I need to understand the basics of "Polynomial Chaos" (http://en.wikipedia.org/wiki/Polynomial_chaos), and I'm having trouble finding a good reference on it. I'm looking for something rigorous enough to satisfy the mathematical mind, but that won't require several weeks of dwelving into the material to get the gist of it.
I borrowed a book from a friend, called "Stochastic Finite Elements: A Spectral Approach", and I confess I didn't find the explanation very clear. Could you point me to other references that might help?
Thanks in advance.
EDIT: Since asking this question (on March 2013) I have done quite a bit of reading on the subject, and the problem I'm studying is the following: Let $X_1, X_2, \ldots, X_n$ be independent and identically distributed variables with a known distribution (Gaussian, Uniform or Exponential, for instance), and let $Y = f(X_1, \ldots, X_n)$, where $f$ is an unknown function of $n$ variables (typically the result of a simulation). We wish to approximate the expectation and variance of $Y$ via Polinomial Chaos Expansions, that is, we first write $$Y = \sum_i \alpha_i P_i(X_1, \ldots, X_n),$$
where the $P_i$ are the Hermite, Legendre or Laguerre polynomials (depending on the distribution of the $X_i$). We know that $E(Y) = \alpha_0$ and $Var(Y) = \sum_{i > 0} \alpha_i^2$, hence the problem reduces to approximating the $\alpha_i$ up to a certain degree using numerical integration methods.
Here $n$ is typically between $5$ and $15$. For $n \leq 4$, Quadrature methods tend to approximate things well with few simulations ($\approx 5^4 = 625$). However, since the number of simulations grows exponentially, other methods must be used, at least for $n \geq 6$, such as Quasi Monte-Carlo Methods (e.g. low discrepancy sequences) and Spare Quadrature Methods. I have tried using the first, without much success for non-uniform distributions, but have not yet begun to study the latter.
Has anyone had experience with this so-called Curse of Dimensionality in this context? Which methods could work for approximating the coefficients in dimensions $5 \leq n \leq 15$ (although, a solution that is good enough for $5 \leq n \leq 10$ would also be considered satisfactory)?