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> because they are actually running inverse problems with the PDEs to fit the data Could you explain this a bit more? Are you talking about data assimilation?

• cultus 7y ago
  Yes, that's right. The issue is that systems like these have a lot of free parameters that have to be estimated from the data. This is basically because you have to make assumptions if you don't want to simulate Earth's systems down to the subatomic level. The inferred values of these parameters are usually very important as well for interpretation. All PDEs are solved numerically by discretizing them into linear or nonlinear systems. If it's convex enough, Tikhonov regularization is often used. Linear PDEs, where the solution depends only linearly on its derivatives, can be discretized into matrices. This gives a system like A m = d, where A is the discretized PDE, m is the "model" or parameter values, and d is the data. This is usually over or under-determined from the data, so it has to be solved by least squares. Since the resulting system is usually almost singular, regularization is applied, usually by penalizing the L2 norm of the model: min || A m -d || + r ||m||^2 If it's nonlinear but convex enough, one can just solve it by fixed point iteration, with the above linear equation being solved at every step (Newton's method often doesn't work well). If it's too nonconvex, then MCMC or other Bayesian methods are better, since a single solution is useless if the parameter values could plausibly around widely separated minima.