American Institute of Mathematical Sciences

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

Factorization method for the inverse Stokes problem

Armin Lechleiter 1, and  Tobias Rienmüller 1,

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.
Keywords: Inverse boundary value problem, factorization method., Stokes equation.
Mathematics Subject Classification: Primary: 35R30, 65N21; Secondary: 76D0.
Citation: Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271
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show all references

References:
[1]

Inverse Problems, 21 (2005), 1531-1552. doi: 10.1088/0266-5611/21/5/003.  Google Scholar

[2]

Journal of Inverse and Ill-Posed Problems, 15 (2007), 461-481. doi: 10.1515/jiip.2007.026.  Google Scholar

[3]

SIAM Journal on Control and Optimization, 48 (2009), 2871-2900. doi: 10.1137/070704332.  Google Scholar

[4]

Applicable Analysis, 92 (2013), 460-481. doi: 10.1080/00036811.2011.628173.  Google Scholar

[5]

M3AS, 21 (2011), 2069-2101. doi: 10.1142/S0218202511005660.  Google Scholar

[6]

Springer Verlag, New York, 2008. doi: 10.1007/978-0-387-75934-0.  Google Scholar

[7]

Inverse Problems, 26 (2010), 095010. doi: 10.1088/0266-5611/26/9/095010.  Google Scholar

[8]

Springer Verlag, New York, 2004.  Google Scholar

[9]

American Mathematical Society, Providence, 1998.  Google Scholar

[10]

Springer Verlag, New York, 2011. doi: 10.1007/978-0-387-09620-9.  Google Scholar

[11]

Inverse Problems, 90 (2003), 65-90. doi: 10.1088/0266-5611/19/6/055.  Google Scholar

[12]

Inverse Problems and Imaging, 1 (2007), 299-317. doi: 10.3934/ipi.2007.1.299.  Google Scholar

[13]

Inverse Problems, 29 (2013), 065009. doi: 10.1088/0266-5611/29/6/065009.  Google Scholar

[14]

Inverse Problems and Imaging, 1 (2007), 63-76. doi: 10.3934/ipi.2007.1.63.  Google Scholar

[15]

Springer Verlag, New York, 2008. doi: 10.1007/978-3-540-68545-6.  Google Scholar

[16]

Oxford University Press, Oxford, 2008.  Google Scholar

[17]

Springer Verlag, New York, 2011. doi: 10.1007/978-1-4419-8474-6.  Google Scholar

[18]

Commun. Comput. Phys., 10 (2011), 1315-1332. Google Scholar

[19]

Gordon and Breach, London, 1969.  Google Scholar

[20]

Annales de l'Institut Henri Poincare - Non Linear Analysis, 28 (2011), 443-469. doi: 10.1016/j.anihpc.2011.03.003.  Google Scholar

[21]

Cambridge University Press, Cambridge, 2000.  Google Scholar

[22]

Mem. Acad. R. Sci. Inst. Fr., 6 (1827), 389-440. Google Scholar

[23]

Commun. Comput. Phys., 6 (2009), 162-184. doi: 10.4208/cicp.2009.v6.p162.  Google Scholar

[24]

Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511624124.  Google Scholar

[25]

PhD thesis, Georg-August-Universität Göttingen, 2003. Google Scholar

[26]

Math. Meth. Appl. Sci., 36 (2013), 1171-1186. doi: 10.1002/mma.2670.  Google Scholar

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