SeAr PC: Sensitivity Enhanced Arbitrary Polynomial Chaos

TL;DR

Uncertainty quantification in high-dimensional spaces is hindered by the curse of dimensionality in polynomial chaos expansions. The paper introduces Sensitivity Enhanced Arbitrary Polynomial Chaos (SeAr-PC), which augments arbitrary Polynomial Chaos with adjoint-derived sensitivity information and employs a D-optimal, coherence-based design of experiments to dramatically reduce the number of required model evaluations. This yields substantial scaling benefits (e.g., decoupling first-order sampling from $n_u$ and, for $p\ge2$, scaling as $n_u^{p-1}$) and extends PCE applicability to multi-modal and fat-tailed distributions beyond the Askey scheme, demonstrated on synthetic high-dimensional tests and a 306-parameter topology-optimization FE problem. Sobol indices derived from the SeAr-PC surrogates enable spatial visualization of uncertainty contributions, supporting robust design and decision making. The approach offers a practical path to deploying PCE-based UQ in industrial contexts such as topology optimization and additive manufacturing.

Abstract

This paper presents a method for performing Uncertainty Quantification in high-dimensional uncertain spaces by combining arbitrary polynomial chaos with a recently proposed scheme for sensitivity enhancement (1). Including available sensitivity information offers a way to mitigate the curse of dimensionality in Polynomial Chaos Expansions (PCEs). Coupling the sensitivity enhancement to arbitrary Polynomial Chaos allows the formulation to be extended to a wide range of stochastic processes, including multi-modal, fat-tailed, and truncated probability distributions. In so doing, this work addresses two of the barriers to widespread industrial application of PCEs. The method is demonstrated for a number of synthetic test cases, including an uncertainty analysis of a Finite Element structure, determined using Topology Optimisation, with 306 uncertain inputs. We demonstrate that by exploiting sensitivity information, PCEs can feasibly be applied to such problems and through the Sobol sensitivity indices, can allow a designer to easily visualise the spatial distribution of the contributions to uncertainty in the structure.

Figures (10)

• Figure 1: Minimum number of model evaluations required as a function of $n_u$ and $p$ for sensitivity enhanced PC, weighted least squares, and Smolyak sampling grid.
• Figure 2: Illustration of the D-optimal sampling points and corresponding diagonal entries in $|R_a|$ for a set of independent Gaussian distributions (top) and Gaussian mixture (bottom).
• Figure 3: Histogram of uni-variate Monte Carlo samples from the multi-modal joint density (top left) with the corresponding histogram found by Monte Carlo sampling for the cubic test function (top right) and non-linear function (bottom). Also indicated are the kernel density estimate of this distribution, together with those of SeAr PC for increasing polynomial order.
• Figure 4: Convergence of the first two statistical moments (top) and KS distance (bottom) with order/number of evaluations for the generalized extreme value test case.
• Figure 5: Histogram of uni-variate Monte Carlo samples from the generalized extreme value joint density (top left), with the corresponding histogram found by Monte Carlo sampling for the cubic test function (top right) and non-linear function (bottom). Also indicated are the kernel density estimate of this distribution, together with those of SeAr PC for increasing polynomial order.