Convolved Orthogonal Expansions on Dynamics and Random Media Problems

Abstract eng:
Conventional probabilistic and reliability methods have been prevalently formulated with a random variable representation of uncertainties for both inputs and outputs through physical, chemical, or biochemical systems. Due to exponential increase of random dimensions in nonlinear propagation of uncertainties, the rv-based probabilistic methods are by nature subjected to the curse-of-dimensionality issue. In (Xu, 2011; Xu and Stefanou, submitted), by constructing an orthogonal functional space about an underlying random process, a novel random field/process based convolved orthogonal expansion (COE) method is proposed, which opens a new direction to break the curse-of-dimensionality in stochastic computation. To model nonlinear systems, an n-th order convolved orthogonal expansion (COE) is formulated as u(t,ϑ)=∑_n=0∑_i=0u_i⁽ⁿ⁾(t)∅_i⁽ⁿ⁾(t,ϑ) ∅_i⁽ⁿ⁾(t,ϑ)=g*g*...*g*∅_i=g^*n*∅_i where g(t,t') is a given kernel, typically a Green's function. The COE method is applied to temporal and spatial uncertainty problems, both of which involve nonlinear propagation of a random process/field input.

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