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

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

www.simulationinformatics.com