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August 2017, 11(4): 623-641. doi: 10.3934/ipi.2017029
A discrete Liouville identity for numerical reconstruction of Schrödinger potentials
Liliana Borcea 1, , Fernando Guevara Vasquez 2,, and Alexander V. Mamonov 3,
Mathematics, University of Michigan, 2074 E Hall, 530 Church St, Ann Arbor, MI 48109-1043, USA
Mathematics, University of Utah, 155 S 1400 E RM 233, Salt Lake City, UT 84112-0090, USA
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
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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.
Keywords: Schrödinger equation, Liouville transform, Dirichlet to Neumann map, absorptive potentials, nonlinear preconditioner.
Mathematics Subject Classification: Primary: 05C22, 35R30, 35J10; Secondary: 05C50, 78A46.
Citation: Liliana Borcea, Fernando Guevara Vasquez, Alexander V. Mamonov. A discrete Liouville identity for numerical reconstruction of Schrödinger potentials. Inverse Problems & Imaging, 2017, 11 (4) : 623-641. doi: 10.3934/ipi.2017029
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Figure 1. 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 $n = 2m+1$
Figure 2. A graph $\mathcal{G}$ in black and its line graph $\widetilde{\mathcal{G}}$ in red
Figure 3. Conductivities $\sigma^{(i)}$, corresponding to different Schrödinger potentials $q^{(i)}$, $i=1, 2$
Figure 4. 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 $2\pi/17$
Figure 5. Sensitivity grids. The ''x'' are for $q=1$ and the ''$\circ$'' for $q=3$
Figure 6. Gauss-Newton iterates for smooth $q$ (sensitivity grid)
Figure 7. Gauss-Newton iterates for piecewise constant $q$ (sensitivity grid)
Figure 8. 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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Liliana Borcea Fernando Guevara Vasquez Alexander V. Mamonov