Determining a distributed conductance parameter for a neuronal cable model defined on a tree graph
Sergei Avdonin Jonathan Bell
Inverse Problems & Imaging 2015, 9(3): 645-659 doi: 10.3934/ipi.2015.9.645
In this paper we solve the inverse problem of recovering a single spatially distributed conductance parameter in a cable equation model (one-dimensional diffusion) defined on a metric tree graph that represents a dendritic tree of a neuron. Dendrites of nerve cells have membranes with spatially distributed densities of ionic channels and hence non-uniform conductances. We employ the boundary control method that gives a unique reconstruction and an algorithmic approach.
keywords: metric graph parameter recovery. inverse problem tree graph boundary control Cable equation
Inverse problems for quantum trees
Sergei Avdonin Pavel Kurasov
Inverse Problems & Imaging 2008, 2(1): 1-21 doi: 10.3934/ipi.2008.2.1
Three different inverse problems for the Schrödinger operator on a metric tree are considered, so far with standard boundary conditions at the vertices. These inverse problems are connected with the matrix Titchmarsh-Weyl function, response operator (dynamic Dirichlet-to-Neumann map) and scattering matrix. Our approach is based on the boundary control (BC) method and in particular on the study of the response operator. It is proven that the response operator determines the quantum tree completely, i.e. its connectivity, lengths of the edges and potentials on them. The same holds if the response operator is known for all but one boundary points, as well as for the Titchmarsh-Weyl function and scattering matrix. If the connectivity of the graph is known, then the lengths of the edges and the corresponding potentials are determined by just the diagonal terms of the data.
keywords: inverse problems wave equation controllability Quantum graphs boundary control. Schrödinger equation
Spectral estimation and inverse initial boundary value problems
Sergei Avdonin Fritz Gesztesy Konstantin A. Makarov
Inverse Problems & Imaging 2010, 4(1): 1-9 doi: 10.3934/ipi.2010.4.1
We extend the classical spectral estimation problem to the infinite-dimensional case and propose a new approach to this problem using the Boundary Control (BC) method. Several applications to inverse problems for partial differential equations are provided.
keywords: Spectral estimation inverse problems boundary control method.
Inverse problems for quantum trees II: Recovering matching conditions for star graphs
Sergei Avdonin Pavel Kurasov Marlena Nowaczyk
Inverse Problems & Imaging 2010, 4(4): 579-598 doi: 10.3934/ipi.2010.4.579
The inverse problem for the Schrödinger operator on a star graph is investigated. It is proven that such Schrödinger operator, i.e. the graph, the real potential on it and the matching conditions at the central vertex, can be reconstructed from the Titchmarsh-Weyl matrix function associated with the graph boundary. The reconstruction is also unique if the spectral data include not the whole Titchmarsh-Weyl function but its principal block (the matrix reduced by one dimension). The same result holds true if instead of the Titchmarsh-Weyl function the dynamical response operator or just its principal block is known.
keywords: matching conditions. quantum graphs inverse problems

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