liu.seSearch for publications in DiVA
Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • oxford
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Reconstruction of velocity data, using optimization
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.
Linköping University, Department of Management and Engineering, Applied Thermodynamics and Fluid Mechanics. Linköping University, The Institute of Technology.ORCID iD: 0000-0001-5526-2399
2003 (English)In: Computational Fluid and Solid Mechanics 2003 / [ed] K.J. Bathe, 2003, 2324-2327 p.Conference 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.

Place, publisher, year, edition, pages
2003. 2324-2327 p.
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-14648DOI: 10.1016/B978-008044046-0.50571-6OAI: oai:DiVA.org:liu-14648DiVA: diva2:24090
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
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
2. Reconstruction of velocity data using adjoint optimization
Open this publication in new window or tab >>Reconstruction of velocity data using adjoint optimization
2004 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

In many application areas there is a growing interest in data assimilation or data reconstruction. Data assimilation is a process for integrating observed or measured data into a physical model. The problem originates from a vast array of different topics: traditionally in metereological and oceanographic modelling, and recently from non-invasive medical measurement devices such as magnetic resonance imaging. The measured data may contain inaccurancies and random noise, given with low spatial and/or temporal resolution.

This thesis presents a method for solving reconstruction problems in fluid dynamics using optimal control theory. The problem considered here includes a known partial differential equation and some spatially and temporarily sparsely distributed data with an unknown initial state. From a given velocity field uδ, a flow field u is determined which satisfies a given system of partial differential equations and minimizes || u - u*|| L2. The function u(x,t) is known at the boundary and the initial condition u0(x) is used as design variable. The optimization problem is solved using adjoint formulation.

Place, publisher, year, edition, pages
Linköping: Linköpings universitet, 2004. 8 p.
Series
Linköping Studies in Science and Technology. Thesis, ISSN 0280-7971 ; 1096
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-23162 (URN)2566 (Local ID)91-7373-969-3 (ISBN)2566 (Archive number)2566 (OAI)
Presentation
2004-05-25, Glashuset, Hus B, Linköpings universitet, Linköpings, 13:15 (Swedish)
Opponent
Available from: 2009-10-07 Created: 2009-10-07 Last updated: 2013-11-04

Open Access in DiVA

No full text

Other links

Publisher's full textLink to Ph.D. Thesis

Authority records BETA

Lundvall, JohanWeinerfelt, PerKarlsson, Matts

Search in DiVA

By author/editor
Lundvall, JohanWeinerfelt, PerKarlsson, Matts
By organisation
Applied MathematicsThe Institute of TechnologyApplied Thermodynamics and Fluid Mechanics
Mathematics

Search outside of DiVA

GoogleGoogle Scholar

doi
urn-nbn

Altmetric score

doi
urn-nbn
Total: 819 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • oxford
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf