# Sampling from elliptic pde solution in high dimensions

What is known about sampling from solutions to elliptic PDE's in high dimensions, where it is computationally infeasible to construct or store the actual solution?

For example, let $u$ solve the Poisson equation $\Delta u = f$ on $\mathbb{R}^n$ with the standard decay condition at infinity, and $n \approx 100$. Assuming $f$ is sufficiently nice so that $u$ can be normalized. How can we draw samples from the normalized version of $u$, interpreted as a probability distribution?

Edit: My first guess would be from analogy with the physical origins of the Poisson equation, where $f$ can be interpreted as a source of fluid that flows out diffusively. The idea would be to interpret $f$ as a probability distribution, randomly draw from $f$, then do a random walk from there for a sufficiently long time to get a draw from $u$.

• Do you have analytic expression of your PDE solution? Commented Nov 29, 2012 at 13:21
• No, nor is it likely there is an analytic solution Commented Nov 29, 2012 at 19:32
• Is there any reason to believe that either $f$ or $u$ are probability density functions? Commented Nov 30, 2012 at 8:05
• I'm considering the normalized version of $u$ and $f$ as probability density functions of interest. From the PDE theory, the solution $u$ can be controlled by the data $f$, even adding 2 extra degrees of smoothness, so $u$ should be normalizable if $f$ is. Commented Nov 30, 2012 at 8:13