I have a heat transfer equation in a cube in $R^{100}$: $[0,1]\times[0,1]\times[0,1]\dots$: $$ \nabla^2 \varphi = f, $$ with boundary conditions set in a form that in the number of points $p_i$, temperature field should least deviate from observed values $o_i$, or in other words that solution of heat equation should minimise: $$ \sum_{k=0}^{m}|\varphi(p_i) - o_i|^2. $$
This would be pretty straightforward problem in 2-3 dimensional case (assuming problem is well-posed), I've solved it with FEM successfully, but for high dimensional case I cannot even build the grid, let alone do any calculations. (I don't store $f$, I can easily calculate it in any point).
It seems, I need to employ some grid-less method. I've skimmed google briefly and found two possible venues: to use radial basis functions or use particle methods. Are they applicable in my case? Do my problem feasible at all?
I've never worked with high dimensional problems before, so I would like to hear all suggestions and references to the relevant and possibly relevant literature.