Stably numerical solving inverse boundary value problem for data assimilation |
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| Affiliation: | 1. Key Laboratory of Transportation Meteorology, China Meteorological Administration, Nanjing 210009, China;2. Jiangsu Institute of Meteorological Sciences, Nanjing 210009, China;3. Nanjing Joint Institute for Atmospheric Sciences, Nanjing 210009, China;4. School of Sciences, Nanjing Tech University, Nanjing 211800, China;5. Jiangsu Air Traffic Management Branch Bureau of CAAC, Nanjing 210000, China;1. School of Energy, Power and Mechanical Engineering, North China Electric Power University, Changping District, Beijing 102206, China;2. Institute of Engineering Thermophysics, Chinese Academy of Sciences, Haidian District, Beijing 100190, China;3. School of Control and Computer Engineering, North China Electric Power University, Changping District, Beijing 102206, China;1. Department of Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University, Menouf, Egypt;2. Department of Mathematics and Statistics, Faculty of Science, Taif University, Taif, Saudi Arabia;3. Department of Basic Engineering Sciences, Faculty of Engineering, Menoufia University, Shibin El-Kom, Egypt;4. Department of Biomedical Engineering, Helwan University, Cairo, Egypt;5. Department of Chemical Engineering, University of Waterloo, Waterloo, ON, Canada;1. Department of Mathematics and Statistics, The University of Missouri-Kansas City, Kansas City MO 64110, USA;2. Department of Mathematics, The University of Kansas, Lawrence, KS 66045, USA |
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| Abstract: | In remote sensing data assimilation, inverse methods are often used to retrieve the boundary conditions from some of the observations in the interior scanning region. Needless to say, the inverse boundary value problem (IBVP) is ill-conditioned in general. Ultimately, the data assimilation must include mechanisms that enable them to overcome the numerical instability for solving IBVP. In this paper, we begin by studying the behavior of ill-conditioned IBVP, and find that the condition number varies with the number of the observations and distribution locations of the observations. Next, we define the number of equivalent independent equations as a novel measurement of ill-conditioning of a problem, which can measure the degree of bad conditions. Furthermore, the novel measure can answer how many additional observations are needed to stabilize the retrieving problem, and where additional observations are fixed up. Finally, we illustrate the proposed methodology by applying it to the study of precipitation data assimilation, with a particular emphasis on the analysis of the effect of the number of observations and their distribution locations. The new methodology appears to be particularly efficient in tackling the instability of the retrieving problem in data assimilation. |
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