In this work, we present an overview of the work developed by the
authors in the context of inverse problems on nite networks. This study performs
an extension of the pioneer studies by E.B. Curtis and J.A. Morrow, and
sets the theoretical basis for solving inverse problems on networks. We present
just a glance of what we call overdetermined partial boundary value problems,
in which any data are not prescribed on a part of the boundary, whereas in
another part of the boundary both the values of...
In this work, we present an overview of the work developed by the
authors in the context of inverse problems on nite networks. This study performs
an extension of the pioneer studies by E.B. Curtis and J.A. Morrow, and
sets the theoretical basis for solving inverse problems on networks. We present
just a glance of what we call overdetermined partial boundary value problems,
in which any data are not prescribed on a part of the boundary, whereas in
another part of the boundary both the values of the function and of its normal
derivative are given. The resolvent kernels associated with these problems are
described and they are the fundamental tool to perform an algorithm for the
recovery of the conductance of a 3{dimensional grid. We strongly believe that
the columns of the partial overdetermined Poisson kernel are the discrete counterpart
of the so{called CGO solutions (complex geometrical optic solutions)
that, in their turn, are the key to solve inverse continuous problems on planar
domains. Finally, we display the steps needed to recover the conductances in
a 3{dimensional grid.
In this work, we present an overview of the work developed by the
authors in the context of inverse problems on nite networks. This study performs
an extension of the pioneer studies by E.B. Curtis and J.A. Morrow, and
sets the theoretical basis for solving inverse problems on networks. We present
just a glance of what we call overdetermined partial boundary value problems,
in which any data are not prescribed on a part of the boundary, whereas in
another part of the boundary both the values of the function and of its normal
derivative are given. The resolvent kernels associated with these problems are
described and they are the fundamental tool to perform an algorithm for the
recovery of the conductance of a 3{dimensional grid. We strongly believe that
the columns of the partial overdetermined Poisson kernel are the discrete counterpart
of the so{called CGO solutions (complex geometrical optic solutions)
that, in their turn, are the key to solve inverse continuous problems on planar
domains. Finally, we display the steps needed to recover the conductances in
a 3{dimensional grid.
Citation
Arauz, C., Carmona, A., Encinas, A., Mitjana, M. Recovering the conductances on grids: A theoretical justification. "Contemporary mathematics", 2016, vol. 658, p. 149-167.