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October  2019, 13(5): 1095-1111. doi: 10.3934/ipi.2019049

## Integral equations for biharmonic data completion

 1 Faculty of Applied Mathematics and Informatics, Ivan Franko National University of Lviv, 79000 Lviv, Ukraine 2 Mathematics, EAS, Aston University, B4 7ET Birmingham, UK

Received  February 2019 Revised  April 2019 Published  July 2019

A boundary integral based method for the stable reconstruction of missing boundary data is presented for the biharmonic equation. The solution (displacement) is known throughout the boundary of an annular domain whilst the normal derivative and bending moment are specified only on the outer boundary curve. A recent iterative method is applied for the data completion solving mixed problems throughout the iterations. The solution to each mixed problem is represented as a biharmonic single-layer potential. Matching against the given boundary data, a system of boundary integrals is obtained to be solved for densities over the boundary. This system is discretised using the Nyström method. A direct approach is also given representing the solution of the ill-posed problem as a biharmonic single-layer potential and applying the similar techniques as for the mixed problems. Tikhonov regularization is employed for the solution of the corresponding discretised system. Numerical results are presented for several annular domains showing the efficiency of both data completion approaches.

Citation: Roman Chapko, B. Tomas Johansson. Integral equations for biharmonic data completion. Inverse Problems and Imaging, 2019, 13 (5) : 1095-1111. doi: 10.3934/ipi.2019049
##### References:
 [1] G. Alessandrini, E. Rosset and S. Vessella, Optimal three spheres inequality at the boundary for the Kirchhoff-Love plate's equation with Dirichlet conditions, Arch. Ration. Mech. Anal., 231 (2019), 1455-1486.  doi: 10.1007/s00205-018-1302-9. [2] A. Beshley, R. Chapko and B. T. Johansson, A boundary-domain integral equation method for an elliptic Cauchy problem with variable coefficients, in Analysis, Probability, Applications, and Computation, Eds. K. Lindahl, T. Lindström, L. G. Rodino, J. Toft, P. Wahlberg, Birkhäuser, 2019,493–501. doi: 10.1007/978-3-030-04459-6_47. [3] I. Borachok, R. Chapko and B. T. Johansson, Numerical solution of an elliptic 3-dimensional Cauchy problem by the alternating method and boundary integral equations, J. Inverse Ill-Posed Probl., 24 (2016), 711-725.  doi: 10.1515/jiip-2015-0053. [4] Y. Boukari and H. Haddar, A convergent data completion algorithm using surface integral equations, Inverse Problems, 31 (2015), 035011, 21pp. doi: 10.1088/0266-5611/31/3/035011. [5] F. Cakoni, G. C. Hsiao and W. L. Wendland, On the boundary integral equation method for a mixed boundary value problem of the biharmonic equation, Complex Var. Theory Appl., 50 (2005), 681-696.  doi: 10.1080/02781070500087394. [6] F. Cakoni and R. Kress, Integral equations for inverse problems in corrosion detection from partial Cauchy data, Inverse Probl. Imaging, 1 (2007), 229-245.  doi: 10.3934/ipi.2007.1.229. [7] R. Chapko and B. T. Johansson, On the numerical solution of a Cauchy problem for the Laplace equation via a direct integral equation approach, Inverse Probl. Imaging, 6 (2012), 25-38.  doi: 10.3934/ipi.2012.6.25. [8] R. Chapko and B. T. Johansson, Boundary-integral approach for the numerical solution of the Cauchy problem for the Laplace equation, Ukr. Mat. Zh., 68 (2016), 1665-1682.  doi: 10.1007/s11253-017-1339-1. [9] R. Chapko and B. T. Johansson, A boundary integral equation method for numerical solution of parabolic and hyperbolic Cauchy problems, Appl. Numer. Math., 129 (2018), 104-119.  doi: 10.1016/j.apnum.2018.03.004. [10] R. Chapko and B. T. Johansson, An iterative regularizing method for an incomplete boundary data problem for the biharmonic equation, ZAMM: Z. Angew. Math. Mech. 98 (2018), 2010–2021. doi: 10.1002/zamm.201800102. [11] G. Chen and J. Zhou, Boundary Element Methods with Applications to Nonlinear Problems, Atlantis Press, Paris, 2010.  doi: 10.2991/978-94-91216-27-5. [12] M. Costabel and M. Dauge, Invertibility of the biharmonic single layer potential operator, Integral Equations Operator Theory, 24 (1996), 46-67.  doi: 10.1007/BF01195484. [13] R. J. Duffin, Continuation of biharmonic functions by reflection, Duke Math. J., 22 (1955), 313-324.  doi: 10.1215/S0012-7094-55-02233-X. [14] R. Farwig, A note on the reflection principle for the biharmonic equation and the Stokes system, Acta Appl. Math., 37 (1994), 41-51.  doi: 10.1007/BF00995128. [15] M. Garshasbi and F. Hassani, Boundary temperature reconstruction in an inverse heat conduction problem using boundary integral equation method, Bull. Iranian Math. Soc., 42 (2016), 1039-1056. [16] G. C. Hsiao and W. L. Wendland, Boundary Integral Equations, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-68545-6. [17] V. A. Kozlov and V. G. Maz;ya, On iterative procedures for solving ill-posed boundary value problems that preserve differential equations, Algebra i Analiz, 1 (1989), 144–170; English transl.: Leningrad Math. J., 1 (1990), 1207–1228. [18] R. Kress, Linear Integral Equations, ed. 3, Springer-Verlag, New York, 2014. doi: 10.1007/978-1-4614-9593-2. [19] D. Lesnic, L. Elliott and D. B. Ingham, An iterative boundary element method for solving numerically the Cauchy problem for the Laplace equation, Eng. Anal. Bound. Elem., 20 (1997), 123-133.  doi: 10.1016/S0955-7997(97)00056-8. [20] J.-C. Liu and Q. G. Zhang, Cauchy problem for the Laplace equation in 2D and 3D doubly connected domains, CMES Comput. Model. Eng. Sci., 93 (2013), 203-219. [21] V. V. Meleshko, Selected topics in the history of the two-dimensional biharmonic problem, Appl. Mech. Rev., 56 (2003), 33-85.  doi: 10.1115/1.1521166. [22] I. Mitrea and M. Mitrea, Multi-layer Potentials and Boundary Problems for Higher-Order Elliptic Systems in Lipschitz Domains, Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-32666-0. [23] T. V. Savina, On the dependence of the reflection operator on boundary conditions for biharmonic functions, J. Math. Anal. Appl., 370 (2010), 716-725.  doi: 10.1016/j.jmaa.2010.04.036. [24] Y. Sun, Indirect boundary integral equation method for the Cauchy problem of the Laplace equation, J. Sci. Comput., 71 (2017), 469-498.  doi: 10.1007/s10915-016-0308-4. [25] Y. Sun, D. Zhang and F. Ma, A potential function method for the Cauchy problem of elliptic operators, J. Math. Anal. Appl., 395 (2012), 164-174.  doi: 10.1016/j.jmaa.2012.05.038. [26] G. C. Verchota, The biharmonic Neumann problem in Lipschitz domains, Acta Math., 194 (2005), 217-279.  doi: 10.1007/BF02393222.

show all references

##### References:
 [1] G. Alessandrini, E. Rosset and S. Vessella, Optimal three spheres inequality at the boundary for the Kirchhoff-Love plate's equation with Dirichlet conditions, Arch. Ration. Mech. Anal., 231 (2019), 1455-1486.  doi: 10.1007/s00205-018-1302-9. [2] A. Beshley, R. Chapko and B. T. Johansson, A boundary-domain integral equation method for an elliptic Cauchy problem with variable coefficients, in Analysis, Probability, Applications, and Computation, Eds. K. Lindahl, T. Lindström, L. G. Rodino, J. Toft, P. Wahlberg, Birkhäuser, 2019,493–501. doi: 10.1007/978-3-030-04459-6_47. [3] I. Borachok, R. Chapko and B. T. Johansson, Numerical solution of an elliptic 3-dimensional Cauchy problem by the alternating method and boundary integral equations, J. Inverse Ill-Posed Probl., 24 (2016), 711-725.  doi: 10.1515/jiip-2015-0053. [4] Y. Boukari and H. Haddar, A convergent data completion algorithm using surface integral equations, Inverse Problems, 31 (2015), 035011, 21pp. doi: 10.1088/0266-5611/31/3/035011. [5] F. Cakoni, G. C. Hsiao and W. L. Wendland, On the boundary integral equation method for a mixed boundary value problem of the biharmonic equation, Complex Var. Theory Appl., 50 (2005), 681-696.  doi: 10.1080/02781070500087394. [6] F. Cakoni and R. Kress, Integral equations for inverse problems in corrosion detection from partial Cauchy data, Inverse Probl. Imaging, 1 (2007), 229-245.  doi: 10.3934/ipi.2007.1.229. [7] R. Chapko and B. T. Johansson, On the numerical solution of a Cauchy problem for the Laplace equation via a direct integral equation approach, Inverse Probl. Imaging, 6 (2012), 25-38.  doi: 10.3934/ipi.2012.6.25. [8] R. Chapko and B. T. Johansson, Boundary-integral approach for the numerical solution of the Cauchy problem for the Laplace equation, Ukr. Mat. Zh., 68 (2016), 1665-1682.  doi: 10.1007/s11253-017-1339-1. [9] R. Chapko and B. T. Johansson, A boundary integral equation method for numerical solution of parabolic and hyperbolic Cauchy problems, Appl. Numer. Math., 129 (2018), 104-119.  doi: 10.1016/j.apnum.2018.03.004. [10] R. Chapko and B. T. Johansson, An iterative regularizing method for an incomplete boundary data problem for the biharmonic equation, ZAMM: Z. Angew. Math. Mech. 98 (2018), 2010–2021. doi: 10.1002/zamm.201800102. [11] G. Chen and J. Zhou, Boundary Element Methods with Applications to Nonlinear Problems, Atlantis Press, Paris, 2010.  doi: 10.2991/978-94-91216-27-5. [12] M. Costabel and M. Dauge, Invertibility of the biharmonic single layer potential operator, Integral Equations Operator Theory, 24 (1996), 46-67.  doi: 10.1007/BF01195484. [13] R. J. Duffin, Continuation of biharmonic functions by reflection, Duke Math. J., 22 (1955), 313-324.  doi: 10.1215/S0012-7094-55-02233-X. [14] R. Farwig, A note on the reflection principle for the biharmonic equation and the Stokes system, Acta Appl. Math., 37 (1994), 41-51.  doi: 10.1007/BF00995128. [15] M. Garshasbi and F. Hassani, Boundary temperature reconstruction in an inverse heat conduction problem using boundary integral equation method, Bull. Iranian Math. Soc., 42 (2016), 1039-1056. [16] G. C. Hsiao and W. L. Wendland, Boundary Integral Equations, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-68545-6. [17] V. A. Kozlov and V. G. Maz;ya, On iterative procedures for solving ill-posed boundary value problems that preserve differential equations, Algebra i Analiz, 1 (1989), 144–170; English transl.: Leningrad Math. J., 1 (1990), 1207–1228. [18] R. Kress, Linear Integral Equations, ed. 3, Springer-Verlag, New York, 2014. doi: 10.1007/978-1-4614-9593-2. [19] D. Lesnic, L. Elliott and D. B. Ingham, An iterative boundary element method for solving numerically the Cauchy problem for the Laplace equation, Eng. Anal. Bound. Elem., 20 (1997), 123-133.  doi: 10.1016/S0955-7997(97)00056-8. [20] J.-C. Liu and Q. G. Zhang, Cauchy problem for the Laplace equation in 2D and 3D doubly connected domains, CMES Comput. Model. Eng. Sci., 93 (2013), 203-219. [21] V. V. Meleshko, Selected topics in the history of the two-dimensional biharmonic problem, Appl. Mech. Rev., 56 (2003), 33-85.  doi: 10.1115/1.1521166. [22] I. Mitrea and M. Mitrea, Multi-layer Potentials and Boundary Problems for Higher-Order Elliptic Systems in Lipschitz Domains, Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-32666-0. [23] T. V. Savina, On the dependence of the reflection operator on boundary conditions for biharmonic functions, J. Math. Anal. Appl., 370 (2010), 716-725.  doi: 10.1016/j.jmaa.2010.04.036. [24] Y. Sun, Indirect boundary integral equation method for the Cauchy problem of the Laplace equation, J. Sci. Comput., 71 (2017), 469-498.  doi: 10.1007/s10915-016-0308-4. [25] Y. Sun, D. Zhang and F. Ma, A potential function method for the Cauchy problem of elliptic operators, J. Math. Anal. Appl., 395 (2012), 164-174.  doi: 10.1016/j.jmaa.2012.05.038. [26] G. C. Verchota, The biharmonic Neumann problem in Lipschitz domains, Acta Math., 194 (2005), 217-279.  doi: 10.1007/BF02393222.
Errors as a function of the number of iteration steps when reconstructing $Nu$ and $Mu$ on the inner boundary $\Gamma_1$ in Ex. 2
The relative $L_2$ error for $Mu$ on $\Gamma_2$ with $u$ solving $\rm(4) $$\rm(5) , and for Mv on \Gamma_1 with v solving \rm(11)$$ \rm(12)$, for the source point $z_1 = (3, \, 0)$ and the setup of Ex 1
 $m$ $e_{2M}(\Gamma_2)$ $\tilde{e}_{2M}(\Gamma_1)$ 8 $0.012728889$ $0.0080868877$ 16 $0.000480837$ $0.0000485082$ 32 $0.000000700$ $0.0000000002$ 64 $0.0$ $0.0$
 $m$ $e_{2M}(\Gamma_2)$ $\tilde{e}_{2M}(\Gamma_1)$ 8 $0.012728889$ $0.0080868877$ 16 $0.000480837$ $0.0000485082$ 32 $0.000000700$ $0.0000000002$ 64 $0.0$ $0.0$
The errors in the second example calculated on $\Gamma_1$ for the "direct" single-layer approach with exact ($\delta = 0\%$) and noisy data ($\delta = 3\%$)
 $\delta = 0\%$ $\delta = 3\%$ $m$ $\alpha$ $e_{2M}$ $e_{2N}$ $\alpha$ $e_{2M}$ $e_{2N}$ 8 1E$-$05 4.0752E$-$02 4.7541E$-$03 1E$-$02 $0.23812$ 0.02612 16 1E$-$07 8.1073E$-$03 2.5088E$-$04 1E$-$02 $0.23534$ 0.02959 32 1E$-$10 3.4936E$-$05 9.9830E$-$07 1E$-$03 $0.23824$ 0.03871 64 1E$-$10 3.4553E$-$05 9.8739E$-$07 1E$-$02 $0.23239$ 0.02678
 $\delta = 0\%$ $\delta = 3\%$ $m$ $\alpha$ $e_{2M}$ $e_{2N}$ $\alpha$ $e_{2M}$ $e_{2N}$ 8 1E$-$05 4.0752E$-$02 4.7541E$-$03 1E$-$02 $0.23812$ 0.02612 16 1E$-$07 8.1073E$-$03 2.5088E$-$04 1E$-$02 $0.23534$ 0.02959 32 1E$-$10 3.4936E$-$05 9.9830E$-$07 1E$-$03 $0.23824$ 0.03871 64 1E$-$10 3.4553E$-$05 9.8739E$-$07 1E$-$02 $0.23239$ 0.02678
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