I am trying to find the domains in physics where solving large systems of equations is computationally expensive. The sparse systems are of my particular interest, where the input matrix A is in GBs (up to 100 GBs).

  • $\begingroup$ What type of equations? $\endgroup$ – Joe Fitzsimons Oct 16 '11 at 13:52
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    $\begingroup$ What would the purpose of such a list? $\endgroup$ – Marcin Kotowski Oct 16 '11 at 15:40
  • $\begingroup$ You should edit your question and expand a bit. In its current form, it is impossible to answer it. Can you provide examples of what you have in mind? $\endgroup$ – András Bátkai Oct 17 '11 at 16:44
  • $\begingroup$ Sure. Say you have PDE equations which you linearize with FEM or FVM. You end up with a system of linear equations that can be later turned into famoous Ax=b where A is huge (tens of GBs) and sparse. You can solve it with direct methods and have to use iterative solvers such as CG/BCGSTAB/GMRES/multi-grid. Does it answer to your question? $\endgroup$ – Lukasz Oct 19 '11 at 4:43
  • $\begingroup$ Marcin, physists would like to have their supercomputers under the desk instead of using clusters with complicated submission process. With GPUs this is possible. $\endgroup$ – Lukasz Oct 19 '11 at 4:47

For one thing, the solution of any PDE using the finite elements method yields a large sparse system of equations. In the nonlinear case the method is iterative so you need to solve a linear system many times. The applications in physics are countless. To name a few:

  • Numerical general relativity
  • Hydrodynamics
  • Magnetohydrodynamics
  • Stellar structure and evolution
  • Scattering and propagation of electromagnetic radiation

Then you have differential-integral equations such as those coming from computational quantum mechanics (Hartree-Fock, Density Functional), electrostatics and countless other places. These transform to systems of linear equations under most numerical methods

All in all, without additional qualifications the list is simply too long. Equation systems are everywhere!


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