Uncertainty quantification for data assimilation in a steady incompressible Navier-Stokes problem

D’Elia, Marta; Veneziani, Alessandro

ESAIM: Mathematical Modelling and Numerical Analysis , Direct and inverse modeling of the cardiovascular and respiratory systems. Numéro spécial, Tome 47 (2013) no. 4, pp. 1037-1057.

The reliable and effective assimilation of measurements and numerical simulations in engineering applications involving computational fluid dynamics is an emerging problem as soon as new devices provide more data. In this paper we are mainly driven by hemodynamics applications, a field where the progressive increment of measures and numerical tools makes this problem particularly up-to-date. We adopt a Bayesian approach to the inclusion of noisy data in the incompressible steady Navier-Stokes equations (NSE). The purpose is the quantification of uncertainty affecting velocity and flow related variables of interest, all treated as random variables. The method consists in the solution of an optimization 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, namely the maximum a posteriori - MAP - and the maximum likelihood - ML - ones. In addition, we obtain confidence regions for velocity and wall shear stress, a flow related variable of medical relevance. Numerical simulations in 2-dimensional and axisymmetric 3-dimensional domains show the gain yielded by the introduction of a complete statistical knowledge in the assimilation process.

[1] C. Bertoglio, P. Moireau and Jean-Frédéric Gerbeau, Sequential parameter estimation for fluid-structure problems. Application to hemodynamics. Inter. J. Numer. Methods Biomed. Eng. 28 (2012) 434-455. RR-7657. | MR

[2] 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).

[3] D.C. Boes, Fa Graybill and A.M. Mood, Introduction to the Theory of Statistics. McGraw-Hill (1974). | Zbl

[4] 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. | MR

[5] M. D'Elia, Ph.D. thesis.

[6] 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.

[7] 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). | Zbl

[8] P.M. Den Reijer, D. Sallee, P. Van Der Velden, E. Zaaijer, W.J. Parks, S. Ramamurthy, T. Robbie, G. Donati C. Lamphier, R. Beekman and M. Brummer, Hemodynamic predictors of aortic dilatation in bicuspid aortic valve by velocity-encoded cardiovascular magnetic resonance. J. Cardiovasc. Magn. Reson. 12 (2010) 4.

[9] H.A. Van Der Vorst and C. Vuik, Gmresr: a family of nested gmres methods. Numer. Linear Algebra Appl. 1 (1994) 369-386. | MR | Zbl

[10] 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.

[11] L. Formaggia, A. Veneziani and C. Vergara. SIAM J. Sci. Comput. (2008).

[12] L. Formaggia, A. Veneziani and C. Vergara. Comput. Methods Appl. Mech. Eng. (2010). | MR

[13] 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. | MR

[14] M.D. Gunzburger, Perspectives in flow control and optimization. Society for Industrial Mathematics 5 (2003). | MR | Zbl

[15] 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). | MR | Zbl

[16] J.J. Heys, T.A. Manteuffel, S.F. Mccormick, M. Milano, J. Westerdale and M. Belohlavek, Weighted least-squares finite elements based on particle imaging velocimetry data. J. Comput. Phys. 229 (2010) 107-118. | MR

[17] J. G. Heywood, R. Rannacher and S. Turek, Artificial boundaries and flux pressure conditions for the incompressible navier-stokes equations. Int. J. Numer. Methods Fluids 22 (1996) 325-352. | MR | Zbl

[18] R.A. Johnson and D.W. Wichern, Applied multivariate statistical analysis. Prentice-Hall, Inc., Upper Saddle River, NJ, USA (1988). | Zbl

[19] J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems. Springer (2005). | MR | Zbl

[20] E.M. Kalman, A new approach to linear filtering and prediction problems. Trans. ASME-J. Basic Eng. 82 (1960) 35-45.

[21] D. Kay, D. Loghin and A. Wathen, A preconditioner for the steady-state navier-stokes equations. SIAM J. Sci. Comput. 24 (2002) 237-256. | MR | Zbl

[22] 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.

[23] P. Moireau and D. Chapelle, Reduced-order unscented kalman filtering with application to parameter identification in large-dimensional systems. ESAIM: COCV 17 (2011) 380-405. | Numdam | MR | Zbl

[24] J. Nocedal and S. Wright, Numerical Optimization. Springer (2000). | MR | Zbl

[25] M. Perego, A. Veneziani and C. Vergara, 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. | MR | Zbl

[26] A. Quarteroni, G. Rozza and A. Manzoni, Certified reduced basis approximation for parametrized partial differential equations and applications. J. Math. Ind. 1 (2011) 3. | MR | Zbl

[27] D. Silvester, H. Elman, D. Kay and A. Wathen, Efficient preconditioning of the linearized navier-stokes equations for incompressible flow. J. Comput. Appl. Math. 128 (2001) 261-279. | MR | Zbl

[28] A. Tarantola, Inverse problem theory and methods for model parameter estimation. Society for Industrial Mathematics (2005). | MR | Zbl

[29] A. Veneziani, Boundary conditions for blood flow problems, in Proc. of ENUMATH97, edited by R. Rannacher et al., World Sci. Publishing (1998). | MR | Zbl

[30] A. Veneziani, Mathematical and Numerical Modeling of Blood flow Problems. Ph.D. thesis, Politecnico di Milano, Italy (1998).

[31] C. Vuik, New insights in gmres-like methods with variable preconditioners. J. Comput. Appl. Math. 61 (1995) 189-204. | MR | Zbl