Polynomial Chaos ExpansionEdit
Polynomial Chaos Expansion (PCE) is a mathematical framework for representing uncertain quantities as a series of orthogonal polynomials in terms of independent random inputs. Originating in the Wiener chaos expansion for Gaussian processes and extended through the Wiener–Askey scheme, PCE provides a compact, analytical surrogate for propagating input uncertainty through complex computer models. It is widely used in uncertainty quantification to transform stochastic problems into deterministic ones, enabling efficient estimation of moments, distributions, and sensitivity measures without requiring brute-force Monte Carlo simulations.
In practical terms, a quantity of interest X that depends on a set of random inputs ξ can be written as a truncated expansion X(ξ) ≈ Σ_{i∈A} c_i φ_i(ξ), where φ_i are polynomials that are orthogonal with respect to the joint probability distribution of ξ, and the index set A controls the order and dimensionality of the expansion. The choice of polynomial family is guided by the input distributions: Hermite polynomials for Gaussian inputs, Legendre polynomials for uniform inputs, and other families from the Askey scheme for common probability densities. This orthogonality makes coefficient calculation straightforward through projection, regression, or spectral collocation, and it yields analytic expressions for statistical moments and, under certain conditions, fast convergence as the number of terms grows.
Overview
Polynomial chaos expansions arise from the idea that smooth, uncertain responses can be well-approximated by a basis of polynomials. The foundational ideas connect to Wiener chaos and to the broader theory of orthogonal polynomials Orthogonal polynomials tied to a given probability measure. When the input randomness has a known distribution, the appropriate orthogonal basis is selected to maximize convergence and minimize the number of terms needed for a given accuracy. The resulting surrogate is a deterministic function of the random inputs, which can be analyzed with standard numerical tools.
A typical setup treats ξ as a vector of independent random variables with known marginal distributions. The expansion coefficients c_i encode how each polynomial mode contributes to the response X. Because the polynomials are orthogonal, the coefficients can be obtained by projecting X onto each φ_i or by solving a regression problem using data from model evaluations. The approach can be non-intrusive (treating the existing model as a black box) or intrusive (modifying the governing equations to enforce the expansion, as in stochastic Galerkin methods).
PCE is a form of surrogate modeling and shares ties with other uncertainty propagation techniques such as Stochastic Galerkin method and Stochastic Collocation; it is often used in conjunction with [ [Surrogate model] ] building to speed up repeated evaluations in optimization, reliability analysis, and design under uncertainty. Its analytic structure makes it particularly attractive for rapid evaluation of moments, probability distributions, and Sobol-type sensitivity indices.
Theory and construction
Basis selection: The philosophy is to choose φ_i to be orthogonal with respect to the joint pdf of ξ. If ξ has a Gaussian distribution and the components are independent, the natural basis is given by Hermite polynomials; for uniform inputs, Legendre polynomials are standard. Other distributions are associated with corresponding families in the Askey scheme.
Expansion form and truncation: The expansion is truncated to limit the number of terms, introducing a trade-off between accuracy and computational cost. In low to moderate dimensions, a full tensor product grid of basis functions might suffice, but in higher dimensions, practitioners turn to sparse approximations (e.g., choosing a subset of multi-index terms) to manage the growth of terms and mitigate the curse of dimensionality.
Coefficient computation: Coefficients can be obtained by projection (analytical or numerical quadrature), by regression against model evaluations, or by compressed sensing techniques when the true model is sparse in the polynomial basis. Non-intrusive approaches are particularly popular because they treat the existing model as a black box and require only input-output evaluations.
Convergence and error: For smooth model responses with respect to the inputs, PCE often exhibits spectral convergence—the error decreases rapidly as the polynomial order increases. In practice, convergence depends on regularity, dimensionality, and the choice of truncation strategy, with higher-dimensional problems benefiting from adaptive or multi-element approaches.
Construction methods and practical considerations
Non-intrusive methods: Stochastic collocation and regression-based approaches enable PCE without altering the underlying solver. Collocation uses model evaluations at selected input points to build the surrogate; regression fits coefficients from a dataset of input-output pairs.
Intrusive methods: Stochastic Galerkin methods incorporate the PCE basis directly into the governing equations, leading to a coupled deterministic system for the expansion coefficients. This can be advantageous for certain types of linear or mildly nonlinear problems but requires deeper integration with the solver.
Dimensionality management: The number of basis functions grows rapidly with the number of input variables and the expansion order. Techniques such as hyperbolic cross truncation, sparse grids, and adaptive basis selection help cope with the dimensionality, guiding where to place computational effort.
Coefficient sparsity: In many applications, the response depends strongly on a subset of the input directions. Sparse PCE techniques (including L1-regularized regression) exploit this by identifying and retaining only the most influential basis terms, improving robustness and efficiency.
Error assessment: Validation against independent model runs or cross-validation is common practice. Goodness-of-fit metrics, along with moment comparisons and, when available, distributional checks, are used to gauge surrogate accuracy.
Variants and extensions
Sparse PCE: Emphasizes parsimony by selecting a small subset of basis functions, often via regularization or greedy selection, to combat the curse of dimensionality.
Local and multi-element PCE: Divide the input space into smaller regions and fit local PCEs, improving accuracy for highly nonlinear or discontinuous responses.
Quantile and distribution-PCE: Extend the framework to capture not just moments but the full distributional features of the output, including tails and quantiles.
Polynomial chaos with dependent inputs: When inputs are not independent, transforms or copula-based approaches modify the construction to maintain orthogonality with respect to the joint distribution.
Hybrid and data-driven approaches: Combine PCE with machine learning ideas, such as kernel methods or neural nets, to exploit strengths from both probabilistic modeling and flexible function approximation.
Applications
Engineering and physics: PCE is used to propagate uncertainties in aerodynamic design, structural analysis, and thermal-fluid problems, often in conjunction with Finite element method or other simulation tools.
Reliability and risk assessment: By providing analytical access to moments and sensitivities, PCE supports quantitative reliability analysis and risk-informed decision making.
Power systems and energy: Uncertainty in loads, weather, and generation can be represented via PCE to study stability, optimization, and planning problems under uncertainty.
Climate and environmental modeling: Surrogate models built with PCE facilitate long-run simulations and scenario analysis where full-resolution simulations would be computationally prohibitive.
Validation and sensitivity analysis: By decomposing variance into contributions from individual inputs and their interactions, practitioners gain insight into system behavior and key drivers of risk.