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Data Assimilation in Fluid Dynamics using Adjoint Optimization
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
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 [en]
Data assimilation, Inverse problem, Adjoint optimization, Burgers' equation, nonlinear, Landweber, Initial pressure, Euler flow
National Category
Computational Mathematics
Identifiers
ISBN: 978-91-85831-21-0 (print)OAI: oai:DiVA.org:liu-9684DiVA, id: diva2:24091
Supervisors
Available from: 2007-09-17 Created: 2007-09-17 Last updated: 2016-03-14
List of papers
1. Iterative Methods for Data Assimilation for Burgers's Equation
Open this publication in new window or tab >>Iterative Methods for Data Assimilation for Burgers's Equation
2006 (English)In: Journal of Inverse and Ill-Posed Problems, ISSN 0928-0219, E-ISSN 1569-3945, Vol. 14, no 5, p. 505-535Article 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
Mathematics
Identifiers
urn:nbn:se:liu:diva-14646 (URN)10.1515/156939406778247589 (DOI)
Available from: 2007-09-17 Created: 2007-09-17 Last updated: 2017-12-13Bibliographically approved
2. Reconstruction of initial state for 3D time dependent Euler flow using adjoint optimization
Open this publication in new window or tab >>Reconstruction of initial state for 3D time dependent Euler flow using adjoint optimization
Identifiers
urn:nbn:se:liu:diva-14647 (URN)
Available from: 2007-09-17 Created: 2007-09-17 Last updated: 2010-01-13
3. Reconstruction of velocity data, using optimization
Open this publication in new window or tab >>Reconstruction of velocity data, using optimization
2003 (English)In: Computational Fluid and Solid Mechanics 2003 / [ed] K.J. Bathe, 2003, p. 2324-2327Conference paper, Published paper (Other academic)
Abstract [en]

From a given velocity field u*, a flow field that satisfies a given differential equation and minimize some norm is determined. The gradient for the optimization is updated using adjoint technique. The numerical solution of the non-linear partial differential equation is done using a multigrid scheme. The test case shows promising results. The method handles missing data as well as disturbances.

This chapter discusses reconstruction of velocity data, using optimization. There is a growing interest in obtaining velocity data on a higher temporal and/or spatial resolution than is currently possible to measure. The problem originates from a vast array of topics—such as meteorology, hydrology, wind tunnel, or water tunnel experiments—and from noninvasive medical measurement devices, such as 3D time-resolved-phase-contrast magnetic resonance imaging. The rapid development in computer performance gave birth to new methods, based on optimization and simultaneous numerical solution of partial differential equations, well-suited for the task of up-sampling. The data may be of several kinds—low spatial and/or temporal resolution with or without areas of missing and/or uncertain data. It determines a flow field that satisfies a given differential equation and minimize some norm from a given velocity field. The gradient for the optimization can be updated through adjoint technique. The numerical solution of the nonlinear partial differential equation can be done through a multigrid scheme. The method handles missing data as well as disturbances.

Mathematics
Identifiers
urn:nbn:se:liu:diva-14648 (URN)10.1016/B978-008044046-0.50571-6 (DOI)
Conference
Proceedings Second MIT Conference on Compurational Fluid and Solid Mechanics June 17–20, 2003
Available from: 2007-09-17 Created: 2007-09-17 Last updated: 2016-03-14

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Lundvall, Johan

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Cite
Citation style
• apa
• harvard1
• ieee
• modern-language-association-8th-edition
• vancouver
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More styles
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