Adaptive Sparse Polynomial Chaos Expansions via Leja Interpolation

Abstract

This work suggests an interpolation-based stochastic collocation method for the non-intrusive and adaptive construction of sparse polynomial chaos expansions (PCEs). Unlike pseudospectral projection and regression-based stochastic collocation methods, the proposed approach results in PCEs featuring one polynomial term per collocation point. Moreover, the resulting PCEs are interpolating, i.e., they are exact on the interpolation nodes/collocation points. Once available, an interpolating PCE can be used as an inexpensive surrogate model, or be postprocessed for the purposes of uncertainty quantification and sensitivity analysis. The main idea is conceptually simple and relies on the use of Leja sequence points as interpolation nodes. Using Newton-like, hierarchical basis polynomials defined upon Leja sequences, a sparse-grid interpolation can be derived, the basis polynomials of which are unique in terms of their multivariate degrees. A dimension-adaptive scheme can be employed for the construction of an anisotropic interpolation. Due to the degree uniqueness, a one-to-one transform to orthogonal polynomials of the exact same degrees is possible and shall result in an interpolating PCE. However, since each Leja node defines a unique Newton basis polynomial, an implicit one-to-one map between Leja nodes and orthogonal basis polynomials exists as well. Therefore, the in-between steps of hierarchical interpolation and basis transform can be discarded altogether, and the interpolating PCE can be computed directly. For directly computed, adaptive, anisotropic interpolating PCEs, the dimension-adaptive algorithm is modified accordingly. A series of numerical experiments verify the suggested approach in both low and moderately high-dimensional settings, as well as for various input distributions.

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

“…All models feature multiple input parameters, hence, we use adaptive algorithms resulting in sparse approximations. The dimension-adaptive Algorithm 1 based on weighted Leja nodes is implemented in our in-house developed DALI (Dimension-Adaptive Leja Interpolation) software [14,37]. The sparse gPC approximations are constructed with a well-established, degree-adaptive algorithm based on least angle regression (LAR) [9] and implemented in the UQLab [50] software.…”

“…Computationally efficient alternatives are spectral UQ methods [3,4], which approximate the functional dependence of a system's quantity of interest (QoI) on its input parameters. The most popular black-box methods of this category employ global polynomial approximations by aid of either Lagrange interpolation schemes [5,6] or generalized polynomial chaos (gPC) [7], where the latter approach is typically based on least squares (LS) regression [8][9][10] or pseudo-spectral projection methods [11][12][13], or, less commonly, on interpolation [14]. Assuming a smooth input-output dependence, spectral UQ methods provide fast convergence rates, in some cases even of exponential order.…”

On the influence of over-parameterization in manifold based surrogates and deep neural operators

“…Several approaches have been proposed for computing the coefficients c s , including pseudospectral projection [39][40][41], interpolation [42,43], and, most commonly, regression [44][45][46][47][48][49][50]. We employ a regression approach in which the PCE coefficients are determined by solving the penalized least squares problem [51] arg min…”

An $hp$-adaptive multi-element stochastic collocation method for surrogate modeling with information re-use

“…Therefore, it is typically beneficial to construct so-called sparse PCEs [59][60][61][62][63] by neglecting non-influential basis terms. Adaptive basis expansion algorithms are often used for that purpose [64][65][66][67][68][69] .…”