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Nash strategies for the inverse inclusion Cauchy-Stokes problem

  • * Corresponding author: A. Habbal

    * Corresponding author: A. Habbal 
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  • We introduce a new algorithm to solve the problem of detecting unknown cavities immersed in a stationary viscous fluid, using partial boundary measurements. The considered fluid obeys a steady Stokes regime, the cavities are inclusions and the boundary measurements are a single compatible pair of Dirichlet and Neumann data, available only on a partial accessible part of the whole boundary. This inverse inclusion Cauchy-Stokes problem is ill-posed for both the cavities and missing data reconstructions, and designing stable and efficient algorithms is not straightforward. We reformulate the problem as a three-player Nash game. Thanks to an identifiability result derived for the Cauchy-Stokes inclusion problem, it is enough to set up two Stokes boundary value problems, then use them as state equations. The Nash game is then set between 3 players, the two first targeting the data completion while the third one targets the inclusion detection. We used a level-set approach to get rid of the tricky control dependence of functional spaces, and we provided the third player with the level-set function as strategy, with a cost functional of Kohn-Vogelius type. We propose an original algorithm, which we implemented using Freefem++. We present 2D numerical experiments for three different test-cases.The obtained results corroborate the efficiency of our 3-player Nash game approach to solve parameter or shape identification for Cauchy problems.

    Mathematics Subject Classification: Primary: 49J20, 65K10; Secondary: 65N06, 90C30.

    Citation:

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  • Figure 1.  An example of the geometric configuration of the problem : the whole domain including cavities is denoted by $ \Omega $. It contains an inclusion $ \omega^* $. The boundary of $ \Omega $ is composed of $ \Gamma_{\!\! c} $, an accessible part where over-specified data are available, and an inaccessible part $ \Gamma_{\!\!i } $ where the data are missing

    Figure 2.  Different situations

    Figure 5.  Test case A. Reconstruction of the inclusion shape and missing boundary data with noise free Dirichlet data over $ \Gamma_{ c}$. (a) initial contour is $\phi^{(0)}_1$ (b) exact inclusion shape -green line- and computed one - blue dashed- (c) exact -line- and computed -dashed line- first component of the velocity over $ \Gamma_{i }$ (d) exact -line- and computed -dashed line- second component of the velocity over $ \Gamma_{i }$ (e) exact -line- and computed -dashed line- first component of the normal stress over $ \Gamma_{i }$ (f) exact -line- and computed -dashed line- second component of the normal stress over $ \Gamma_{i }$

    Figure 6.  Test case A. Reconstruction of the inclusion shape and missing boundary data with noise free Dirichlet data over $ \Gamma_{ c}$.(a) initial contour is $\phi^{(0)}_2$ (b) exact inclusion shape -green line- and computed one - blue dashed- (c) exact -line- and computed -dashed line- first component of the velocity over $ \Gamma_{i }$ (d) exact -line- and computed -dashed line- second component of the velocity over $ \Gamma_{i }$ (e) exact -line- and computed -dashed line- first component of the normal stress over $ \Gamma_{i }$ (f) exact -line- and computed -dashed line- second component of the normal stress over $ \Gamma_{i }$

    Figure 7.  Test case A. Reconstruction of the inclusion shape and missing boundary data with noisy Dirichlet data over $ \Gamma_{ c}$ with noise levels $\sigma = \lbrace 1\%, 3\%, 5\% \rbrace$.(a) initial contour is $\phi^{(0)}_2$ (b) exact inclusion shape -green line- and computed ones for different noise levels (c) exact and computed first components of the velocity over $ \Gamma_{i }$ (d) exact and computed second components of the velocity over $ \Gamma_{i }$ (e) exact and computed first components of the normal stress over $ \Gamma_{i }$ (f) exact and computed second components of the normal stress over $ \Gamma_{i }$

    Figure 8.  Test case B. Reconstruction of the inclusion shape and missing boundary data with noise free Dirichlet data over $ \Gamma_{ c}$. (a) initial contour is $\phi^{(0)}_2$ (b) exact inclusion shape -green line- and computed one - blue dashed- (c) exact -line- and computed -dashed line- first component of the velocity over $ \Gamma_{i }$ (d) exact -line- and computed -dashed line- second component of the velocity over $ \Gamma_{i }$ (e) exact -line- and computed -dashed line- first component of the normal stress over $ \Gamma_{i }$ (f) exact -line- and computed -dashed line- second component of the normal stress over $ \Gamma_{i }$

    Figure 9.  Test case B. Reconstruction of the inclusion shape and missing boundary data with noisy Dirichlet data over $ \Gamma_{ c}$ with levels $\sigma = \lbrace 1\%, 3\%, 5\% \rbrace$.(a) initial contour is $\phi^{(0)}_2$ (b) exact inclusion shape -green line- and computed ones for different noise levels (c) exact and computed first components of the velocity over $ \Gamma_{i }$ (d) exact and computed second components of the velocity over $ \Gamma_{i }$ (e) exact and computed first components of the normal stress over $ \Gamma_{i }$ (f) exact and computed second components of the normal stress over $ \Gamma_{i }$

    Figure 10.  Test case C. Reconstruction of the inclusion shape and missing boundary data with noise free Dirichlet data over $ \Gamma_{ c}$. (a) initial contour is $\phi^{(0)}_2$ (b) exact inclusion shape -green line- and computed one - blue dashed- (c) exact -line- and computed -dashed line- first component of the velocity over $ \Gamma_{i }$ (d) exact -line- and computed -dashed line- second component of the velocity over $ \Gamma_{i }$ (e) exact -line- and computed -dashed line- first component of the normal stress over $ \Gamma_{i }$ (f) exact -line- and computed -dashed line- second component of the normal stress over $ \Gamma_{i }$

    Figure 11.  Test case C. Reconstruction of the inclusion shape and missing boundary data with noisy Dirichlet data over $ \Gamma_{ c}$ with levels $\sigma = \lbrace 1\%, 3\%, 5\% \rbrace$.(a) initial contour is $\phi^{(0)}_2$ (b) exact inclusion shape -green line- and computed ones for different noise levels (c) exact and computed first components of the velocity over $ \Gamma_{i }$ (d) exact and computed second components of the velocity over $ \Gamma_{i }$ (e) exact and computed first components of the normal stress over $ \Gamma_{i }$ (f) exact and computed second components of the normal stress over $ \Gamma_{i }$

    Figure 3.  Test case A. (Left) sensitivity of the reconstruction w.r.t. the mesh size (on abscissae : the number of F.E. nodes on the boundary $\partial \Omega$). (Right) sensitivity of the reconstruction w.r.t. the distance to the inaccessible boundary $ \Gamma_{i }$ (on abscissae : the distance of the center of the circular inclusion from $ \Gamma_{i }$)

    Figure 4.  Test case C. (Left) Mesh used for solving the direct problem with the P1bubble-P1 finite element, in order to construct the synthetic data. (Right) Mesh used for solving the coupled inverse problem with P2-P1 finite element, using the P1 bubble-P1 synthetic data

    Figure 12.  Assessing Inverse-Crime-Free reconstruction. Test case C. Top: initial and optimal contour. Middle: the two components of the velocity on $\Gamma_i$. Bottom: the two components of the normal stress on $\Gamma_i$ ($err_D = 0.0615048,$ $err_N = 0.124296,$ and $err_O = 0.113156)$

    Table 1.  Test-case A. $L^2$ relative errors on missing data on $\Gamma_i$ (on Dirichlet and Neumann data), and the error between the reconstructed and the real shape of the inclusion for various noise levels

    Noise level $\sigma=0\%$ $\sigma=1\%$ $\sigma=3\%$ $\sigma=5\%$
    $err_D$ 0.010 0.015 0.039 0.063
    $err_N$ 0.031 0.033 0.051 0.07
    $err_O$ 0.032 0.043 0.066 0.117
     | Show Table
    DownLoad: CSV

    Table 2.  Test-case C. $L^2$-errors on missing data over $ \Gamma_{i }$ (on Dirichlet and Neumann data), and the error between the reconstructed and the real shape for various noise levels

    Noise level $\sigma=0\%$ $\sigma=1\%$ $\sigma=3\%$ $\sigma=5\%$
    $err_D$ 0.042 0.044 0.046 0.08
    $err_N$ 0.095 0.1 0.13 0.16
    $err_O$ 0.099 0.11 0.13 0.15
     | Show Table
    DownLoad: CSV

    Table 3.  Relative errors on the reconstructed missing data and inclusion shape for the Stokes problem (with noise free measurements), compared for a classical Nash algorithm and Algorithm 2: (left) test-case A (right) test-case C

    Case A Classical algorithm Algorithm 2
    $err_D$ 0.058 0.033
    $err_N$ 0.106 0.032
    $err_O$ 0.358 0.140
    Case C Classical algorithm Algorithm 2
    $err_D$ 0.067 0.058
    $err_N$ 0.208 0.122
    $err_O$ 0.566 0.167
     | Show Table
    DownLoad: CSV
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