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Scaling SIR to geophysical fluids

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This research is rooted in a desire to apply provably consistent Bayesian methods to select models for nonlinear multiscale dynamics that must be observed in high resolution. As a motivating example, we describe a geophysical mystery (the Madden-Julian Oscillation, MJO) for which it is reasonable to incorporate a half billion observations -- far beyond the computational reach of naive MCMC methods to obtain posterior model probabilities, but we can reduce the burden on MCMC by performing data assimilation to estimate the system state first. Unfortunately, such high dimensionality has also been considered far beyond the reach of Sequential Importance sampling with Resampling (SIR, or "particle filtering"), the unique data assimilation technique flexible enough to obtain consistent estimates of the system state in the presence of strong nonlinearities inherent to the MJO and many other geophysical fluids problems. But we introduce a technique that lessens the computational cost of SIR, improving uncertainty quantification for fixed computational cost, by judiciously choosing a strange model for observational error. This talk will then describe ongoing work to implement our method with a multiresolution approximation that is fast and applicable to scattered observations, by taking advantage of a connection between our strange error model and elliptic PDEs. We conclude with plans to test the method on plausible models of the MJO as a step toward applying provably consistent Bayesian techniques to select between geophysical fluid models.