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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.
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Keywords:
- data assimilation /
- dual-number theory /
- gradient analysis /
- optimization
[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 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 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 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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[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 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 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 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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