensional Analytic Functions

In this work we develop a dynamically adaptive sparse grids (SG) method for quasi-optimal interpolation of multidimensional analytic functions defined over a product of one dimensional bounded domains. The goal of such approach is to construct an interpolant in space that corresponds to the "best $M$-terms" based on sharp a priori estimate of polynomial coefficients. In the past, SG methods have been successful in achieving this, with a traditional construction that relies on the solution to a Knapsack problem: only the most profitable hierarchical surpluses are added to the SG. However, this approach requires additional sharp estimates related to the size of the analytic region and the norm of the interpolation operator, i.e., the Lebesgue constant. Instead, we present an iterative SG procedure that adaptively refines an estimate of the region and accounts for the effects of the Lebesgue constant. Our approach does not require any a priori knowledge of the analyticity or operator norm, is easily generalized to both affine and non-affine analytic functions, and can be applied to sparse grids build from one dimensional rules with arbitrary growth of the number of nodes. In several numerical examples, we utilize our dynamically adaptive SG to interpolate quantities of interest related to the solutions of parametrized elliptic and hyperbolic PDEs, and compare the performance of our quasi-optimal interpolant to several alternative SG schemes

Multilevel Monte Carlo covariance estimation for the computation of Sobol' indices

International audienceCrude and quasi Monte Carlo (MC) sampling techniques are common tools dedicated to estimating statistics (expectation, variance, covariance) of a random quantity of interest.We focus here on the uncertainty quantification framework where the quantity of interest is the output of a numerical simulator fed with uncertain input parameters.Then, sampling the output involves running the simulator for different samples of the inputs, which may be computationally time-consuming.To reduce the cost of sampling, a first approach consists in replacing the numerical simulator by a surrogate model that is cheaper to evaluate, thus making it possible to generate more samples of the output and therefore leading to a lower sampling error.However, this approach adds to the sampling error an unavoidable model error.Another approach, which does not introduce any model error, is the so-called multilevel MC (MLMC) method.Given a sequence of levels corresponding to numerical simulators with increasing accuracy and computational cost, MLMC combines samples obtained at different levels to construct an estimator at a reduced cost compared to standard MC sampling.In this paper, we derive and analyze multilevel covariance estimators and adapt the MLMC convergence theorem in terms of the corresponding covariances and fourth order moments. We propose a multilevel algorithm driven by a target cost as an alternative to typical algorithms driven by a target accuracy.These results are used in a sensitivity analysis context in order to derive a multilevel estimation of Sobol' indices, whose building blocks can be written as covariance terms in a pick-and-freeze formulation.These contributions are successfully tested on an initial value problem with random parameters

MATHICSE Technical Report : Convergence of quasi-optimal sparse grid approximation of Hilbert-valued functions: application to random elliptic PDEs

In this work we provide a convergence analysis for the quasi-optimal version of the Stochastic Sparse Grid Collocation method we had presented in our previous work \On the optimal polynomial approximation of Stochastic PDEs by Galerkin and Collocation methods" [6]. Here the construction of a sparse grid is recast into a knapsack problem: a profit is assigned to each hi- erarchical surplus and only the most profitable ones are added to the sparse grid. The convergence rate of the sparse grid approximation error with respect to the number of points in the grid is then shown to depend on weighted summability properties of the sequence of profits. This argument is very gen- eral and can be applied to sparse grids built with any uni-variate family of points, both nested and non-nested. As an example, we apply such quasi-optimal sparse grid to the solution of a particular elliptic PDE with stochastic diffusion coefficients, namely the \inclusions problem": we detail the conver- gence estimate obtained in this case, using polynomial interpolation on either nested (Clenshaw{Curtis) or non-nested (Gauss{Legendre) abscissas, verify its sharpness numerically, and compare the performance of the resulting quasi- optimal grids with a few alternative sparse grids construction schemes recently proposed in literature