Polynomial chaos

Polynomial chaos (PC), also called Polynomial chaos expansion (PCE) or Wiener chaos expansion, is a method for representing a random variable in terms of a polynomial function of other random variables. The polynomials are chosen to be orthogonal with respect to the joint probability distribution of these random variables. PCE can be used, e.g., to determine the evolution of uncertainty in a dynamical system when there is probabilistic uncertainty in the system parameters. Note that despite its name, PCE has no immediate connections to chaos theory.

PCE was first introduced in 1938 by Norbert Wiener using Hermite polynomials to model stochastic processes with Gaussian random variables. It was introduced to the physics and engineering community by R. Ghanem and P. D. Spanos in 1991 and generalized to other orthogonal polynomial families by D. Xiu and G. E. Karniadakis in 2002. Mathematically rigorous proofs of existence and convergence of generalized PCE were given by O. G. Ernst and coworkers in 2012.

PCE has found widespread use in engineering and the applied sciences because it makes it possible to efficiently deal with probabilistic uncertainty in the parameters of a system. It is widely used in stochastic finite element analysis and as a surrogate model to facilitate uncertainty quantification analyses.

Main principles

Polynomial chaos expansion (PCE) provides a way to represent a random variable $Y$ with finite variance (i.e., $\operatorname {Var} (Y)<\infty$) as a function of an $M$-dimensional random vector $\mathbf {X}$, using a polynomial basis that is orthogonal to the distribution of this random vector. The prototypical PCE can be written as:

$Y=\sum _{i\in \mathbb {N} }c_{i}\Psi _{i}(\mathbf {X} ).$

In this expression, $c_{i}$ is a coefficient and $\Psi _{i}$ denotes a polynomial basis function. Depending on the distribution of $\mathbf {X}$, different PCE types are distinguished.

Generalized polynomial chaos

Xiu (in his PhD under Karniadakis at Brown University) generalized the result of Cameron–Martin to various continuous and discrete distributions using orthogonal polynomials from the so-called Askey-scheme and demonstrated $L_{2}$ convergence in the corresponding Hilbert functional space. This is popularly known as the generalized polynomial chaos (gPC) framework. The gPC framework has been applied to applications including stochastic fluid dynamics, stochastic finite elements, solid mechanics, nonlinear estimation, the evaluation of finite word-length effects in non-linear fixed-point digital systems and probabilistic robust control. It has been demonstrated that gPC based methods are computationally superior to Monte-Carlo based methods in a number of applications. However, the method has a notable limitation. For large numbers of random variables, polynomial chaos becomes very computationally expensive and Monte-Carlo methods are typically more feasible .

Arbitrary polynomial chaos

Recently chaos expansion received a generalization towards the arbitrary polynomial chaos expansion (aPC), which is a so-called data-driven generalization of the PC. Like all polynomial chaos expansion techniques, aPC approximates the dependence of simulation model output on model parameters by expansion in an orthogonal polynomial basis. The aPC generalizes chaos expansion techniques towards arbitrary distributions with arbitrary probability measures, which can be either discrete, continuous, or discretized continuous and can be specified either analytically (as probability density/cumulative distribution functions), numerically as histogram or as raw data sets. The aPC at finite expansion order only demands the existence of a finite number of moments and does not require the complete knowledge or even existence of a probability density function. This avoids the necessity to assign parametric probability distributions that are not sufficiently supported by limited available data. Alternatively, it allows modellers to choose freely of technical constraints the shapes of their statistical assumptions. Investigations indicate that the aPC shows an exponential convergence rate and converges faster than classical polynomial chaos expansion techniques. Yet these techniques are in progress but the impact of them on CFD models is quite impressionable.

Polynomial chaos & incomplete statistical information

In many practical situations, only incomplete and inaccurate statistical knowledge on uncertain input parameters are available. Fortunately, to construct a finite-order expansion, only some partial information on the probability measure is required that can be simply represented by a finite number of statistical moments. Any order of expansion is only justified if accompanied by reliable statistical information on input data. Thus, incomplete statistical information limits the utility of high-order polynomial chaos expansions.

Polynomial chaos & non-linear prediction

Polynomial chaos can be utilized in the prediction of non-linear functionals of Gaussian stationary increment processes conditioned on their past realizations. Specifically, such prediction is obtained by deriving the chaos expansion of the functional with respect to a special basis for the Gaussian Hilbert space generated by the process that with the property that each basis element is either measurable or independent with respect to the given samples. For example, this approach leads to an easy prediction formula for the Fractional Brownian motion.

See also

• Orthogonal polynomials
• Surrogate model
• Variance-based sensitivity analysis
• Karhunen–Loève theorem
• Hilbert space
• Proper orthogonal decomposition