Given :
function $g : \mathbb{R}^d \to \mathbb{R}$, $d$ dimensional Gaussian random variable $Y$ with density $f(Y)$
A function $L : \mathbb{R}^d \to \mathbb{R}$ :
$$L(x) = \int g(x*Y)f(Y)dY$$
Here $*$ denotes multiplication element by element : $[x_1,x_2,...,x_d]*[y_1,y_2,...,y_d] = [x_1y_1,x_2y_2,...,x_dy_d]$
Function $L$ is very difficult to evaluate. To evaluate $L$ at given $x$ I will approximate it numerically by sampling $y_1,...y_n$ from density of $f(Y)$ and then take average of $g(x*y_i)$. So I want to have some approximation of $L$ in terms of functions that are easier to compute.
My plan:
1) Calculate $L(x_1),L(x_2),...L(x_i)$ for some vectors $x_i$
2) Chose basis functions that are easy to calculate $\psi_1,\psi_2,...,\psi_m$
3) Make regression of $L(x_1),L(x_2),...L(x_i)$ on $\psi$ to obtain representation of $L(x) = \sum_{i=1}^ma_i\psi_i(x)$
My question is how to chose basis functions $\psi$?
I would be happy if you pointed me to some articles or anything where similar questions were investigated
What I have found so far is related to Laplace approximation of integrals. Since $Y$ is Gaussian its density is exponential and if I understood correctly Laplace method, there should be some $y_0$ such that $\psi(x) = g(x*y_0)f(y_0)$ would be good choice of basis function. Right now I am testing this idea