Starting with an example:

I want to apply a numerical method (e.g. the finite difference method) to find a solution of a partial differential equation

$$ \nabla^2 f(\mathbf{x}) = k f(\mathbf{x}) $$

where $\mathbf{x} \in R^D$, $\nabla^2 = \sum_{d=1}^{D} \frac{\partial^2}{\partial x_d ^2}$ and $D >> 3$.

Applying an ordinary finite difference scheme would result in the exponentially-growing number of grid points.

Is there any approximate numerical method that can handle such exponential increase of computation in higher dimension?

Moreover, how can boundary conditions be specified in such a approximate method?

  • $\begingroup$ Any refs and tips on solving PDE in higher dimensions are also welcome! $\endgroup$ Dec 31, 2017 at 7:50
  • $\begingroup$ This is a very hard problem in general with not many great solutions. For linear PDEs, you can look at spectral methods (the idea is the solution may have a sparse representation in some appropriate basis). For fully nonlinear equations there is often little hope. $\endgroup$
    – Jeff
    Jan 1, 2018 at 15:30


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