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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 $u$.

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  • $\begingroup$ Do you have analytic expression of your PDE solution? $\endgroup$
    – Tomas
    Nov 29, 2012 at 13:21
  • $\begingroup$ No, nor is it likely there is an analytic solution $\endgroup$
    – Nick Alger
    Nov 29, 2012 at 19:32
  • $\begingroup$ Is there any reason to believe that either $f$ or $u$ are probability density functions? $\endgroup$
    – Xi'an
    Nov 30, 2012 at 8:05
  • $\begingroup$ 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. $\endgroup$
    – Nick Alger
    Nov 30, 2012 at 8:13

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