Reference on Polynomial Chaos 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)?
 A: Dongbin Xiu and George Karniadakis also published a very good summary on gPC for stochastic differential equations entitled "The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations." It is publicly available online, and summarizes the approach very well.
Of course, feel free to ask questions here; I deal with polynomial chaos quite a bit in my work.

As far as the basics go, let's just look at a single random variable with an arbitrary distribution.
We know that this r.v. has certain properties, e.g. a pdf/pmf that is non-negative and integrates/sums to one monotonically, etc.
When looking at arbitrary variables, it is a natural question ask if we can represent it by expanding about some chosen basis of some other variable whose properties we know and/or like better, and whether that expansion converges in some meaningful way.
For the random variable, call it $y$, we might want to find some expansion about a basis in terms of some other random variable, $\zeta$, whose properties are better understood.
For instance, we can very easily generate uniform and normal random variables numerically, and doing so is usually very stable and very fast -- the algorithms are ubiquitous and well-studied. So maybe we want to represent $y$ in terms of some expansion of $\zeta$.
Usually, in such expansions, we want a pairwise-orthogonal basis function on our support (typically orthogonal in $L^2$). As it turns out, when you look at the weighting functions for polynomials orthogonal in $L^2$, you find a close coupling to the structure of the probability density/mass functions of your random variables. For instance, the weighting function of the Hermite polynomials is of the form $e^{-x^2}$. This is, within a scaling, the pdf of a normal random variable!
So, based on some complicated math, we determine that as long as $y$ has finite second moments, that we can write
$$y = \sum_{i=0}^\infty y_i \Phi_i(\zeta)$$
where the $y_i$ are deterministic coefficients, and the $\Phi_i$ are orthogonal polynomials associated with the choice of r.v. $\zeta$. Convergence, in this case, means limit in the mean convergence.
As with any series expansion, it is useful most often when it can be truncated. By truncating the series expansion and applying the Galerkin method -- i.e. projecting the expansion onto each basis function and letting orthogonality annihilate mixed terms -- we can compute the $y_i$ coefficients in a straightforward manner.
Solving this system is often less computationally intensive than Monte-Carlo simulation.
For instance, if I wanted to solve the stochastic differential equation
$$x' = kx$$ for some random parameter $k$, I could inject $k$'s gPC representation into the equation, treat $x$ as a random process and perform a similar gPC expansion, and compute the coefficients $k_i$ and $x_i$. The $k_i$ coefficients are usually analytically solvable -- they come down to a scaling of a Kronecker delta (read the Xiu and Karniadakis paper) -- so the only numerical challenge is computing $x_i$.
But if we have a $P$ term expansion, we need only compute $P+1$ deterministic coefficients. Then we can simulate our SDE simply by computing $P+1$ terms, generating a large vector of $\zeta$'s using some efficient algorithm, and BAM! we've got a full Monte-Carlo simulation with only $P+1$ computations of the system.
Not bad at all.
A: Dongbin Xiu is one of the most prominent PC researchers currently. He has a lot of publications and this book
Xiu, Dongbin. 2010. Numerical methods for stochastic computations: a spectral method approach. Princeton, N.J.: Princeton University Press.
Worldcat link
