Hi Richard. That's very interesting, I'm eager to read your answer!

Thanks a lot for the pointers! I'm going to look into them more in detail and see what I can get out of it.

Thanks a lot for your comments. What I'm trying to achieve here is the computation of the heat kernel for a given point of a 2d compact manifold $M$, restricted to the time domain. In particular, given the eigen-decomposition $\Delta_M \phi = \lambda \phi$ of the Laplace-Beltrami operator $\Delta_M$ on $M$, I'm interested in computing the quantity $\sum_{i=1}^m e^{-\lambda_i t} \phi_i(x)^2$ for a given $x\in M$, $t \in \mathbb{R}$ and $m \in \mathbb{N}$. I need to do it very efficiently, that's why I'd like to compute $\phi_i(x)$ instead of the whole $\phi_i$.