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Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations

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  • This paper concerns the numerical resolution of a data completion problem for the time-harmonic Maxwell equations in the electric field. The aim is to recover the missing data on the inaccessible part of the boundary of a bounded domain from measured data on the accessible part. The non-iterative quasi-reversibility method is studied and different mixed variational formulations are proposed. Well-posedness, convergence and regularity results are proved. Discretization is performed by means of edge finite elements. Various two- and three-dimensional numerical simulations attest the efficiency of the method, in particular for noisy data.

    Mathematics Subject Classification: Primary: 65N20, 35Q61; Secondary: 65N21.

    Citation:

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  • Figure 1.  Choice of the accessible part $ \Gamma_0 $ (grey line). Left: configuration G34. Right: configuration GE37

    Figure 2.  Unit disc. QR method. Relative error $ \frac{\| {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta\|_{0, \Omega}}{\| {{\boldsymbol{E}}}\|_{0, \Omega}} $ with respect to the regularization parameter $ \delta $. Left: configuration G34. Right: configuration GE37

    Figure 3.  Unit disc. QR method. Modulus of the error $ | {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_{\delta}| $. Left: configuration G34. Right: configuration GE37

    Figure 4.  Choice of the accessible part $ \Gamma_0 $ (grey line). Left: configuration GExt ($ \Gamma_0 $ covers 57% of the ring boundary). Middle: configuration G34 (43%). Right: configuration GE37 (47%)

    Figure 5.  Ring. QR method. Relative error $ \frac{\| {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta\|_{0, \Omega}}{\| {{\boldsymbol{E}}}\|_{0, \Omega}} $ with respect to the regularization parameter $ \delta $. Left: configuration GExt. Middle: configuration G34. Right: configuration GE37

    Figure 6.  Ring. QR method. Modulus of the error $ | {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta| $. Left: configuration GExt. Middle: configuration G34. Right: configuration GE37

    Figure 7.  Ring. QR method. Extension/restriction method. Relative error $ \frac{\| {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta\|_{0, \Omega}}{\| {{\boldsymbol{E}}}\|_{0, \Omega}} $ with respect to the regularization parameter $ \delta $. Left: configuration GExt. Middle: configuration G34. Right: configuration GE37

    Figure 8.  Ring. QR method. Extension/restriction method. Modulus of the error $ | {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta| $. Left: configuration GExt. Middle: configuration G34. Right: configuration GE37

    Figure 9.  Unit disc. 5% noisy data. RR-QR method. Relative error $ \frac{\| {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\beta\|_{0, \Omega}}{\| {{\boldsymbol{E}}}\|_{0, \Omega}} $ with respect to the regularization parameter $ \delta $ at fixed $ \eta $ and $ \nu = \delta $. Left: configuration G34. Right: configuration GE37

    Figure 10.  Unit disc. 5% noisy data. RR-QR method. Modulus of the error $ | {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\beta| $. Left: configuration G34. Right: configuration GE37

    Figure 11.  Ring. 5% noisy data. RR-QR method with extension/restriction. Relative error for $ {{\boldsymbol{E}}} $ in $ L^2(\Omega) $-norm with respect to $ \delta $ for $ \nu = \delta $. $ \eta $ automatically fixed from noise level

    Figure 12.  Ring. 5% noisy data. RR-QR method with extension/restriction. Modulus of the error in the approximation of $ {{\boldsymbol{E}}} $ for optimal $ \delta $ and $ \nu = \delta $. $ \eta $ automatically fixed from noise level. Left: configuration GExt. Middle: configuration G34. Right: configuration GE37

    Figure 13.  Accessible part $ \Gamma_0 $ in 3D configurations. Set of 128 electrodes

    Figure 14.  Relative error $ \frac{\| {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta\|_{0, \Omega}}{\| {{\boldsymbol{E}}}\|_{0, \Omega}} $ with respect to the regularization parameter $ \delta $. Left: Unit ball of $ {\mathbb{R}}^3 $. Right: 3D ring of internal radius 0.7

    Figure 15.  Error in the three-dimensional ring. Left: view of the external boundary. Right: cut showing the internal boundary

    Figure 16.  L-curve corresponding to the RR-QR method with extension/restriction in the ring. 5% noisy data. Left: configuration G37 (2D). Right: 3D ring

    Table 1.  Unit disc. QR method. Errors for $ \delta = 9.103\text{e}-7 $

    Configuration $ \frac{\| {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta\|_{0, \Omega}}{\| {{\boldsymbol{E}}}\|_{0, \Omega}} $ $ \frac{\|( {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta) \times {{\boldsymbol{n}}}\|_{0, \Gamma_1}}{\| {{\boldsymbol{E}}} \times {{\boldsymbol{n}}}\|_{0, \Gamma_1}} $ $ \| {{\boldsymbol{F}}}_\delta\|_{0, \Omega} $
    G34 1.1244e-02 1.7302e-01 1.8374e-04
    GE37 7.6288e-04 1.5185e-02 1.8224e-04
     | Show Table
    DownLoad: CSV

    Table 2.  Ring. QR method. Errors

    Configuration $ \frac{\| {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta\|_{0, \Omega}}{\| {{\boldsymbol{E}}}\|_{0, \Omega}} $ $ \frac{\|( {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta) \times {{\boldsymbol{n}}}\|_{0, \Gamma_1}}{\| {{\boldsymbol{E}}} \times {{\boldsymbol{n}}}\|_{0, \Gamma_1}} $ $ \frac{\|( {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta) \times {{\boldsymbol{n}}}\|_{0, \Gamma_\text{i}}}{\| {{\boldsymbol{E}}} \times {{\boldsymbol{n}}}\|_{0, \Gamma_\text{i}}} $ $ \| {{\boldsymbol{F}}}_\delta\|_{0, \Omega} $
    GExt 3.9423e-04 6.6661e-04 6.6661e-04 1.7964e-04
    G34 3.5093e-01 6.9598e-01 4.8011e-01 8.1894e-05
    GE37 9.9784e-03 3.9068e-02 3.8737e-02 8.4337e-05
     | Show Table
    DownLoad: CSV

    Table 3.  Ring. QR method. Extension/restriction method. Errors

    Configuration $ \frac{\| {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta\|_{0, \Omega}}{\| {{\boldsymbol{E}}}\|_{0, \Omega}} $ $ \frac{\|( {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta) \times {{\boldsymbol{n}}}\|_{0, \Gamma_1}}{\| {{\boldsymbol{E}}} \times {{\boldsymbol{n}}}\|_{0, \Gamma_1}} $ $ \frac{\|( {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta) \times {{\boldsymbol{n}}}\|_{0, \Gamma_\text{i}}}{\| {{\boldsymbol{E}}} \times {{\boldsymbol{n}}}\|_{0, \Gamma_\text{i}}} $ $ \| {{\boldsymbol{F}}}_\delta\|_{0, \Omega} $
    GExt 3.7745e-04 1.5665e-04 1.5665e-04 3.1125e-04
    G34 3.4345e-02 1.2558e-01 8.1477e-03 2.6564e-04
    GE37 1.7728e-03 1.1968e-02 2.9520e-04 9.0131e-05
     | Show Table
    DownLoad: CSV

    Table 4.  Unit disc. 5% noisy data. RR-QR method. Errors in the approximation of $ {{\boldsymbol{E}}} $ for $ \nu = \delta $ at optimal $ \delta $ and automatically fixed $ \eta $

    Configuration $ \frac{\| {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta\|_{0, \Omega}}{\| {{\boldsymbol{E}}}\|_{0, \Omega}} $ $ \frac{\|( {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta) \times {{\boldsymbol{n}}}\|_{0, \Gamma_0}}{\| {{\boldsymbol{E}}} \times {{\boldsymbol{n}}}\|_{0, \Gamma_0}} $ $ \frac{\|( {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta) \times {{\boldsymbol{n}}}\|_{0, \Gamma_1}}{\| {{\boldsymbol{E}}} \times {{\boldsymbol{n}}}\|_{0, \Gamma_1}} $ $ \| {{\boldsymbol{F}}}_\delta\|_{0, \Omega} $
    G34 3.4698e-02 2.9145e-02 3.5686e-01 7.0667e-03
    GE37 1.8550e-02 3.2848e-02 1.7154e-01 1.1857e-02
     | Show Table
    DownLoad: CSV

    Table 5.  Ring. 5% noisy data. RR-QR method with extension/restriction. Errors in the approximation of $ {{\boldsymbol{E}}} $ for optimal $ \delta $ and $ \nu = \delta $. $ \eta $ automatically fixed from noise level. Errors on parts of the boundary refer to the $ L^2 $-norm of the tangential trace

    Configuration $ \frac{\| {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta\|_{0, \Omega}}{\| {{\boldsymbol{E}}}\|_{0, \Omega}} $ $ \Gamma_0 $ $ \Gamma_1 $ $ \Gamma_\text{i} $ $ \| {{\boldsymbol{F}}}_\delta\|_{0, \Omega} $
    GExt 9.2775e-03 3.3871e-02 4.2425e-03 4.2425e-03 1.0602e-02
    G34 7.9236e-02 3.0600e-02 1.5695e-01 4.2697e-02 8.7901e-03
    GE37 2.6712e-02 3.3503e-02 7.7043e-02 1.6996e-02 1.3596e-02
     | Show Table
    DownLoad: CSV

    Table 6.  Errors in the unit ball of $ {\mathbb{R}}^3 $ and in a three dimensional ring. Errors on parts of the boundary refer to the $ L^2 $-norm of the tangential trace

    Domain $ \frac{\| {{\boldsymbol{E}}} - {{\boldsymbol{E}}}_\delta\|_{0, \Omega}}{\| {{\boldsymbol{E}}}\|_{0, \Omega}} $ $ \Gamma_0 $ $ \Gamma_1 $ $ \Gamma_\text{i} $ $ \| {{\boldsymbol{F}}}_\delta\|_{0, \Omega} $
    Ball 1.0834e-01 2.3864e-02 4.0949e-01 $ \cdot $ 2.9336e-03
    Ring 1.2414e-01 1.1567e-02 2.7969e-01 5.6528e-02 1.6497e-03
     | Show Table
    DownLoad: CSV
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