title: Data assimilation for massive autonomous systems based on a second-order adjoint method.
abstract: Data assimilation (DA) is a fundamental computational technique that integrates numerical simulation models and observation data on the basis of Bayesian statistics. Originally developed for meteorology, especially weather forecasting, DA is now an accepted technique in various scientific fields. One key issue that remains controversial is the implementation of DA in massive simulation models under the constraints of limited computation time and resources. In this paper, we propose an adjoint-based DA method for massive autonomous models that produces optimum estimates and their uncertainties within reasonable computation time and resource constraints. The uncertainties are given as several diagonal elements of an inverse Hessian matrix, which is the covariance matrix of a normal distribution that approximates the target posterior probability density function in the neighborhood of the optimum. Conventional algorithms for deriving 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 given autonomous system and C is the number of computations needed to simulate time series of suitable length. The proposed method using a second-order adjoint method allows us to directly evaluate the diagonal elements of the inverse Hessian matrix without computing all of its elements. This drastically reduces the number of computations to O(C) and the amount of memory to O(N) for each diagonal element. The proposed method is validated through numerical tests using a massive two-dimensional Kobayashi phase-field model. We confirm that the proposed method correctly reproduces the parameter and initial state assumed in advance, and successfully evaluates the uncertainty of the parameter. Such information regarding uncertainty is valuable, as it can be used to optimize the design of experiments.
authors.affiliation: Earthquake Research Institute, The University of Tokyo, 1-1-1, Yayoi, Bunkyo-ku, Tokyo 113-0032, Japan.
authors.forename: Shin-Ichi
authors.initials: SI
authors.lastname: Ito
authors.affiliation: Earthquake Research Institute, The University of Tokyo, 1-1-1, Yayoi, Bunkyo-ku, Tokyo 113-0032, Japan.|Graduate School of Information Science and Technology, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, 113-8656 Tokyo, Japan.
authors.forename: Hiromichi
authors.initials: H
authors.lastname: Nagao
authors.affiliation: Division of Advanced Mechanical Systems Engineering, Institute of Engineering, Tokyo University of Agriculture and Technology, 2-24-16, Naka-cho, Koganei, Tokyo 184-8588, Japan.
authors.forename: Akinori
authors.initials: A
authors.lastname: Yamanaka
authors.affiliation: Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi 464-8603, Japan.
authors.forename: Yuhki
authors.initials: Y
authors.lastname: Tsukada
authors.affiliation: Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi 464-8603, Japan.
authors.forename: Toshiyuki
authors.initials: T
authors.lastname: Koyama
authors.affiliation: Earthquake Research Institute, The University of Tokyo, 1-1-1, Yayoi, Bunkyo-ku, Tokyo 113-0032, Japan.
authors.forename: Masayuki
authors.initials: M
authors.lastname: Kano
authors.affiliation: Research Center for Advanced Science and Technology, The University of Tokyo, 4-6-1, Komaba, Meguro-ku, Tokyo 153-8904, Japan.
authors.forename: Junya
authors.initials: J
authors.lastname: Inoue