A dynamically adaptive sparse grids method for quasi-optimal interpolation of multidimensional functions

Abstract

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.

Approximation and Uncertainty Quantification of Systems with Arbitrary Parameter Distributions Using Weighted Leja Interpolation

“…Compared to isotropic schemes, anisotropic interpolations typically result in great computational savings for the same approximation accuracy, or, equivalently, in greater interpolation accuracy for an equal cost. Since, in most cases, the parameter anisotropy cannot be a priori estimated in an accurate way [46], anisotropic interpolations are usually constructed using greedy, adaptive algorithms [18,47,48]. For the particular case of Leja nodes, such an algorithm is discussed in Section 2.4.2.…”

“…However, their performance decreases rapidly for an increasing number of input parameters due to the curse of dimensionality [15]. To overcome this bottleneck, state-of-the-art approaches rely on algorithms for the adaptive construction of sparse approximations [8][9][10][16][17][18].Adaptive sparse approximation algorithms for Lagrange interpolation methods typically employ nested sequences of univariate interpolation nodes. While not strictly necessary [19], nested node sequences are helpful for the efficient construction of sparse grid interpolation algorithms.…”

“…However, their performance decreases rapidly for an increasing number of input parameters due to the curse of dimensionality [15]. To overcome this bottleneck, state-of-the-art approaches rely on algorithms for the adaptive construction of sparse approximations [8][9][10][16][17][18].…”

Engine power measurement without external load based on vibration signal

“…A smooth curve is constructed using ordered value points so that the curve passes sequentially through the sampling points. There are many methods for curve interpolation, such as polynomial curve interpolation, cubic Hermit interpolation, cubic parametric spline interpolation, multinode spline interpolation, algebraic interpolation, circular spline interpolation, and quasi-interpolation [16][17][18]. The adaptive cubic B-spline curve interpolation based on error control can adjust the number of interpolation points according to a preset error [19].…”

Ensemble Grouping Strategies for Embedded Stochastic Collocation Methods Applied to Anisotropic Diffusion Problems

“…Accordingly, numerous uncertainty quantification (UQ) methods have been developed and studied in the literature, including random sampling [19,29,37,38,39], stochastic collocation [1,41,40,55], and stochastic Galerkin [23,24,56], with an emphasis on applying UQ methods to problems relevant to large-scale scientific computing. Frequently these problems exhibit highdimensional uncertain input spaces and localized or nonsmooth behavior, which has motivated research on reducing the number of samples needed, e.g., locally adaptive sampling methods [21,27,52], multilevel methods that exploit a hierarchy of physical and temporal discretizations [5,6,7,13,25], and methods that attempt to construct minimal or optimal uncertainty representations such as compressed sensing [16,36] and tensor methods [1,2,3,14,20,22,41,40,48,55].…”