Chaos expansions

Polynomial chaos, generalized polynomial chaos, arbitrary chaos etc

May 21, 2020 — February 15, 2021

Figure 1

Placeholder, for a topic which has a slightly confusing name. To explore: Connection to/difference from other methods of keeping track of evolution of uncertainty in dynamical systems. C&C Gaussian process regression as used in Gratiet, Marelli, and Sudret (2016), functional data analysis etc.

This is not the same thing as chaos in the sense of the deterministic chaos made famous by dynamical systems theory and fractal t-shirts. Different term

1 Polynomial chaos expansion

Ground Zero, the original famous one. Wikipedia has an inscrutable introduction which I can make no sense of.

Popular introductions to the concept seem to be (R. Ghanem and Spanos 1990; R. G. Ghanem and Spanos 2003; Witteveen and Bijl 2006; Xiu and Karniadakis 2002; Kim et al. 2013). For my money, the most direct is O’Hagan (2013) which runs thusly: We define an inner product \[ \left\langle \psi, \phi\right\rangle=\int \psi(\xi) \phi(\xi) p_{\xi}(\xi) d \xi \] with respect to the probability density function \(p_{\xi}\) of some random variate \(\Xi\) which we call the germ.

Suppose we have chosen a functional basis comprising polynomials \(\psi_{0}=1, \psi_{1}, \psi_{2}, \ldots,\) where \(\psi_{j}\) is a polynomial of order \(j\) and where they satisfy the orthogonality condition that for all \(j \neq k\) \[ \left\langle\psi_{j}, \psi_{k}\right\rangle=0. \]

We write \[ X=f(\Xi)=\sum_{j=0}^{\infty} x_{j} \psi_{j}(\Xi) \] The combination of \(x_{j}\) (the mode strength) and \(\psi_{j}\) (the mode function) is called the \(j\) -th mode. By orthogonality, given \(f\) and the \(\psi_{j}\) s there is a unique expansion in which the mode strengths are given by \[ x_{j}=\langle f, \psi j\rangle /\left\langle\psi_{j}, \psi_{j}\right\rangle \] An expansion of \(X\) in this form is called a polynomial chaos expansion. Since there are many possible functions \(f\) for given \(X\) and \(\Xi\) distributions, there are many possible expansions of a given \(X\) using a given germ. They will differ in the mode strengths.

In practice we truncate the expansions to a finite number \(p\) of terms, \[ X_{p}=f_{p}(\Xi)=\sum_{j=0}^{p} x_{j} \psi_{j}(\Xi). \]

C&C Karhunen—Loève expansion.

It is not clear immediately, but this gives us a tool to track propagation of error through a model.

There is a veritable zoo of bases to consider.

Here associated with means orthogonal with respect to.

2 “Generalized” chaos expansion

Wikipedia credits Xiu (2010) with the particular generalisation which apparently got naming rights for generalized chaos expansion, in the teeth of my private campaign for a moratorium on naming anything generalized [whatever]. I think it is about expanding the list of acceptable polynomial bases? TBC.

3 Arbitrary chaos expansion

Learnable sparse basis-style chaos expansions. See (Witteveen and Bijl 2006; Zhang et al. 2019; Lei et al. 2018; Oladyshkin and Nowak 2012; Wan and Karniadakis 2006; Zheng, Wan, and Karniadakis 2015).

As far as I can tell from brusque mentions about the place, these methods construct polynomial basis expansions weighted by the empirical distribution of observationss. That is, instead of taking a given germ, we use an empirical estimate of a germ and calculate a basis over it by the Gram-Schmidt procedure.

4 References

Alexanderian. 2015. A Brief Note on the Karhunen-Loève Expansion.” arXiv:1509.07526 [Math].
Alpay, and Kipnis. 2015. Wiener Chaos Approach to Optimal Prediction.” Numerical Functional Analysis and Optimization.
Calatayud Gregori, Chen-Charpentier, Cortés López, et al. 2019. Combining Polynomial Chaos Expansions and the Random Variable Transformation Technique to Approximate the Density Function of Stochastic Problems, Including Some Epidemiological Models.” Symmetry.
Cameron, and Martin. 1944. Transformations of Weiner Integrals Under Translations.” The Annals of Mathematics.
Franceschini, and Giardinà. 2017. Stochastic Duality and Orthogonal Polynomials.” arXiv:1701.09115 [Math-Ph].
Ghanem, Roger, and Red-Horse. 2017. Polynomial Chaos: Modeling, Estimation, and Approximation.” In Handbook of Uncertainty Quantification.
Ghanem, Roger, and Spanos. 1990. Polynomial Chaos in Stochastic Finite Elements.” Journal of Applied Mechanics.
Ghanem, Roger G., and Spanos. 2003. Stochastic Finite Elements: A Spectral Approach.
Gratiet, Marelli, and Sudret. 2016. Metamodel-Based Sensitivity Analysis: Polynomial Chaos Expansions and Gaussian Processes.” In Handbook of Uncertainty Quantification.
Kim, and Braatz. 2013. Generalised Polynomial Chaos Expansion Approaches to Approximate Stochastic Model Predictive Control .” International Journal of Control.
Kim, Shen, Nagy, et al. 2013. Wiener’s Polynomial Chaos for the Analysis and Control of Nonlinear Dynamical Systems with Probabilistic Uncertainties [Historical Perspectives].” IEEE Control Systems Magazine.
Lei, Li, Gao, et al. 2018. A Data-Driven Framework for Sparsity-Enhanced Surrogates with Arbitrary Mutually Dependent Randomness.”
Levajkovic, and Selesi. 2011. Chaos Expansion Methods for Stochastic Differential Equations Involving the Malliavin Derivative, Part I.” Publications de l’Institut Mathematique.
Luo. 2006. Wiener Chaos Expansion and Numerical Solutions of Stochastic Partial Differential Equations.”
Noorshams, and Wainwright. 2013. “Belief Propagation for Continuous State Spaces: Stochastic Message-Passing with Quantitative Guarantees.” The Journal of Machine Learning Research.
Nualart, and Schoutens. 2000. Chaotic and Predictable Representations for Lévy Processes.” Stochastic Processes and Their Applications.
O’Hagan. 2013. “Polynomial Chaos: A Tutorial and Critique from a Statistician’s Perspective.”
Oladyshkin, and Nowak. 2012. Data-Driven Uncertainty Quantification Using the Arbitrary Polynomial Chaos Expansion.” Reliability Engineering & System Safety.
Rahman. 2017. Wiener-Hermite Polynomial Expansion for Multivariate Gaussian Probability Measures.” arXiv:1704.07912 [Math].
Schoutens. 2000. Stochastic Processes and Orthogonal Polynomials. Lecture Notes in Statistics.
———. 2001. Orthogonal Polynomials in Stein’s Method.” Journal of Mathematical Analysis and Applications.
Wan, and Karniadakis. 2006. Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures.” SIAM Journal on Scientific Computing.
Wiener. 1938. The Homogeneous Chaos.” American Journal of Mathematics.
Witteveen, and Bijl. 2006. Modeling Arbitrary Uncertainties Using Gram-Schmidt Polynomial Chaos.” In 44th AIAA Aerospace Sciences Meeting and Exhibit.
Xiu. 2010. Numerical Methods for Stochastic Computations: A Spectral Method Approach.
Xiu, and Hesthaven. 2005. High-Order Collocation Methods for Differential Equations with Random Inputs.” SIAM Journal on Scientific Computing.
Xiu, and Karniadakis. 2002. The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations.” SIAM Journal on Scientific Computing.
Zhang, Lu, Guo, et al. 2019. Quantifying Total Uncertainty in Physics-Informed Neural Networks for Solving Forward and Inverse Stochastic Problems.” Journal of Computational Physics.
Zheng, Wan, and Karniadakis. 2015. Adaptive Multi-Element Polynomial Chaos with Discrete Measure: Algorithms and Application to SPDEs.” Applied Numerical Mathematics.