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SIAM Annual Meeting 2013

My presentation at the SIAM Annual Meeting during the session "Sensitivity Analysis and Uncertainty Quantification in Chaotic Systems."
by

Paul Constantine

on 11 August 2013

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Transcript of SIAM Annual Meeting 2013

How well does polynomial chaos model chaos?
Paul Constantine (Stanford -> Colorado School of Mines)
Qiqi Wang (MIT)
There is nothing
chaotic*
about
polynomial chaos
.
"Of all the forms of chaos occurring in physics, there is
only one class which has been studied with anything approaching completeness. This is the class of types of chaos connected with the theory of Brownian motion."
"Publish the same results several times."
- Gian-Carlo Rota, Ten Things I Wish I Had Been Taught (1997)
Wiener Chaos
"The homogeneous chaos." (1938)
Connecting harmonic analysis and stochastic processes.
"The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals." (1947)
Square-integrable functionals of Brownian motion admit a convergent series of Hermite polynomials of Gaussian random variables. (But like, infinity to the infinity power.)
"Stochastic Finite Elements: A Spectral Approach." (1990)
If the input of a physics (structural mechanics) model is a stochastic process, and the output is square integrable, then the Cameron-Martin theory applies!
"... approaches transform the original stochastic problem into a deterministic one with a large dimensional parameter, ..."
- Babuska, Tempone, & Zouraris "Galerkin finite element approximations to stochastic elliptic partial differential equations." (2004)
"The Wiener-Askey polynomial chaos for stochastic differential equations." (2002)
The connection between Gaussian measures and Hermite polynomials extends to other measures and associated orthogonal polynomials. (The GENERALIZED polynomial chaos or gPC.)
(1) Represent uncertain model inputs with a set of parameters.
(2) Approximate model output by a polynomial of parameters.
Norbert Wiener
Cameron & Martin
Ghanem & Spanos
Xiu & Karniadakis
Polynomial Approximation
Chaos & Polynomials
Lorenz system
Least Squares Formulation
Given:
Minimize
Subject to
It's
all*
about smoothness!
THANKS!

Wang, Gomez, Blonigan, Alastair, Qian. "Towards scalable parallel-in-time turbulent flow simulations." arxiv.org/abs/1211.2437


@DrPaulynomial
www.stanford.edu/~paulcon
www.simulationinformatics.com
Full transcript