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Cryptanalysis and imge processing the best of application in Mathematics

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Computers helped me in solving nonlinear equations, which otherwise remained unsolved.

Perhaps the best way to answer this is to look at what limitations computer simulations currently have - then seeing what areas of applied mathematics can be used to leverage an advantage.

Arguably, time complexity and accuracy are probably the biggest limitations. Numerical accuracy (and convergence) is generally well understood and can be relatively simple to define (with conditions / constraints). However, time complexity always appears to be a more significant constraint.

In my opinion, the best improvements in computational mathematics come from better heuristics or short-cuts that allow solutions to emerge faster. As we can't model everything in fine detail, adaptive or statistical methods are certainly the best compromise. That is, applying computational effort in the best 'location' (I.E. Adaptive mesh generation algorithms for FEM/FVM) or abstracting a complex mechanism into a simplified statistical distribution / estimation model and reducing the degrees of freedom (I.E. Parameter estimation methods, Lumped parameter analysis etc...).

Therefore I would lean heavily on uncertainty, system identification and computational statistics as being hugely critical areas of applied mathematics. Another interesting conversation could be had on optimisation techniques (for the same reasons), as they power a lot of applications of computer modelling / prediction.