# American Institute of Mathematical Sciences

November  2013, 7(4): 1271-1293. doi: 10.3934/ipi.2013.7.1271

## Factorization method for the inverse Stokes problem

 1 Zentrum für Technomathematik, Universität Bremen, 28359 Bremen, Germany, Germany

Received  January 2013 Revised  August 2013 Published  November 2013

We propose an imaging technique for the detection of porous inclusions in a stationary flow based on the Factorization method. The stationary flow is described by the Stokes-Brinkmann equations, a standard model for a flow through a (partially) porous medium, involving the deformation tensor of the flow and the permeability tensor of the porous inclusion. On the boundary of the domain we prescribe Robin boundary conditions that provide the freedom to model, e.g., in- or outlets for the flow. The direct Stokes-Brinkmann problem to find a velocity field and a pressure for given boundary data is a mixed variational problem lacking coercivity due to the indefinite pressure part. It is well-known that indefinite problems are difficult to tackle theoretically using Factorization methods. Interestingly, the Factorization method can nevertheless be applied to this non-coercive problem, as long as one uses data consisting only of velocity measurements. We provide numerical experiments showing the feasibility of the proposed technique.
Citation: Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems and Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271
##### References:
 [1] C. Alvarez, C. Conca, L. Friz, O. Kavian and J. H. Ortega, Identification of immersed obstacles via boundary measurements, Inverse Problems, 21 (2005), 1531-1552. doi: 10.1088/0266-5611/21/5/003. [2] C. J. Alves, R. Kress and A. L. Silvestre, Integral equations for an inverse boundary value problem for the two-dimensional stokes equations, Journal of Inverse and Ill-Posed Problems, 15 (2007), 461-481. doi: 10.1515/jiip.2007.026. [3] A. Ben Abda, M. Hassine, M. Jaoua and M. Masmoudi, Topological sensitivity analysis for the location of small cavities in stokes flows. SIAM Journal on Control and Optimization, 48 (2009), 2871-2900. doi: 10.1137/070704332. [4] A. Ballerini, Stable determination of a body immersed in a fluid: The nonlinear stationary case, Applicable Analysis, 92 (2013), 460-481. doi: 10.1080/00036811.2011.628173. [5] M. Badra, F. Caubet and M. Dambrine, Detecting an obstacle immersed in a fluid by shape optimization methods, M3AS, 21 (2011), 2069-2101. doi: 10.1142/S0218202511005660. [6] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Springer Verlag, New York, 2008. doi: 10.1007/978-0-387-75934-0. [7] C. Conca, M. Malik and A. Munnier, Detection of a moving rigid solid in a perfect fluid, Inverse Problems, 26 (2010), 095010. doi: 10.1088/0266-5611/26/9/095010. [8] A. Ern and J. L. Guermond, Theory and Practice of Finite Elements, Springer Verlag, New York, 2004. [9] L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, 1998. [10] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems, Springer Verlag, New York, 2011. doi: 10.1007/978-0-387-09620-9. [11] M. Hanke and M. Brühl, Recent progress in electrical impedance tomography, Inverse Problems, 90 (2003), 65-90. doi: 10.1088/0266-5611/19/6/055. [12] N. Hyvönen, H. Hakula and S. Pursiainen, Numerical implementation of the factorization method within the complete electrode model of impedance tomography, Inverse Problems and Imaging, 1 (2007), 299-317. doi: 10.3934/ipi.2007.1.299. [13] H. Haddar and G. Migliorati, Numerical analysis of the Factorization Method for EIT with piecewise constant uncertain background, Inverse Problems, 29 (2013), 065009. doi: 10.1088/0266-5611/29/6/065009. [14] H. Heck, G. Uhlmann and J.-N. Wang, Reconstruction of obstacles immersed in an incompressible fluid, Inverse Problems and Imaging, 1 (2007), 63-76. doi: 10.3934/ipi.2007.1.63. [15] G. C. Hsiao and W. L. Wendland, Boundary Integral Equations, Springer Verlag, New York, 2008. doi: 10.1007/978-3-540-68545-6. [16] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford University Press, Oxford, 2008. [17] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer Verlag, New York, 2011. doi: 10.1007/978-1-4419-8474-6. [18] M. Krotkiewski, I. Ligaarden, K.-A. Lie and D. W. Schmid, On the importance of the Stokes-Brinkman equations for computing effective permeability in carbonate karst reservoirs, Commun. Comput. Phys., 10 (2011), 1315-1332. [19] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, London, 1969. [20] M. Lewicka and S. Müller, The uniform Korn-Poincaré inequality in thin domains, Annales de l'Institut Henri Poincare - Non Linear Analysis, 28 (2011), 443-469. doi: 10.1016/j.anihpc.2011.03.003. [21] W. C. H. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. [22] C. L. M. H. Navier, Sur les lois du mouvement des fluides, Mem. Acad. R. Sci. Inst. Fr., 6 (1827), 389-440. [23] P. Popov, Y. Efendiev and G. Qin, Multiscale modeling and simulations of flows in naturally fractured karst reservoirs, Commun. Comput. Phys., 6 (2009), 162-184. doi: 10.4208/cicp.2009.v6.p162. [24] C. Pozrikidis, Boundary Integral and Singularity Methods for Linearized Viscous Flow, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511624124. [25] V. Tsiporin, Charakterisierung Eines Gebiets Durch Spektraldaten eines Dirichletproblems zur Stokesgleichung, (German) [Characterization of a Domain via the Spectral data of a Dirichlet Problem for the Stokes Equation], PhD thesis, Georg-August-Universität Göttingen, 2003. [26] Q. M. Z. Zia and R. Potthast, Flow and shape reconstructions from remote measurements, Math. Meth. Appl. Sci., 36 (2013), 1171-1186. doi: 10.1002/mma.2670.

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##### References:
 [1] C. Alvarez, C. Conca, L. Friz, O. Kavian and J. H. Ortega, Identification of immersed obstacles via boundary measurements, Inverse Problems, 21 (2005), 1531-1552. doi: 10.1088/0266-5611/21/5/003. [2] C. J. Alves, R. Kress and A. L. Silvestre, Integral equations for an inverse boundary value problem for the two-dimensional stokes equations, Journal of Inverse and Ill-Posed Problems, 15 (2007), 461-481. doi: 10.1515/jiip.2007.026. [3] A. Ben Abda, M. Hassine, M. Jaoua and M. Masmoudi, Topological sensitivity analysis for the location of small cavities in stokes flows. SIAM Journal on Control and Optimization, 48 (2009), 2871-2900. doi: 10.1137/070704332. [4] A. Ballerini, Stable determination of a body immersed in a fluid: The nonlinear stationary case, Applicable Analysis, 92 (2013), 460-481. doi: 10.1080/00036811.2011.628173. [5] M. Badra, F. Caubet and M. Dambrine, Detecting an obstacle immersed in a fluid by shape optimization methods, M3AS, 21 (2011), 2069-2101. doi: 10.1142/S0218202511005660. [6] S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Springer Verlag, New York, 2008. doi: 10.1007/978-0-387-75934-0. [7] C. Conca, M. Malik and A. Munnier, Detection of a moving rigid solid in a perfect fluid, Inverse Problems, 26 (2010), 095010. doi: 10.1088/0266-5611/26/9/095010. [8] A. Ern and J. L. Guermond, Theory and Practice of Finite Elements, Springer Verlag, New York, 2004. [9] L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, 1998. [10] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems, Springer Verlag, New York, 2011. doi: 10.1007/978-0-387-09620-9. [11] M. Hanke and M. Brühl, Recent progress in electrical impedance tomography, Inverse Problems, 90 (2003), 65-90. doi: 10.1088/0266-5611/19/6/055. [12] N. Hyvönen, H. Hakula and S. Pursiainen, Numerical implementation of the factorization method within the complete electrode model of impedance tomography, Inverse Problems and Imaging, 1 (2007), 299-317. doi: 10.3934/ipi.2007.1.299. [13] H. Haddar and G. Migliorati, Numerical analysis of the Factorization Method for EIT with piecewise constant uncertain background, Inverse Problems, 29 (2013), 065009. doi: 10.1088/0266-5611/29/6/065009. [14] H. Heck, G. Uhlmann and J.-N. Wang, Reconstruction of obstacles immersed in an incompressible fluid, Inverse Problems and Imaging, 1 (2007), 63-76. doi: 10.3934/ipi.2007.1.63. [15] G. C. Hsiao and W. L. Wendland, Boundary Integral Equations, Springer Verlag, New York, 2008. doi: 10.1007/978-3-540-68545-6. [16] A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford University Press, Oxford, 2008. [17] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer Verlag, New York, 2011. doi: 10.1007/978-1-4419-8474-6. [18] M. Krotkiewski, I. Ligaarden, K.-A. Lie and D. W. Schmid, On the importance of the Stokes-Brinkman equations for computing effective permeability in carbonate karst reservoirs, Commun. Comput. Phys., 10 (2011), 1315-1332. [19] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, London, 1969. [20] M. Lewicka and S. Müller, The uniform Korn-Poincaré inequality in thin domains, Annales de l'Institut Henri Poincare - Non Linear Analysis, 28 (2011), 443-469. doi: 10.1016/j.anihpc.2011.03.003. [21] W. C. H. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. [22] C. L. M. H. Navier, Sur les lois du mouvement des fluides, Mem. Acad. R. Sci. Inst. Fr., 6 (1827), 389-440. [23] P. Popov, Y. Efendiev and G. Qin, Multiscale modeling and simulations of flows in naturally fractured karst reservoirs, Commun. Comput. Phys., 6 (2009), 162-184. doi: 10.4208/cicp.2009.v6.p162. [24] C. Pozrikidis, Boundary Integral and Singularity Methods for Linearized Viscous Flow, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511624124. [25] V. Tsiporin, Charakterisierung Eines Gebiets Durch Spektraldaten eines Dirichletproblems zur Stokesgleichung, (German) [Characterization of a Domain via the Spectral data of a Dirichlet Problem for the Stokes Equation], PhD thesis, Georg-August-Universität Göttingen, 2003. [26] Q. M. Z. Zia and R. Potthast, Flow and shape reconstructions from remote measurements, Math. Meth. Appl. Sci., 36 (2013), 1171-1186. doi: 10.1002/mma.2670.
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