Data assimilation for massive autonomous systems based on second-order adjoint method

摘要

Data assimilation (DA) is a fundamental computational technique thatintegrates numerical simulation models and observation data on the basis ofBayesian statistics. Originally developed for meteorology, especially weatherforecasting, DA is now an accepted technique in various scientific fields. Onekey issue that remains controversial is the implementation of DA in massivesimulation models under limited computation time and resources. In this paper,we propose an adjoint-based DA method for massive autonomous models thatproduces optimum estimates and their uncertainties within practical computationtime and resource constraints. The uncertainties are given as several diagonalcomponents of an inverse Hessian matrix, which is the covariance matrix of anormal distribution that approximates the target posterior probability densityfunction in the neighborhood of the optimum. Conventional algorithms forderiving the inverse Hessian matrix require $O(CN^2+N^3)$ computations and$O(N^2)$ memory, where $N$ is the number of degrees of freedom of a givenautonomous system and $C$ is the number of computations needed to simulate timeseries of suitable length. The proposed method using a second-order adjointmethod allows us to directly evaluate the diagonal components of the inverseHessian matrix without computing all of its components. This drasticallyreduces the number of computations to $O(C)$ and the amount of memory to $O(N)$for each diagonal component. The proposed method is validated through numericaltests using a massive two-dimensional Kobayashi's phase-field model. We confirmthat the proposed method correctly reproduces the parameter and initial stateassumed in advance, and successfully evaluates the uncertainty of theparameter. Such information regarding uncertainty is valuable, as it can beused to optimize the design of experiments.