Polynomial chaos

Polynomial chaos (PC), also called polynomial chaos expansion (PCE) and 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.[1]

PCE was first introduced in 1938 by Norbert Wiener using Hermite polynomials to model stochastic processes with Gaussian random variables.[2] It was introduced to the physics and engineering community by R. Ghanem and P. D. Spanos in 1991[3] and generalized to other orthogonal polynomial families by D. Xiu and G. E. Karniadakis in 2002.[4] Mathematically rigorous proofs of existence and convergence of generalized PCE were given by O. G. Ernst and coworkers in 2011.[5]

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[3] and as a surrogate model[6] to facilitate uncertainty quantification analyses.

Main principles

Polynomial chaos expansion (PCE) provides a way to represent a random variable with finite variance (i.e., ) as a function of an -dimensional random vector , using a polynomial basis that is orthogonal to the distribution of this random vector. The prototypical PCE can be written as:

In this expression, is a coefficient and denotes a polynomial basis function. Depending on the distribution of , different PCE types are distinguished.

Hermite polynomial chaos

The original PCE formulation used by Norbert Wiener[2] was limited to the case where is a random vector with a Gaussian distribution. Considering only the one-dimensional case (i.e., and ), the polynomial basis function orthogonal w.r.t. the Gaussian distribution are the set of -th degree Hermite polynomials . The PCE of can then be written as:

.

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 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.[7] 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),[8] 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.[9]

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.[10] 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.

Bayesian polynomial chaos

In a non-intrusive setting, the estimation of the expansion coefficients for a given set of basis functions can considered a Bayesian regression problem by constructing a surrogate model. This approach has benefits in that analytical expressions for the data evidence (in the sense of Bayesian inference) as well as the uncertainty of the expansion coefficients are available.[11] The evidence then can be used as a measure for the selection of expansion terms and pruning of the series (see also Bayesian model comparison). The uncertainty of the expansion coefficients can be used to assess the quality and trustworthiness of the PCE, and furthermore the impact of this assessment on the actual quantity of interest .

Let be a set of pairs of input-output data that is used to estimate the expansion coefficients . Let be the data matrix with elements , let be the set of output data written in vector form, and let be the set of expansion coefficients in vector form. Under the assumption that the uncertainty of the PCE is of Gaussian type with unknown variance and a scale-invariant prior, the expectation value for the expansion coefficients is

With , then the covariance of the coefficients is[11]

where is the minimal misfit and is the identity matrix. The uncertainty of the estimate for the coefficient is then given by .Thus the uncertainty of the estimate for expansion coefficients can be obtained with simple vector-matrix multiplications. For a given input propability density function , it was shown the second moment for the quantity of interest then simply is[11]

This equation amounts the matrix-vector multiplications above plus the marginalization with respect to . The first term determines the primary uncertainty of the quantity of interest , as obtained based on the PCE used as a surrogate. The second term constitutes an additional inferential uncertainty (often of mixed aleatoric-epistemic type) in the quantity of interest that is due to a finite uncertainty of the PCE.[11] If enough data is available, in terms of quality and quantity, it can be shown that becomes negligibly small and becomes small [11] This can be judged by simply building the ratios of the two terms, e.g. .This ratio quantifies the amount of the PCE's own uncertainty in the total uncertainty and is in the interval . E.g., if , then half of the uncertainty stems from the PCE itself, and actions to improve the PCE can be taken or gather more data. If, then the PCE's uncertainty is low and the PCE may be deemed trustworthy.

In a Bayesian surrogate model selection, the probability for a particular surrogate model, i.e. a particular set of expansion coefficients and basis functions , is given by the evidence of the data ,

where is the Gamma-function, is the determinant of , is the number of data, and is the solid angle in dimensions, where is the number of terms in the PCE.

Analogous findings can be transferred to the computation of PCE-based sensitivity indices . Similar results can be obtained for Kriging.[11]

See also

References

  1. The use of the word "chaos" by Norbert Wiener in his 1938 publication precedes the use of "chaos" in the branch of mathematics called chaos theory by almost 40 years.
  2. Wiener, Norbert (1938). "The Homogeneous Chaos". American Journal of Mathematics. 60 (4): 897–936. doi:10.2307/2371268. JSTOR 2371268.
  3. Ghanem, Roger G.; Spanos, Pol D. (1991), "Stochastic Finite Element Method: Response Statistics", Stochastic Finite Elements: A Spectral Approach, New York, NY: Springer New York, pp. 101–119, doi:10.1007/978-1-4612-3094-6_4, ISBN 978-1-4612-7795-8, retrieved 2021-09-29
  4. Xiu, Dongbin; Karniadakis, George Em (2002). "The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations". SIAM Journal on Scientific Computing. 24 (2): 619–644. Bibcode:2002SJSC...24..619X. doi:10.1137/s1064827501387826. ISSN 1064-8275. S2CID 10358251.
  5. Ernst, Oliver G.; Mugler, Antje; Starkloff, Hans-Jörg; Ullmann, Elisabeth (2011-10-12). "On the convergence of generalized polynomial chaos expansions". ESAIM: Mathematical Modelling and Numerical Analysis. 46 (2): 317–339. doi:10.1051/m2an/2011045. ISSN 0764-583X.
  6. Soize, Christian; Ghanem, Roger (2004). "Physical Systems with Random Uncertainties: Chaos Representations with Arbitrary Probability Measure". SIAM Journal on Scientific Computing. 26 (2): 395–410. Bibcode:2004SJSC...26..395S. doi:10.1137/s1064827503424505. ISSN 1064-8275. S2CID 39569403.
  7. Enstedt, Mattias; Wellander, Niklas (2016). "Uncertainty Quantification of Radio Propagation Using Polynomial Chaos" (PDF). Progress in Electromagnetics Research M. 50: 205–213. doi:10.2528/PIERM16062101.
  8. Oladyshkin, S.; Nowak, W. (2012). "Data-driven uncertainty quantification using the arbitrary polynomial chaos expansion". Reliability Engineering & System Safety. 106: 179–190. doi:10.1016/j.ress.2012.05.002.
  9. Oladyshkin, Sergey; Nowak, Wolfgang (2018). "Incomplete statistical information limits the utility of high-order polynomial chaos expansions". Reliability Engineering & System Safety. 169: 137–148. doi:10.1016/j.ress.2017.08.010.
  10. Alpay, Daniel; Kipnis, Alon (2015). "Wiener Chaos Approach to Optimal Prediction". Numerical Functional Analysis and Optimization. 36 (10): 1286–1306. doi:10.1080/01630563.2015.1065273. S2CID 54744829.
  11. Ranftl, Sascha; von der Linden, Wolfgang (2021-11-13). "Bayesian Surrogate Analysis and Uncertainty Propagation". Physical Sciences Forum. 3 (1): 6. doi:10.3390/psf2021003006. ISSN 2673-9984.
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