Discrete and Continuous Dirichlet-to-Neumann Maps in the Layered Case

Ingerman, David

doi:10.1137/s0036141097326581

Cited by 18 publications

(32 citation statements)

References 9 publications

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“…Thus, we have established the link between discrete inversion, such as inverse spectral problems for Jacobi matrices [18] or resistor network tomography [22,45,48] and continuous inversion.…”

Section: Discussionmentioning

confidence: 99%

“…To our knowledge, this is the first established link between discrete inversion, such as inverse spectral problems for Jacobi matrices [18] or the impedance tomography problem for graphs [20,21,22,40,45,48], and continuous inversion. Discrete studies consider k fixed and, in general, it is not known if the limit k → ∞ gives convergence to the continuous solution.…”

Section: Introductionmentioning

confidence: 94%

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On the continuum limit of a discrete inverse spectral problem on optimal finite difference grids

Borcea

Druskin

Knizhnerman

2005

Comm Pure Appl Math

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We consider finite difference approximations of solutions of inverse Sturm-Liouville problems in bounded intervals. Using three-point finite difference schemes, we discretize the equations on so-called optimal grids constructed as follows: For a staggered grid with 2k points, we ask that the finite difference operator (a k × k Jacobi matrix) and the Sturm-Liouville differential operator share the k lowest eigenvalues and the values of the orthonormal eigenfunctions at one end of the interval. This requirement determines uniquely the entries in the Jacobi matrix, which are grid cell averages of the coefficients in the continuum problem. If these coefficients are known, we can find the grid, which we call optimal because it gives, by design, a finite difference operator with a prescribed spectral measure. We focus attention on the inverse problem, where neither the coefficients nor the grid are known.A key question in inversion is how to parametrize the coefficients, i.e., how to choose the grid. It is clear that, to be successful, this grid must be close to the optimal one, which is unknown. Fortunately, as we show here, the grid dependence on the unknown coefficients is weak, so the inversion can be done on a precomputed grid for an a priori guess of the unknown coefficients. This observation leads to a simple yet efficient inversion algorithm, which gives coefficients that converge pointwise to the true solution as the number k of data points tends to infinity. The cornerstone of our convergence proof is showing that optimal grids provide an implicit, natural regularization of the inverse problem, by giving reconstructions with uniformly bounded total variation. The analysis is based on a novel, explicit perturbation analysis of Lanczos recursions and on a discrete Gel fand-Levitan formulation.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 94%

On the continuum limit of a discrete inverse spectral problem on optimal finite difference grids

Borcea

Druskin

Knizhnerman

2005

Comm Pure Appl Math

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“…We begin in section 2.1 with the formulation of the discrete EIT problem for networks, and we cite from [14,15,29,17,18] the necessary and sufficient conditions for its unique solvability. We motivate the networks in the context of finite volume discretizations of (1.1)-(1.2) in section 2.2.…”

Section: Electrical Impedance Tomography With Resistor Networkmentioning

confidence: 99%

Pyramidal resistor networks for electrical impedance tomography with partial boundary measurements

et al. 2010

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Abstract. We introduce an inversion algorithm for electrical impedance tomography (EIT) with partial boundary measurements, in two dimensions. It gives stable and fast reconstructions using sparse parameterizations of the unknown conductivity on optimal grids that are computed as part of the inversion. We follow the approach in [8,27] that connects inverse discrete problems for resistor networks to continuum EIT problems, using optimal grids. The algorithm in [8,27] is based on circular resistor networks, and solves the EIT problem with full boundary measurements. It is extended in [11] to EIT with partial boundary measurements, using extremal quasiconformal mappings that transform the problem to one with full boundary measurements. Here we introduce a different class of optimal grids, based on resistor networks with pyramidal topology, that is better suited for the partial measurements setup. We prove the unique solvability of the discrete inverse problem for these networks, and develop an algorithm for finding them from the measurements of the DtN map. Then, we show how to use the networks to define the optimal grids and to approximate the unknown conductivity. We assess the performance of our approach with numerical simulations and compare the results with those in [11].

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“…They are resistor networks constructed from very accurate approximations of the DtN map and the resistances in the network play the role of the parameters in the parametric model reduction. These resistor network reduced models rely on the graph theory developed in [12,14,15,16,17,18,19,20,27].…”

Section: Introductionmentioning

confidence: 99%

A discrete Liouville identity for numerical reconstruction of Schrödinger potentials

Borcea¹,

Vasquez

Mamonov

2017

Inverse Problems &Amp; Imaging

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We propose a discrete approach for solving an inverse problem for Schrödinger's equation in two dimensions, where the unknown potential is to be determined from boundary measurements of the Dirichlet to Neumann map. For absorptive potentials, and in the continuum, it is known that by using the Liouville identity we obtain an inverse conductivity problem. Its discrete analogue is to find a resistor network that matches the measurements, and is well understood. Here we show how to 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 optimization formulation of the inverse Schrödinger problem. Instead of minimizing the boundary measurement misfit, we minimize the misfit between the discrete Schrödinger potentials. This results in a better behaved optimization problem that converges in a single Gauss-Newton iteration, and gives good quality reconstructions of the potential, as illustrated by the numerical results.

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Discrete and Continuous Dirichlet-to-Neumann Maps in the Layered Case

Cited by 18 publications

References 9 publications

On the continuum limit of a discrete inverse spectral problem on optimal finite difference grids

On the continuum limit of a discrete inverse spectral problem on optimal finite difference grids

Pyramidal resistor networks for electrical impedance tomography with partial boundary measurements

A discrete Liouville identity for numerical reconstruction of Schrödinger potentials

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