> Professional simulation software with finite element methods and the like I learned a little about the theory behind this a decade ago, and how to implement

> Professional simulation software with finite element methods and the like I learned a little about the theory behind this a decade ago, and how to implement it. For a simple PDE you discretise the domain, apply something like the Galerkin method [1] to convert your true problem on the continuous domain to an approximate problem on the finite-dimensional discretised domain. For particularly simple PDE the approximate problem turns into a gigantic linear system of equations, A x = b. The derivations to figure out the exact form of A and b might be fiddly, but not fundamentally challenging. Then you solve for x. Large-scale linear algebra is relatively well understood and industrialised. As dimensions of problem are so large, iterative methods [2] are used to obtain an approximation to x without needing to compute an explicit inverse matrix A^-1. Discretised PDE generally produce sparse matrices A, which are efficient to compute with, whereas A^-1 will be dense, and you do not want to try to work with A^-1 explicitly. Theoretically, given certain assumptions you can prove desirable properties about finite element schemes. One highly desirable property is that the sequence of approximate solutions with smaller and smaller mesh sizes converge to the true not-approximated continuous solution in the limit as the mesh size goes to zero. Some of the assumptions required to guarantee convergence might place restrictions on the shape of your mesh -- e.g. if the mesh is triangulated and some mesh elements become arbitrarily thin and pointy, the approximation may not be guaranteed to converge. [1] https://en.wikipedia.org/wiki/Galerkin_method https://en.wikipedia.org/wiki/Galerkin_method [2] e.g. https://en.wikipedia.org/wiki/Krylov_subspace https://en.wikipedia.org/wiki/Krylov_subspace > MRI and similar scanners I have no idea how the physics, sensors and hardware side of MRI works, but once you have the data and equations from some physical model linking the state you want to infer to the observed data, it is some kind of mathematical inverse problem to estimate and infer the state from the observation, and we're back in applied math & industrial linear algebra territory. No magic.