Stochastic Estimation via Polynomial Chaos
Interim rept. 20 Apr-7 Aug 2015
AIR FORCE RESEARCH LAB EGLIN AFB FL MUNITIONS DIR/ORDNANCE DIVISION
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This expository report discusses fundamental aspects of the polynomial chaos method for representing the properties of second order stochastic processes. As originally developed by Norbert Weiner, a polynomial chaos represents key properties of a stochastic process through the application of finite series of orthogonal polynomials. The attendant polynomial expansion is used to describe the statistical properties of a stochastic process based upon an input uncertainty. The statistics of a random process is given by evaluating the appropriate polynomial chaos for an input uncertainty represented by one or more random variables. An evolved application of this idea applies a polynomial chaos to represent uncertainties in boundary or initial conditions for partial differential equations. Here, the elementary theory of the polynomial chaos is presented followed by the details of a number of example calculations where the statistical mean and standard deviation are compared against exact solutions. The Legendre chaos is described in some detail for uniformly distributed input random variables. Also, the Hermite chaos is discussed.
- Numerical Mathematics
- Statistics and Probability