Contents

Polynomial ChaosEdit

Polynomial Chaos is a mathematical framework for representing and propagating uncertainty in complex systems. By expressing random inputs and responses as series of orthogonal polynomials, this approach enables efficient analysis of how variability in parameters or governing conditions affects model outputs. From its origins in probabilistic analysis to modern applications in engineering, physics, and finance, polynomial chaos provides a structured way to build surrogate models, quantify risk, and perform sensitivity studies without resorting to brute-force sampling.

The method is particularly powerful for problems described by differential or algebraic equations where inputs are uncertain but statistically characterized. It complements more traditional techniques such as Monte Carlo methods by turning a stochastic problem into a deterministic one in a higher-dimensional but structured space of polynomial coefficients. The resulting surrogates can yield moments, probability distributions, and error bounds with far fewer forward evaluations than naive sampling.

Core concepts

Random variables and distributions

Polynomial chaos relies on representing uncertain inputs as a collection of random variables with known probability distributions. When these inputs are independent, one can construct a corresponding family of orthogonal polynomials. See random variable and probability distribution for foundational notions.

Orthogonal polynomials

The choice of polynomials is tied to the input distributions. Classic families include Hermite polynomials for Gaussian inputs, Legendre polynomials for uniform inputs, and other families linked by the Askey scheme. The orthogonality of these polynomials with respect to the input distributions underpins how coefficients are determined and how moments of the response are recovered.

Polynomial chaos expansion

The central idea is to approximate the model output Y as a finite sum of orthogonal polynomial basis functions of the input random variables: Y ≈ Σ a_i Φ_i(ξ), where Φ_i are orthogonal polynomials and a_i are coefficients to be determined. This expansion is the polynomial chaos expansion, also called a type of surrogate model for the system. See polynomial chaos expansion for details.

Generalized polynomial chaos

Generalized polynomial chaos (gPC) extends the classic approach to arbitrary, possibly non-Gaussian inputs by using appropriate orthogonal polynomials for each distribution. This generalization broadens applicability beyond Gaussian assumptions. See Generalized polynomial chaos.

Types of polynomial chaos

Wiener chaos and Hermite polynomials

When inputs are modeled with Gaussian distributions, Hermite polynomials provide an natural basis. This setting is often referred to as Wiener chaos in stochastic analysis, connecting with a long tradition of expanding functionals of Gaussian processes. See Hermite polynomials and Wiener chaos.

Generalized polynomial chaos (gPC)

For non-Gaussian uncertainties, gPC chooses a different orthogonal polynomial family corresponding to the input distribution. The construction remains similar, but the basis aligns with the statistics of the inputs. See Generalized polynomial chaos and orthogonal polynomials.

Arbitrary distributions and the Askey scheme

Beyond the standard Gaussian or uniform cases, distributions from the Askey scheme can be accommodated by selecting the matching orthogonal polynomials. This allows polynomial chaos to model a wide range of uncertainty types found in engineering and science. See Askey scheme.

Numerical methods

Stochastic Galerkin (intrusive)

In the intrusive stochastic Galerkin approach, the governing equations are projected onto the polynomial chaos basis, producing a larger, deterministic system for the expansion coefficients. This method can yield fast convergence for smooth responses but requires modification of the original equations. See stochastic Galerkin.

Stochastic collocation (non-intrusive)

Stochastic collocation uses samples of the input variables and evaluates the deterministic model at those points, followed by a polynomial regression or interpolation to obtain the coefficients. It is non-intrusive, meaning it can be applied to existing models without altering their internals. See stochastic collocation and surrogate model.

Sparse and adaptive approaches

High-dimensional problems can lead to a large number of basis terms. Sparse expansions and adaptive basis selection reduce computational cost by retaining only the most influential terms. See sparse polynomial chaos and compressive sensing in the context of surrogate modeling.

Applications

Engineering and physics

Polynomial chaos is widely used to quantify uncertainty in simulations of fluid dynamics, structural mechanics, aeroelasticity, and heat transfer. It enables prediction of statistics for quantities of interest and supports design optimization under uncertainty. See uncertainty quantification and partial differential equation modeling.

Climate and earth systems

In climate modeling and geophysics, gPC-type expansions help propagate parametric uncertainty through complex climate models, providing insights into risk and range of outcomes. See climate modeling and earth system model.

Finance and risk assessment

Uncertainty representations via polynomial chaos can be applied to price paths, risk metrics, and scenario analysis in financial engineering, especially when models exhibit nonlinear dependence on uncertain factors. See financial mathematics and risk assessment.

Controversies and limitations

Curse of dimensionality

As the number of uncertain inputs grows, the number of basis terms increases combinatorially, which can erode efficiency gains. Researchers address this with sparse, adaptive, or low-rank representations and by coupling with other uncertainty techniques. See curse of dimensionality.

Dependence between inputs

The standard construction assumes independent inputs. When dependencies exist, one must either transform the inputs (e.g., through copulas) or use more sophisticated bases, which can complicate the methodology. See copula (probability theory).

Non-smooth or highly nonlinear responses

If the model response is highly non-smooth with respect to inputs, convergence of the polynomial expansion can be slow or unstable. Alternative methods or hybrid approaches may be preferable in such cases. See nonlinear systems.

Intrusive vs non-intrusive trade-offs

Intrusive methods (like stochastic Galerkin) can yield fast results but require altering the original model equations, which may be impractical for legacy codes. Non-intrusive methods (like stochastic collocation) are easier to apply but can demand more forward evaluations. See intrusive method and non-intrusive method.

See also