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基于对偶数理论的资料同化新方法

曹小群 皇群博 刘柏年 朱孟斌 余意

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基于对偶数理论的资料同化新方法

曹小群, 皇群博, 刘柏年, 朱孟斌, 余意

A new data assimilation method based on dual-number theory

Cao Xiao-Qun, Huang Qun-Bo, Liu Bai-Nian, Zhu Meng-Bin, Yu Yi
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  • 针对变分资料同化中目标泛函梯度计算精度不高且复杂等问题, 提出了一种基于对偶数理论的资料同化新方法, 主要优点是: 能避免复杂的伴随模式开发及其逆向积分, 只需在对偶数空间通过正向积分就能同时计算出目标泛函和梯度向量的值. 首先利用对偶数理论把梯度分析过程转换为对偶数空间中目标泛函计算过程, 简单、高效和高精度地获得梯度向量值; 其次结合典型的最优化方法, 给出了非线性物理系统资料同化问题的新求解算法; 最后对Lorenz 63混沌系统、包含开关的不可微物理模型和抛物型偏微分方程分别进行了资料同化数值实验, 结果表明: 新方法能有效和准确地估计出预报模式的初始条件或物理参数值.
    In gradient computations of the variational data assimilation (VDA) by the adjoint method, in order to overcome a lot of shortcomings such as low accuracy, difficult implementation, and great complexity, etc., a novel data assimilation method is proposed based on the dual-number theory. The important advantages are that the coding of adjoint models and reverse integrations are not necessary any more, and the values of cost functional and its corresponding gradient vectors can be attained simultaneously only by one forward computation in dual-number space. Furthermore, the accuracy of gradient can be close to the computer machine precision without other error sources. The paper is organised as follows. Firstly, the dual-number theory and algorithm rules are introduced. Then, the issues of gradient analysis and computation in VDA are transformed into the processes of calculating the cost functional numerically in dual-number space, and the gradient vectors can be obtained at the same time in an easy, efficient and accurate way. Secondly, the new algorithm for data assimilation in nonlinear physical systems is developed by combining accurate gradient information from the dual-number method with classical optimization algorithm. Thirdly, numerical experiments on sensitivity analysis for an ENSO nonlinear air-sea coupled oscillator are implemented, and the results are presented to demonstrate the important advantages of the dual-number method in the calculation of derivative information. Finally, numerical simulations for data assimilation are carried out respectively for the typical Lorenz 63 chaotic systems, the specific humidity evolving equation with physical “on-off” process at a single grid point, and a parabolic partial differential equation. Some conclusions can be drawn from the numerical experiments. The newly proposed method may be suited to many kinds of optimization problems with ordinary or partial differential equations as constraints, such as data assimilation, parameter estimation, inverse problems, sensitivity analysis etc. Results show that the new method can reconstruct the initial conditions or parameters of a nonlinear dynamical system very conveniently and accurately. Its another advantage is being very easy to implement with a high accuracy in gradient computation, so it is robust in the process of numerical optimization. The estimated initial states or parameters are convergent to real value in the cost of no more computations, when there are noises in the observations. But many tests are still needed to demonstrate the validity and advantages of the new data assimilation method, especially in more complex and realistic numerical prediction models of atmosphere and ocean.
    • 基金项目: 国家自然科学基金(批准号:41475094,41105063,41375105)和高分青年创新基金(批准号:GFZX04060103-5-19)资助的课题.
    • Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 41475094, 41105063, 41375105), and the Young Innovation Science Foundation of CHREO (Grant Nos. GFZX04060103-5-19).
    [1]

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    [2]

    Zou X L 2009 Data Assimilation-Theory and Application (Vol. 1) (Beijing: Meteorology Press) p43 (in Chinese) [邹晓蕾 2009 资料同化-理论与应用(上册) (北京: 气象出版社) 第 43 页]

    [3]

    Evensen G 1994 J. Geophys. Res. 99 10143

    [4]

    Talagrand O, Courtier P 1987 Q. J. R. Meteorol. Soc 113 1311

    [5]

    Rabier, F., Jarvinen, H., Klinker, E. and Mahfouf, J. F 2000 Q. J. R. Meteorol. Soc. 126 1148

    [6]

    Cao X Q, Huang S X, Du H D 2008 Acta Phys. Sin. 57 1984 (in Chinese) [曹小群, 黄思训, 杜华栋 2008 物理学报 57 1984]

    [7]

    Cao X Q, Song J Q, Zhang W M 2013 Acta Phys. Sin. 62 170504 (in Chinese) [曹小群, 宋君强, 张卫民 2013 物理学报 62 170504]

    [8]

    Zhang W M, Cao X Q, Song J Q 2012 Acta Phys. Sin. 61 249202 (in Chinese) [张卫民, 曹小群, 宋君强 2012 物理学报 61 249202]

    [9]

    Giering R 1998 ACM Trans. On Math. Software 24 437

    [10]

    Cheng Q, Zhang H B, Wang B 2009 Mathematica Numerica Sinica 31 15 (in Chinese) [程强、张海斌、王斌 2009 计算数学 31 15]

    [11]

    Lyness J N, Moler C B 1967 SIAM Journal of Numerical Analysis 4 202

    [12]

    Martins J R R A 2002 A Coupled-adjoint method for highfidelity aero-structural optimization Ph. D. Dissertation. (Stanford: Stanford University)

    [13]

    Martins J R R A, Kroo I M, Alonso J J 2000 Proceedings of the 38th Aerospace Sciences Meeting, Reno, NV, January 2-5, AIAA Paper 2000-0689

    [14]

    Gao X W, Liu D D, Chen P C 2002 Computational Mechanics 28 40

    [15]

    Guo L, Gao X W 2008 journal of southeast university (Natural Science Edition) 38 141 (in Chinese) [郭力、高效伟 2008 东南大学学报: 自然科学版 38 141]

    [16]

    Clifford W K 1871 Proceedings of the London M athematical Society London, U. K., April 13-15, 1871 p381

    [17]

    Brodsky V, Shoham M 1999 Mechanism and Machine Theory 34 693

    [18]

    WANG J Y, LIANG H Z, SUN Z W 2010 Journal of Astronautics 31 1711 (in Chinese) [王剑颖、梁海朝、孙兆伟 2010 宇航学报 31 1711]

    [19]

    Spall R, Yu W 2013 Journal of Fluids Engineering 135 014501

    [20]

    Wenbin Yu, Maxwell Blair 2013 Computer Physics Communications 184 1446

    [21]

    Mo J Q, Lin W T, Zhu J 2006 Adv. Math. 35 232

    [22]

    He J H 2008 Int. J. Modern. Phys. B 22 3487

    [23]

    He J H, Lee E. W. M 2009 Phys. Lett. A 373 1644

    [24]

    Wu G C 2012 Chin. Phys. B 21 120504

    [25]

    Lorenz E N 1963 J. Atmos. Sci. 20 130

    [26]

    Mu M, Zheng Q 2005 Mon Wea. Rev. 133 2711

    [27]

    Wang J F, Mu M, Zheng Q 2005 Tellus 57A 736

    [28]

    Zheng Q, Sha J X, Fang C L 2012 Sci. China Earth Sci. 42 458

    [29]

    Isakov V, Kindermann S 2000 Inverse Problems 16 665

  • [1]

    Huang S X, Wu R S 2001 Mathematical Physics Problems in Atmosphere Science (Beijing: Meteorology Press) (in Chinese) p460 [黄思训、伍荣生 2001 大气科学中的数学物理问题(北京: 气象出版社) 第 460 页]

    [2]

    Zou X L 2009 Data Assimilation-Theory and Application (Vol. 1) (Beijing: Meteorology Press) p43 (in Chinese) [邹晓蕾 2009 资料同化-理论与应用(上册) (北京: 气象出版社) 第 43 页]

    [3]

    Evensen G 1994 J. Geophys. Res. 99 10143

    [4]

    Talagrand O, Courtier P 1987 Q. J. R. Meteorol. Soc 113 1311

    [5]

    Rabier, F., Jarvinen, H., Klinker, E. and Mahfouf, J. F 2000 Q. J. R. Meteorol. Soc. 126 1148

    [6]

    Cao X Q, Huang S X, Du H D 2008 Acta Phys. Sin. 57 1984 (in Chinese) [曹小群, 黄思训, 杜华栋 2008 物理学报 57 1984]

    [7]

    Cao X Q, Song J Q, Zhang W M 2013 Acta Phys. Sin. 62 170504 (in Chinese) [曹小群, 宋君强, 张卫民 2013 物理学报 62 170504]

    [8]

    Zhang W M, Cao X Q, Song J Q 2012 Acta Phys. Sin. 61 249202 (in Chinese) [张卫民, 曹小群, 宋君强 2012 物理学报 61 249202]

    [9]

    Giering R 1998 ACM Trans. On Math. Software 24 437

    [10]

    Cheng Q, Zhang H B, Wang B 2009 Mathematica Numerica Sinica 31 15 (in Chinese) [程强、张海斌、王斌 2009 计算数学 31 15]

    [11]

    Lyness J N, Moler C B 1967 SIAM Journal of Numerical Analysis 4 202

    [12]

    Martins J R R A 2002 A Coupled-adjoint method for highfidelity aero-structural optimization Ph. D. Dissertation. (Stanford: Stanford University)

    [13]

    Martins J R R A, Kroo I M, Alonso J J 2000 Proceedings of the 38th Aerospace Sciences Meeting, Reno, NV, January 2-5, AIAA Paper 2000-0689

    [14]

    Gao X W, Liu D D, Chen P C 2002 Computational Mechanics 28 40

    [15]

    Guo L, Gao X W 2008 journal of southeast university (Natural Science Edition) 38 141 (in Chinese) [郭力、高效伟 2008 东南大学学报: 自然科学版 38 141]

    [16]

    Clifford W K 1871 Proceedings of the London M athematical Society London, U. K., April 13-15, 1871 p381

    [17]

    Brodsky V, Shoham M 1999 Mechanism and Machine Theory 34 693

    [18]

    WANG J Y, LIANG H Z, SUN Z W 2010 Journal of Astronautics 31 1711 (in Chinese) [王剑颖、梁海朝、孙兆伟 2010 宇航学报 31 1711]

    [19]

    Spall R, Yu W 2013 Journal of Fluids Engineering 135 014501

    [20]

    Wenbin Yu, Maxwell Blair 2013 Computer Physics Communications 184 1446

    [21]

    Mo J Q, Lin W T, Zhu J 2006 Adv. Math. 35 232

    [22]

    He J H 2008 Int. J. Modern. Phys. B 22 3487

    [23]

    He J H, Lee E. W. M 2009 Phys. Lett. A 373 1644

    [24]

    Wu G C 2012 Chin. Phys. B 21 120504

    [25]

    Lorenz E N 1963 J. Atmos. Sci. 20 130

    [26]

    Mu M, Zheng Q 2005 Mon Wea. Rev. 133 2711

    [27]

    Wang J F, Mu M, Zheng Q 2005 Tellus 57A 736

    [28]

    Zheng Q, Sha J X, Fang C L 2012 Sci. China Earth Sci. 42 458

    [29]

    Isakov V, Kindermann S 2000 Inverse Problems 16 665

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出版历程
  • 收稿日期:  2014-10-26
  • 修回日期:  2015-01-30
  • 刊出日期:  2015-07-05

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