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).
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:
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!