Recovering boundary data: The Cauchy Stokes system

Abstract This paper is concerned with the severely ill-posed Cauchy–Stokes problem. We are interested in a data completion problem which is exploited to detect small leaks to control water loss Kim et al. (2008) [1] . This inverse problem is rephrased into an optimization one: An energy-like error functional is introduced. We prove that the optimality condition of the first order is equivalent to solving an interfacial equation which turns out to be a Cauchy-Steklov-Poincare operator. Numerical trials highlight the efficiency of the method.

[1]  R. Kohn,et al.  Determining conductivity by boundary measurements II. Interior results , 1985 .

[2]  Daniel Lesnic,et al.  An alternating method for the stationary Stokes system , 2006 .

[3]  Thouraya Baranger,et al.  Solving Cauchy problems by minimizing an energy-like functional , 2006 .

[4]  Faker Ben Belgacem,et al.  On Cauchy's problem: I. A variational Steklov–Poincaré theory , 2005 .

[5]  Daniel Lesnic,et al.  An iterative method for the reconstruction of a stationary flow , 2007 .

[6]  Amel Ben Abda,et al.  Data completion for the Stokes system , 2009 .

[7]  Amel Ben Abda,et al.  Data completion via an energy error functional , 2005 .

[8]  Robert V. Kohn,et al.  Determining conductivity by boundary measurements , 1984 .

[9]  Vladimir Maz’ya,et al.  An iterative method for solving the Cauchy problem for elliptic equations , 1991 .

[10]  Derek B. Ingham,et al.  Boundary element two-dimensional solution of an inverse Stokes problem , 2000 .

[11]  A. Fursikov.  Optimal Control of Distributed Systems: Theory and Applications , 2000 .

[12]  Daniel Lesnic,et al.  Reconstruction of a stationary flow from incomplete boundary data using iterative methods , 2006, European Journal of Applied Mathematics.

[13]  Derek B. Ingham,et al.  An inverse Stokes problem using interior pressure data , 2002 .

[14]  A variational conjugate gradient method for determining the fluid velocity of a slow viscous flow , 2006 .

[15]  Hiroshi Tsukamoto,et al.  Leak detection in pipe using transient flow and genetic algorithm , 2008 .

[16]  A. B. Abda,et al.  Missing boundary data reconstruction by the factorization method , 2009 .

[17]  Alfio Quarteroni,et al.  An Iterative Procedure with Interface Relaxation for Domain Decomposition Methods , 1988 .