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A new method for data assimilation is proposed using complex-variable differentiation (CVD), which can be used to estimate the initial conditions of the nonlinear physical system governed by the following equation:. Firstly, the gradient analysis of cost function in variational data assimilation is transformed into function numerical computation in complex domain, and the value of gradient is computed more efficiently and exactly. Secondly, the new algorithm of data assimilation is developed by combining an accurate gradient information from CVD with the classical optimization method. Finally, numerical simulations of typical chaotic systems and a humidity evolution equation with physical on-off process show that the new data assimilation method can reconstruct initial conditions of the nonlinear dynamical system very conveniently and accurately.
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Keywords:
- data assimilation /
- complex-variable differentiation /
- nonlinear physical system /
- gradient analysis
[1] 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]
[2] 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]
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[6] Talagrand O, Courtier P 1987 Q. J. R. Meteorol. Soc. 113 1311
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[8] Andersson E, Coauthors 1998 Q. J. R. Meteorol. Soc. 124 1831
[9] 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]
[10] 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]
[11] Rabier F, Jarvinen H, Klinker E, Mahfouf J F 2000 Q. J. R. Meteorol. Soc. 126 1148
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[13] He J H 2000 Appl. Math. Mech. 21 797
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[15] He J H, Lee E W M 2009 Phys. Lett. A 373 1644
[16] Wu G C 2012 Chin. Phys. B 21 120504
[17] Wu G C, Dumitru B 2013 Appl. Math. Model. 37 6183
[18] Giering R 1998 ACM Trans. On Math. Software 24 437
[19] Cheng Q, Zhang H B, Wang B 2009 Math. Numer. Sin. 31 15 (in Chinese) [程强, 张海斌, 王斌 2009 计算数学 31 15]
[20] Lyness J N, Moler C B 1967 SIAM J. Numer. Anal. 4 202
[21] Martins J R R A 2002 A Coupled-adjoint method for highfidelity aero-structural optimization Ph. D. Dissertation. (Stanford: Stanford University)
[22] 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
[23] Gao X W, Liu D D, Chen P C 2002 Comput. Mech. 28 40
[24] Guo L, Gao X W 2008 Journal of Southeast Univerity (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]
[25] Mu M, Zheng Q 2005 Mon Wea. Rev. 133 2711
[26] Wang J F, Mu M, Zheng Q 2005 Tellus 57A 736
[27] Zheng Q, Sha J X, Fang C L 2012 Sci. China Earth Sci. 42 458
[28] Zheng Q, Dai Y 2009 Progress in Natural Science 19 81
[29] Han Y Q, Zhong Z, Wang Y F, Du H D 2013 Acta Phys. Sin. 62 049201 (in Chinese) [韩月琪, 钟中, 王云峰, 杜华栋 2013 62 049201]
[30] Lorenz E N 1963 J. Atmos. Sci. 20 130
[31] Liu F C, Liang X M 2005 Acta Phys. Sin. 54 4584 (in Chinese) [刘福才, 梁晓明 2005 54 4584]
[32] Li L X, Peng H P, Lu H B, Guan X P 2001 Acta Phys. Sin. 50 629 (in Chinese) [李丽香, 彭海朋, 卢辉斌, 关新平 2001 50 629]
[33] Dai D, Ma X K, Li F C, You Y 2002 Acta Phys. Sin. 51 2459 (in Chinese) [戴栋, 马西奎, 李富才, 尤勇 2002 51 2459]
[34] Gao F, Tong H Q 2006 Acta Phys. Sin. 55 577 (in Chinese) [高飞, 童恒庆 2006 55 577]
[35] Cai G L, Huang J J 2006 Acta Phys. Sin. 55 3997 (in Chinese) [蔡国梁, 黄娟娟 2006 55 3997]
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[1] 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]
[2] 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]
[3] Evensen G 1994 J. Geophys. Res. 99 10143
[4] Evensen G, Van Leeuwen P J 2000 Mon. Wea. Rev. 128 1852
[5] Houtekamer P L, Mitchell H L 2001 Mon. Wea. Rev.129 123
[6] Talagrand O, Courtier P 1987 Q. J. R. Meteorol. Soc. 113 1311
[7] Lorenc A C 2003 Q. J. R. Meteorol. Soc. 129 3183
[8] Andersson E, Coauthors 1998 Q. J. R. Meteorol. Soc. 124 1831
[9] 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]
[10] 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]
[11] Rabier F, Jarvinen H, Klinker E, Mahfouf J F 2000 Q. J. R. Meteorol. Soc. 126 1148
[12] He J H 2008 Int. J. Modern. Phys. B. 22 3487
[13] He J H 2000 Appl. Math. Mech. 21 797
[14] He J H 2001 Int. J. Nonlin. Sci. Numer. 2 309
[15] He J H, Lee E W M 2009 Phys. Lett. A 373 1644
[16] Wu G C 2012 Chin. Phys. B 21 120504
[17] Wu G C, Dumitru B 2013 Appl. Math. Model. 37 6183
[18] Giering R 1998 ACM Trans. On Math. Software 24 437
[19] Cheng Q, Zhang H B, Wang B 2009 Math. Numer. Sin. 31 15 (in Chinese) [程强, 张海斌, 王斌 2009 计算数学 31 15]
[20] Lyness J N, Moler C B 1967 SIAM J. Numer. Anal. 4 202
[21] Martins J R R A 2002 A Coupled-adjoint method for highfidelity aero-structural optimization Ph. D. Dissertation. (Stanford: Stanford University)
[22] 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
[23] Gao X W, Liu D D, Chen P C 2002 Comput. Mech. 28 40
[24] Guo L, Gao X W 2008 Journal of Southeast Univerity (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]
[25] Mu M, Zheng Q 2005 Mon Wea. Rev. 133 2711
[26] Wang J F, Mu M, Zheng Q 2005 Tellus 57A 736
[27] Zheng Q, Sha J X, Fang C L 2012 Sci. China Earth Sci. 42 458
[28] Zheng Q, Dai Y 2009 Progress in Natural Science 19 81
[29] Han Y Q, Zhong Z, Wang Y F, Du H D 2013 Acta Phys. Sin. 62 049201 (in Chinese) [韩月琪, 钟中, 王云峰, 杜华栋 2013 62 049201]
[30] Lorenz E N 1963 J. Atmos. Sci. 20 130
[31] Liu F C, Liang X M 2005 Acta Phys. Sin. 54 4584 (in Chinese) [刘福才, 梁晓明 2005 54 4584]
[32] Li L X, Peng H P, Lu H B, Guan X P 2001 Acta Phys. Sin. 50 629 (in Chinese) [李丽香, 彭海朋, 卢辉斌, 关新平 2001 50 629]
[33] Dai D, Ma X K, Li F C, You Y 2002 Acta Phys. Sin. 51 2459 (in Chinese) [戴栋, 马西奎, 李富才, 尤勇 2002 51 2459]
[34] Gao F, Tong H Q 2006 Acta Phys. Sin. 55 577 (in Chinese) [高飞, 童恒庆 2006 55 577]
[35] Cai G L, Huang J J 2006 Acta Phys. Sin. 55 3997 (in Chinese) [蔡国梁, 黄娟娟 2006 55 3997]
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