THESIS
2019
xi, 45 pages : illustrations ; 30 cm
Abstract
In this thesis, we propose a simple algorithm for solving an inverse problem for the Schrödinger
equation. The idea is to apply the gradient descent and the adjoint state technique. We
observe that since the forward operator is self-adjoint, the approach simply requires to solve
the same partial differential equation for both the forward problem and the adjoint problem.
To speed up the computations, we also develop a cascadic initialization strategy to provide
a better initial condition for the inversion process. To be more realistic for real life applications,
we incorporate techniques from the level set method to handle cases with only a set
of finite number of Dirichlet-to-Neumann (DN) measurements. Moreover, based on a usual
reduction, this inverse problem can be linked to...[
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In this thesis, we propose a simple algorithm for solving an inverse problem for the Schrödinger
equation. The idea is to apply the gradient descent and the adjoint state technique. We
observe that since the forward operator is self-adjoint, the approach simply requires to solve
the same partial differential equation for both the forward problem and the adjoint problem.
To speed up the computations, we also develop a cascadic initialization strategy to provide
a better initial condition for the inversion process. To be more realistic for real life applications,
we incorporate techniques from the level set method to handle cases with only a set
of finite number of Dirichlet-to-Neumann (DN) measurements. Moreover, based on a usual
reduction, this inverse problem can be linked to the standard Calderón inverse problem for
the electrical impedance tomography (EIT). Therefore, our approach might provide a simple
numerical alternative to solve the EIT problem. Numerical examples will demonstrate that
the new formulation is effective and robust.
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