2013/07/23 by Nikolas Kantas, Alexandros Beskos, Kantas, Nikolas +3 · 5 citations
Computer Science · Decision Sciences · Earth and Planetary Sciences · Environmental Science · #Computation (stat.CO) #FOS: Computer and information sciences #FOS: Mathematics #Gaussian Processes and Bayesian Inference #Meteorological Phenomena and Simulations #Numerical Analysis (math.NA) #Probabilistic and Robust Engineering Design #Wind and Air Flow Studies
paper · pdf · doi:10.48550/arxiv.1307.6127
openalex publication_date 2013/07/23 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
We consider the inverse problem of estimating the initial condition of a\npartial differential equation, which is only observed through noisy\nmeasurements at discrete time intervals. In particular, we focus on the case\nwhere Eulerian measurements are obtained from the time and space evolving\nvector field, whose evolution obeys the two-dimensional Navier-Stokes equations\ndefined on a torus. This context is particularly relevant to the area of\nnumerical weather forecasting and data assimilation. We will adopt a Bayesian\nformulation resulting from a particular regularization that ensures the problem\nis well posed. In the context of Monte Carlo based inference, it is a\nchallenging task to obtain samples from the resulting high dimensional\nposterior on the initial condition. In real data assimilation applications it\nis common for computational methods to invoke the use of heuristics and\nGaussian approximations. The resulting inferences are biased and not\nwell-justified in the presence of non-linear dynamics and observations. On the\nother hand, Monte Carlo methods can be used to assimilate data in a principled\nmanner, but are often perceived as inefficient in this context due to the\nhigh-dimensionality of the problem. In this work we will propose a generic\nSequential Monte Carlo (SMC) sampling approach for high dimensional inverse\nproblems that overcomes these difficulties. The method builds upon Markov chain\nMonte Carlo (MCMC) techniques, which are currently considered as benchmarks for\nevaluating data assimilation algorithms used in practice. In our numerical\nexamples, the proposed SMC approach achieves the same accuracy as MCMC but in a\nmuch more efficient manner.\n