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Reconstruction of velocity data using adjoint optimization
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
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: urn:nbn:se:liu:diva-23162Local ID: 2566ISBN: 91-7373-969-3 (print)OAI: oai:DiVA.org:liu-23162DiVA: diva2:243476
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
List of papers
1. Reconstruction of velocity data, using the viscous Burgers' equation and adjoint optimization
Open this publication in new window or tab >>Reconstruction of velocity data, using the viscous Burgers' equation and adjoint optimization
(English)Manuscript (preprint) (Other academic)
Abstract [en]

For a given field u*, a field u governed by a nonlinear partial differential equation and minimizing ||u - u*||2 is determined. The initial condition is used as control variable and the gradient for the minimization is based on adjoint technique. Continuous and discrete gradient formulations are compared. The method handles missing as well as noisy data.

Keyword
adjoint technique, data reconstruction, optimal flow control, optimization, uncertain velocity data
National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-100374 (URN)
Available from: 2013-11-04 Created: 2013-11-04 Last updated: 2016-03-14
2. On the solution of the viscous Burgers' equation with nonlinear viscosity
Open this publication in new window or tab >>On the solution of the viscous Burgers' equation with nonlinear viscosity
2004 (English)Report (Other academic)
Abstract [en]

The viscous Burgers' equation with nonlinear viscosity is considered. The equation is written as a quasilinear parabolic equation in divergence form, and existence of a weak solution is shown. The proof is based on Galerkin approximations which converges in a suitable Banach space. Finally, the Cole-Hopf transformation is used to derive an analytical solution in the case when the viscosity is constant. This solution turns out to be very ill-conditioned for numerical evaluations. The solution can be rewritten with the Poisson summation formula. Comparisons to a finite difference solution are done.

Place, publisher, year, edition, pages
Linköping: Linköpings universitet, 2004. 18 p.
Series
LiTH-MAT-R, ISSN 0348-2960
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-22770 (URN)LITH-MAT-R-2004-07 (ISRN)2097 (Local ID)2097 (Archive number)2097 (OAI)
Available from: 2009-10-07 Created: 2009-10-07 Last updated: 2013-11-04
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, 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.

National Category
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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