Solving Poisson equation while suffering from the curse of dimensionality - Computational Science Stack Exchange most recent 30 from scicomp.stackexchange.com 2022-01-27T02:54:36Z https://scicomp.stackexchange.com/feeds/question/25728 https://creativecommons.org/licenses/by-sa/4.0/rdf https://scicomp.stackexchange.com/q/25728 0 Solving Poisson equation while suffering from the curse of dimensionality Moonwalker https://scicomp.stackexchange.com/users/16620 2016-12-06T00:40:31Z 2016-12-06T16:16:44Z <p>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: $$\sum_{k=0}^{m}|\varphi(p_i) - o_i|^2.$$ </p> <p>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).</p> <p>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? </p> <p>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.</p> https://scicomp.stackexchange.com/questions/25728/-/25733#25733 2 Answer by Wolfgang Bangerth for Solving Poisson equation while suffering from the curse of dimensionality Wolfgang Bangerth https://scicomp.stackexchange.com/users/393 2016-12-06T16:16:44Z 2016-12-06T16:16:44Z <p>The short answer is that you can't do this -- it's outside our computational power today. To explain why, think of just building the box itself, where you have one degree of freedom on each vertex. In 100 dimensions, there are $2^{100}\approx 10^{10}$ vertices. That's not far from the size of the biggest finite element computations, and you'll need on the order of $10^5-10^6$ processors to do that. At the same time, all you could do on this one cube is represent the solution as some kind of linear function in each direction -- that's not going to tell you anything about what is really going on; in fact, all it really does is interpolate the boundary values.</p> <p>In other words, try smaller problems. You probably want to look into sparse grids in that case.</p>