# Stochastic Collocation for time evolving ODE

For an Stochastic Differential Equation, e.g.,

$$\frac{du}{dt} = \alpha*\sin(u*t)$$ where $\alpha$ is normally distributed with nonzero mean, I am trying to use a stochastic collocation approach to find simple statistics such as the mean and the variance.

From what I understand, $\alpha$ is sampled a few times, and the deterministic solution is solved from $t_0$ to $t_F$. Each of these samples is used in a collocation scheme to form the interpolating polynomial. Since in this case, there are multiple time steps between $t_0$ and $t_F$, don't I need to form (possibly) different interpolating polynomials for each of the time points chosen between $t_0$ and $t_F$? Would this not be very slow/inefficient?

Thank you,

Raghu