# American Institute of Mathematical Sciences

August  2017, 11(4): 623-641. doi: 10.3934/ipi.2017029

## A discrete Liouville identity for numerical reconstruction of Schrödinger potentials

 1 Mathematics, University of Michigan, 2074 E Hall, 530 Church St, Ann Arbor, MI 48109-1043, USA 2 Mathematics, University of Utah, 155 S 1400 E RM 233, Salt Lake City, UT 84112-0090, USA 3 Mathematics, University of Houston, 4800 Calhoun Rd., Houston, TX, 77004, USA

* Corresponding author: F. Guevara Vasquez

Received  January 2016 Revised  April 2017 Published  June 2017

Fund Project: LB acknowledges partial support from ONR Grant N00014-14-1-0077 and NSF grant DMS-1510429. The work of FGV was partially supported by the National Science Foundation grant DMS-1411577. The work of AVM was partially supported by the National Science Foundation grant DMS-1619821.

We propose a discrete approach for solving an inverse problem for the two-dimensional Schrödinger equation, where the unknown potential is to be determined from the Dirichlet to Neumann map. In the continuum, the problem for absorptive potentials can be transformed with the Liouville identity into a conductivity inverse problem. Its discrete analogue is to find a resistor network matching the measurements, and is well understood. Here we use a discrete Liouville identity to transform its solution to that of Schrödinger's problem. The discrete Schrödinger potential given by the discrete Liouville identity can be used to reconstruct the potential in the continuum in two ways. First, we can obtain a direct but coarse reconstruction by interpreting the values of the discrete Schrödinger potential as averages of the continuum Schrödinger potential on a special sensitivity grid. Second, the discrete Schrödinger potential may be used to reformulate the conventional nonlinear output least squares formulation of the inverse Schrödinger problem. Instead of minimizing the boundary measurement misfit, we minimize the misfit between discrete Schrödinger potentials. This results in a better behaved optimization problem converging in a single Gauss-Newton iteration, and gives good quality reconstructions of the potential, as illustrated by the numerical results.

Citation: Liliana Borcea, Fernando Guevara Vasquez, Alexander V. Mamonov. A discrete Liouville identity for numerical reconstruction of Schrödinger potentials. Inverse Problems and Imaging, 2017, 11 (4) : 623-641. doi: 10.3934/ipi.2017029
##### References:
 [1] C. Araúz, A. Carmona and A. Encinas, Dirichlet-to-Robin matrix on networks, Electronic Notes in Discrete Mathematics, 46 (2014), 65–72, Jornadas de Matemática Discreta y Algorítmica. doi: 10.1016/j.endm.2014.08.010. [2] C. Araúz, A. Carmona and A. Encinas, Overdetermined partial boundary value problems on finite networks, Journal of Mathematical Analysis and Applications, 423 (2015), 191-207.  doi: 10.1016/j.jmaa.2014.09.025. [3] G. Bal, Optical tomography for small volume absorbing inclusions, Inverse Problems, 19 (2003), 371-386.  doi: 10.1088/0266-5611/19/2/308. [4] P. Benner, S. Gugercin and K. Willcox, A survey of projection-based model reduction methods for parametric dynamical systems, SIAM Review, 57 (2015), 483-531.  doi: 10.1137/130932715. [5] L. Borcea, V. Druskin, A. Mamonov and F. Guevara Vasquez, Pyramidal resistor networks for electrical impedance tomography with partial boundary measurements Inverse Problems, 26 (2010), 105009, 36pp. doi: 10.1088/0266-5611/26/10/105009. [6] L. Borcea, Electrical impedance tomography, Inverse Problems, 18 (2002), R99-R136, Topical Review.  doi: 10.1088/0266-5611/18/6/201. [7] L. Borcea, V. Druskin and F. Guevara Vasquez, Electrical impedance tomography with resistor networks Inverse Problems, 24 (2008), 035013, 31pp. doi: 10.1088/0266-5611/24/3/035013. [8] L. Borcea, V. Druskin, F. Guevara Vasquez and A. V. Mamonov, Resistor network approaches to electrical impedance tomography, Inside Out Ⅱ (ed. G. Uhlmann), vol. 60, MSRI Publications, 2012. [9] L. Borcea, V. Druskin and L. Knizhnerman, On the continuum limit of a discrete inverse spectral problem on optimal finite difference grids, Communications on Pure and Applied Mathematics, 58 (2005), 1231-1279.  doi: 10.1002/cpa.20073. [10] L. Borcea, V. Druskin, A. V. Mamonov and M. Zaslavsky, A model reduction approach to numerical inversion for a parabolic partial differential equation Inverse Problems, 30 (2014), 125011, 30pp. doi: 10.1088/0266-5611/30/12/125011. [11] L. Borcea, V. Druskin and A. Mamonov, Circular resistor networks for electrical impedance tomography with partial boundary measurements, Inverse Problems, 26 (2010), 045010, 30pp. doi: 10.1088/0266-5611/26/4/045010. [12] K. Chadan, D. Colton, L. Päivärinta and W. Rundell, An Introduction to Inverse Scattering and Inverse Spectral Problems, SIAM Monographs on Mathematical Modeling and Computation, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1997, With a foreword by Margaret Cheney. doi: 10.1137/1.9780898719710. [13] F. R. K. Chung, Spectral Graph Theory, vol. 92 of CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1997. [14] Y. Colin de Verdiére, Réseaux électriques planaires. Ⅰ, Comment. Math. Helv., 69 (1994), 351-374.  doi: 10.1007/BF02564493. [15] Y. Colin de Verdiére, Spectres de Graphes vol. 4 of Cours Spécialisés [Specialized Courses], Société Mathématique de France, Paris, 1998. [16] Y. Colin de Verdiére, I. Gitler and D. Vertigan, Réseaux électriques planaires. Ⅱ, Comment. Math. Helv., 71 (1996), 144-167.  doi: 10.1007/BF02566413. [17] E. Curtis, T. Edens and J. Morrow, Calculating resistors in a network, Engineering in Medicine and Biology Society, 1989. Images of the Twenty-First Century. Proceedings of the Annual International Conference of the IEEE Engineering in, 2 (1989), 451-452.  doi: 10.1109/IEMBS.1989.95813. [18] E. Curtis, E. Mooers and J. Morrow, Finding the conductors in circular networks from boundary measurements, RAIRO Modél. Math. Anal. Numér., 28 (1994), 781-814.  doi: 10.1051/m2an/1994280707811. [19] E. B. Curtis, D. Ingerman and J. A. Morrow, Circular planar graphs and resistor networks, Linear Algebra Appl., 283 (1998), 115-150.  doi: 10.1016/S0024-3795(98)10087-3. [20] E. B. Curtis and J. A. Morrow, Determining the resistors in a network, SIAM J. Appl. Math., 50 (1990), 918-930.  doi: 10.1137/0150055. [21] E. B. Curtis and J. A. Morrow, The Dirichlet to Neumann map for a resistor network, SIAM J. Appl. Math., 51 (1991), 1011-1029.  doi: 10.1137/0151051. [22] E. B. Curtis and J. A. Morrow, Inverse Problems for Electrical Networks, vol. 13 of Series on Applied Mathematics, World Scientific, 2000. [23] V. Druskin, V. Simoncini and M. Zaslavsky, Solution of the time-domain inverse resistivity problem in the model reduction framework part Ⅰ. one-dimensional problem with SISO data, SIAM Journal on Scientific Computing, 35 (2013), A1621-A1640.  doi: 10.1137/110852607. [24] L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1998. [25] D. Gisser, D. Isaacson and J. Newell, Electric current computed tomography and eigenvalues, SIAM Journal on Applied Mathematics, 50 (1990), 1623-1634.  doi: 10.1137/0150096. [26] G. H. Golub and C. F. Van Loan, Matrix Computations, 4th edition, Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 2013. [27] R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd edition, Cambridge University Press, Cambridge, 2013. [28] D. Ingerman, Schrodinger equation, 2012, http://en.wikibooks.org/wiki/User:Daviddaved/Schrodinger_equation, Retrieved November 2013. [29] D. V. Ingerman, Discrete and continuous Dirichlet-to-Neumann maps in the layered case, SIAM Journal on Mathematical Analysis, 31 (2000), 1214-1234.  doi: 10.1137/S0036141097326581. [30] A. V. Mamonov, V. Druskin and M. Zaslavsky, Nonlinear seismic imaging via reduced order model backprojection, SEG Technical Program Expanded Abstracts, (2015), 4375–4379, arXiv: 1504.00094. doi: 10.1190/segam2015-5830429.1. [31] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math.(2), 125 (1987), 153-169.  doi: 10.2307/1971291.

show all references

##### References:
 [1] C. Araúz, A. Carmona and A. Encinas, Dirichlet-to-Robin matrix on networks, Electronic Notes in Discrete Mathematics, 46 (2014), 65–72, Jornadas de Matemática Discreta y Algorítmica. doi: 10.1016/j.endm.2014.08.010. [2] C. Araúz, A. Carmona and A. Encinas, Overdetermined partial boundary value problems on finite networks, Journal of Mathematical Analysis and Applications, 423 (2015), 191-207.  doi: 10.1016/j.jmaa.2014.09.025. [3] G. Bal, Optical tomography for small volume absorbing inclusions, Inverse Problems, 19 (2003), 371-386.  doi: 10.1088/0266-5611/19/2/308. [4] P. Benner, S. Gugercin and K. Willcox, A survey of projection-based model reduction methods for parametric dynamical systems, SIAM Review, 57 (2015), 483-531.  doi: 10.1137/130932715. [5] L. Borcea, V. Druskin, A. Mamonov and F. Guevara Vasquez, Pyramidal resistor networks for electrical impedance tomography with partial boundary measurements Inverse Problems, 26 (2010), 105009, 36pp. doi: 10.1088/0266-5611/26/10/105009. [6] L. Borcea, Electrical impedance tomography, Inverse Problems, 18 (2002), R99-R136, Topical Review.  doi: 10.1088/0266-5611/18/6/201. [7] L. Borcea, V. Druskin and F. Guevara Vasquez, Electrical impedance tomography with resistor networks Inverse Problems, 24 (2008), 035013, 31pp. doi: 10.1088/0266-5611/24/3/035013. [8] L. Borcea, V. Druskin, F. Guevara Vasquez and A. V. Mamonov, Resistor network approaches to electrical impedance tomography, Inside Out Ⅱ (ed. G. Uhlmann), vol. 60, MSRI Publications, 2012. [9] L. Borcea, V. Druskin and L. Knizhnerman, On the continuum limit of a discrete inverse spectral problem on optimal finite difference grids, Communications on Pure and Applied Mathematics, 58 (2005), 1231-1279.  doi: 10.1002/cpa.20073. [10] L. Borcea, V. Druskin, A. V. Mamonov and M. Zaslavsky, A model reduction approach to numerical inversion for a parabolic partial differential equation Inverse Problems, 30 (2014), 125011, 30pp. doi: 10.1088/0266-5611/30/12/125011. [11] L. Borcea, V. Druskin and A. Mamonov, Circular resistor networks for electrical impedance tomography with partial boundary measurements, Inverse Problems, 26 (2010), 045010, 30pp. doi: 10.1088/0266-5611/26/4/045010. [12] K. Chadan, D. Colton, L. Päivärinta and W. Rundell, An Introduction to Inverse Scattering and Inverse Spectral Problems, SIAM Monographs on Mathematical Modeling and Computation, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1997, With a foreword by Margaret Cheney. doi: 10.1137/1.9780898719710. [13] F. R. K. Chung, Spectral Graph Theory, vol. 92 of CBMS Regional Conference Series in Mathematics, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1997. [14] Y. Colin de Verdiére, Réseaux électriques planaires. Ⅰ, Comment. Math. Helv., 69 (1994), 351-374.  doi: 10.1007/BF02564493. [15] Y. Colin de Verdiére, Spectres de Graphes vol. 4 of Cours Spécialisés [Specialized Courses], Société Mathématique de France, Paris, 1998. [16] Y. Colin de Verdiére, I. Gitler and D. Vertigan, Réseaux électriques planaires. Ⅱ, Comment. Math. Helv., 71 (1996), 144-167.  doi: 10.1007/BF02566413. [17] E. Curtis, T. Edens and J. Morrow, Calculating resistors in a network, Engineering in Medicine and Biology Society, 1989. Images of the Twenty-First Century. Proceedings of the Annual International Conference of the IEEE Engineering in, 2 (1989), 451-452.  doi: 10.1109/IEMBS.1989.95813. [18] E. Curtis, E. Mooers and J. Morrow, Finding the conductors in circular networks from boundary measurements, RAIRO Modél. Math. Anal. Numér., 28 (1994), 781-814.  doi: 10.1051/m2an/1994280707811. [19] E. B. Curtis, D. Ingerman and J. A. Morrow, Circular planar graphs and resistor networks, Linear Algebra Appl., 283 (1998), 115-150.  doi: 10.1016/S0024-3795(98)10087-3. [20] E. B. Curtis and J. A. Morrow, Determining the resistors in a network, SIAM J. Appl. Math., 50 (1990), 918-930.  doi: 10.1137/0150055. [21] E. B. Curtis and J. A. Morrow, The Dirichlet to Neumann map for a resistor network, SIAM J. Appl. Math., 51 (1991), 1011-1029.  doi: 10.1137/0151051. [22] E. B. Curtis and J. A. Morrow, Inverse Problems for Electrical Networks, vol. 13 of Series on Applied Mathematics, World Scientific, 2000. [23] V. Druskin, V. Simoncini and M. Zaslavsky, Solution of the time-domain inverse resistivity problem in the model reduction framework part Ⅰ. one-dimensional problem with SISO data, SIAM Journal on Scientific Computing, 35 (2013), A1621-A1640.  doi: 10.1137/110852607. [24] L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1998. [25] D. Gisser, D. Isaacson and J. Newell, Electric current computed tomography and eigenvalues, SIAM Journal on Applied Mathematics, 50 (1990), 1623-1634.  doi: 10.1137/0150096. [26] G. H. Golub and C. F. Van Loan, Matrix Computations, 4th edition, Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 2013. [27] R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd edition, Cambridge University Press, Cambridge, 2013. [28] D. Ingerman, Schrodinger equation, 2012, http://en.wikibooks.org/wiki/User:Daviddaved/Schrodinger_equation, Retrieved November 2013. [29] D. V. Ingerman, Discrete and continuous Dirichlet-to-Neumann maps in the layered case, SIAM Journal on Mathematical Analysis, 31 (2000), 1214-1234.  doi: 10.1137/S0036141097326581. [30] A. V. Mamonov, V. Druskin and M. Zaslavsky, Nonlinear seismic imaging via reduced order model backprojection, SEG Technical Program Expanded Abstracts, (2015), 4375–4379, arXiv: 1504.00094. doi: 10.1190/segam2015-5830429.1. [31] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math.(2), 125 (1987), 153-169.  doi: 10.2307/1971291.
Graphs of class $C(m, n)$, where $n$ the number of boundary nodes and $m$ is equal to the number of edges from the center in a fixed radial direction, plus the number of concentric layers of $n$ edges each. The boundary nodes are shown as circles and the interior nodes as filled (black) circles. Both graphs are critical since $n = 2m+1$
A graph $\mathcal{G}$ in black and its line graph $\widetilde{\mathcal{G}}$ in red
Conductivities $\sigma^{(i)}$, corresponding to different Schrödinger potentials $q^{(i)}$, $i=1, 2$
Sensitivity functions for $n=17$, $q = 1$, i.e. the ''rows'' of $D\mathit{\boldsymbol{Q}}[\mathit{\boldsymbol{M}}(q)] D\mathit{\boldsymbol{M}}[q]$. The other sensitivity functions can be obtained by rotations of integer multiples of $2\pi/17$
Sensitivity grids. The ''x'' are for $q=1$ and the ''$\circ$'' for $q=3$
Gauss-Newton iterates for smooth $q$ (sensitivity grid)
Gauss-Newton iterates for piecewise constant $q$ (sensitivity grid)
A typical convergence history for the preconditioned Gauss-Newton method. We show convergence in terms of the unpreconditioned residual (green), the preconditioned residual (red) and the projected preconditioned residual (blue)
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