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Uncertainty quantification for data assimilation in a steadyincompressible Navier-Stokes problem

Published online by Cambridge University Press:  13 June 2013

Marta D’Elia  and
Alessandro Veneziani
Marta D’Elia
Affiliation:
Dept. of Mathematics and Computer Science, Emory University, 400 Dowman Drive, Atlanta, GA 30322, USA.. mdelia2@mathcs.emory.edu; ale@mathcs.emory.edu
Alessandro Veneziani
Affiliation:
Dept. of Mathematics and Computer Science, Emory University, 400 Dowman Drive, Atlanta, GA 30322, USA.. mdelia2@mathcs.emory.edu; ale@mathcs.emory.edu
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Abstract

The reliable and effective assimilation of measurements and numerical simulations inengineering applications involving computational fluid dynamics is an emerging problem assoon as new devices provide more data. In this paper we are mainly driven by hemodynamicsapplications, a field where the progressive increment of measures and numerical toolsmakes this problem particularly up-to-date. We adopt a Bayesian approach to the inclusionof noisy data in the incompressible steady Navier-Stokes equations (NSE). The purpose isthe quantification of uncertainty affecting velocity and flow related variables ofinterest, all treated as random variables. The method consists in the solution of anoptimization problem where the misfit between data and velocity - in a convenient norm -is minimized under the constraint of the NSE. We derive classical point estimators, namelythe maximum a posteriori – MAP – and the maximum likelihood – ML – ones.In addition, we obtain confidence regions for velocity and wall shear stress, a flowrelated variable of medical relevance. Numerical simulations in 2-dimensional andaxisymmetric 3-dimensional domains show the gain yielded by the introduction of a completestatistical knowledge in the assimilation process.

Keywords

Computational fluid dynamicsoptimizationuncertainty quantificationstatistical inverse problemsdata assimilationhemodynamics
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 4: Direct and inverse modeling of the cardiovascular and respiratory systems , July 2013 , pp. 1037 - 1057
Copyright
© EDP Sciences, SMAI, 2013

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References

Bertoglio, C., Moireau, P. and Gerbeau, Jean-Frédéric, Sequential parameter estimation for fluid-structure problems. Application to hemodynamics. Inter. J. Numer. Methods Biomed. Eng. 28 (2012) 434–455. RR-7657. Google Scholar
J. Blum, F.X. Le Dimet and I.M. Navon, Data Assimilation for Geophysical Fluids, Handbook of numerical analysis, vol. XIV, chapter 9. Elsevier (2005).
D.C. Boes, FA Graybill and A.M. Mood, Introduction to the Theory of Statistics. McGraw-Hill (1974).
D. Calvetti and E. Somersalo, Subjective knowledge or objective belief? an oblique look to bayesian methods, in Large-Scale Inverse Problems and Quantification of Uncertainty, edited by G. Biros et al. Wiley Online Library (2011) 33–70.
M. D’Elia, Ph.D. thesis.
M. D’Elia, L. Mirabella, T. Passerini, M. Perego, M. Piccinelli, C. Vergara and A. Veneziani, Applications of Variational Data Assimilation in Computational Hemodynamics, chapter 12. MS & A. Springer (2011) 363–394.
M. D’Elia, M. Perego and A. Veneziani, A variational Data Assimilation procedure for the incompressible Navier-Stokes equations in hemodynamics. Technical Report TR-2010-19, Department of Mathematics and Computer Science, Emory University, To appear in J. Sci. Comput. Available on www.mathcs.emory.edu (2010).
den Reijer, P.M., Sallee, D., van der Velden, P., Zaaijer, E., Parks, W.J., Ramamurthy, S., Robbie, T., Donati, G. Lamphier, C., Beekman, R. and Brummer, M., Hemodynamic predictors of aortic dilatation in bicuspid aortic valve by velocity-encoded cardiovascular magnetic resonance. J. Cardiovasc. Magn. Reson. 12 (2010) 4. Google ScholarPubMed
Van der Vorst, H.A. and Vuik, C., Gmresr: a family of nested gmres methods. Numer. Linear Algebra Appl. 1 (1994) 369–386. Google Scholar
R.P. Dwight, Bayesian inference for data assimilation using Least-Squares Finite Element methods, in IOP Conf. Ser. Mat. Sci. Eng., vol. 10. IOP Publishing (2010) 012224.
L. Formaggia, A. Veneziani and C. Vergara. SIAM J. Sci. Comput. (2008).
L. Formaggia, A. Veneziani and C. Vergara. Comput. Methods Appl. Mech. Eng. (2010).
M. Frangos, Y. Marzouk, K. Willcox and B. van Bloemen Waanders, Surrogate and reduced-order modeling: A comparison of approaches for large-scale statistical inverse problems. Large-Scale Inverse Problems and Quantification of Uncertainty (2010) 123–149.
Gunzburger, M.D., Perspectives in flow control and optimization. Society for Industrial Mathematics 5 (2003). Google Scholar
Per Christian Hansen, Rank-deficient and discrete ill-posed problems. SIAM Monographs on Mathematical Modeling and Computation. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1998).
Heys, J.J., Manteuffel, T.A., McCormick, S.F., Milano, M., Westerdale, J. and Belohlavek, M., Weighted least-squares finite elements based on particle imaging velocimetry data. J. Comput. Phys. 229 (2010) 107–118. Google Scholar
Heywood, J. G., Rannacher, R. and Turek, S., Artificial boundaries and flux pressure conditions for the incompressible navier-stokes equations. Int. J. Numer. Methods Fluids 22 (1996) 325–352. Google Scholar
R.A. Johnson and D.W. Wichern, Applied multivariate statistical analysis. Prentice-Hall, Inc., Upper Saddle River, NJ, USA (1988).
J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems. Springer (2005).
Kalman, E.M., A new approach to linear filtering and prediction problems. Trans. ASME-J. Basic Eng. 82 (1960) 35–45. Google Scholar
Kay, D., Loghin, D. and Wathen, A., A preconditioner for the steady-state navier–stokes equations. SIAM J. Sci. Comput. 24 (2002) 237–256. Google Scholar
P. Moireau, C. Bertoglio, N. Xiao, C. Figueroa, C. Taylor, D. Chapelle and J.-F. Gerbeau, Sequential identification of boundary support parameters in a fluid-structure vascular model using patient image data. Biomechanics and Modeling in Mechanobiology. Published Online (2012) 1–22.
Moireau, P. and Chapelle, D., Reduced-order unscented kalman filtering with application to parameter identification in large-dimensional systems. ESAIM: COCV 17 (2011) 380–405. Google Scholar
J. Nocedal and S. Wright, Numerical Optimization. Springer (2000).
Perego, M., Veneziani, A. and Vergara, C., A variational approach for estimating the compliance of the cardiovascular tissue: An inverse fluid-structure interaction problem. SIAM J. Sci. Comput. 33 (2011) 1181–1211. Google Scholar
Quarteroni, A., Rozza, G. and Manzoni, A., Certified reduced basis approximation for parametrized partial differential equations and applications. J. Math. Ind. 1 (2011) 3. Google Scholar
Silvester, D., Elman, H., Kay, D. and Wathen, A., Efficient preconditioning of the linearized navier-stokes equations for incompressible flow. J. Comput. Appl. Math. 128 (2001) 261–279. Google Scholar
A. Tarantola, Inverse problem theory and methods for model parameter estimation. Society for Industrial Mathematics (2005).
A. Veneziani, Boundary conditions for blood flow problems, in Proc. of ENUMATH97, edited by R. Rannacher et al., World Sci. Publishing (1998).
A. Veneziani, Mathematical and Numerical Modeling of Blood flow Problems. Ph.D. thesis, Politecnico di Milano, Italy (1998).
Vuik, C., New insights in gmres-like methods with variable preconditioners. J. Comput. Appl. Math. 61 (1995) 189–204. Google Scholar