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Iterative Methods for Data Assimilation for Burgers's Equation
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
Aeronautical Engineering, Saab Aerosystems, Linköping, Sweden.
2006 (English)In: Journal of Inverse and Ill-Posed Problems, ISSN 0928-0219, E-ISSN 1569-3945, Vol. 14, no 5, 505-535 p.Article in journal (Refereed) Published
Abstract [en]

In this paper we consider one-dimensional flow governed by Burgers' equation. We analyze two iterative methods for data assimilation problem for this equation. One of them so called adjoint optimization method, is based on minimization in L 2-norm. We show that this minimization problem is ill-posed but the adjoint optimization iterative method is regularizing, and represents the well-known Landweber method in inverse problems. The second method is based on L 2-minimization of the gradient. We prove that this problem always has a solution. We present numerical comparisons of these two methods.

Place, publisher, year, edition, pages
Walter de Gruyter, 2006. Vol. 14, no 5, 505-535 p.
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-14646DOI: 10.1515/156939406778247589OAI: oai:DiVA.org:liu-14646DiVA: diva2:24088
Available from: 2007-09-17 Created: 2007-09-17 Last updated: 2017-12-13Bibliographically approved
In thesis
1. Data Assimilation in Fluid Dynamics using Adjoint Optimization
Open this publication in new window or tab >>Data Assimilation in Fluid Dynamics using Adjoint Optimization
2007 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Data assimilation arises in a vast array of different topics: traditionally in meteorological and oceanographic modelling, wind tunnel or water tunnel experiments and recently from biomedical engineering. Data assimilation is a process for combine measured or observed data with a mathematical model, to obtain estimates of the expected data. The measured data usually contains inaccuracies and is given with low spatial and/or temporal resolution.

In this thesis data assimilation for time dependent fluid flow is considered. The flow is assumed to satisfy a given partial differential equation, representing the mathematical model. The problem is to determine the initial state which leads to a flow field which satisfies the flow equation and is close to the given data.

In the first part we consider one-dimensional flow governed by Burgers’ equation. We analyze two iterative methods for data assimilation problem for this equation. One of them so called adjoint optimization method, is based on minimization in L2-norm. We show that this minimization problem is ill-posed but the adjoint optimization iterative method is regularizing, and represents the well-known Landweber method in inverse problems. The second method is based on L2-minimization of the gradient. We prove that this problem always has a solution. We present numerical comparisons of these two methods.

In the second part three-dimensional inviscid compressible flow represented by the Euler equations is considered. Adjoint technique is used to obtain an explicit formula for the gradient to the optimization problem. The gradient is used in combination with a quasi-Newton method to obtain a solution. The main focus regards the derivation of the adjoint equations with boundary conditions. An existing flow solver EDGE has been modified to solve the adjoint Euler equations and the gradient computations are validated numerically. The proposed iteration method are applied to a test problem where the initial pressure state is reconstructed, for exact data as well as when disturbances in data are present. The numerical convergence and the result are satisfying.

Place, publisher, year, edition, pages
Matematiska institutionen, 2007
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1121
Keyword
Data assimilation, Inverse problem, Adjoint optimization, Burgers' equation, nonlinear, Landweber, Initial pressure, Euler flow
National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-9684 (URN)978-91-85831-21-0 (ISBN)
Public defence
2007-09-14, Glashuset, Hus B, Campus Valla, Linköpings universitet, Linköping, 10:15 (English)
Opponent
Supervisors
Available from: 2007-09-17 Created: 2007-09-17 Last updated: 2016-03-14

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Lundvall, JohanKozlov, VladimirWeinerfelt, Per

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