Inverse problem on the semi-axis: local approach

Avdonin, Sergei; Belinskiy, Boris P.; Matthews, John V.

doi:10.5556/j.tkjm.42.2011.275-293

Cited by 4 publications

(1 citation statement)

References 24 publications

(40 reference statements)

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“…To date, there are few papers presenting methods apt for numerical solution of inverse problems on quantum graphs [7,[12][13][14]. Since the problems of space discretization of differential equations on metric graphs turn out to be very difficult, and even the forward boundary value problems on graphs present considerable numerical challenges (see, e.g.…”

Section: Introductionmentioning

confidence: 99%

Reconstruction techniques for quantum trees

Avdonin,

Khmelnytskaya,

Kravchenko

2024

Math Methods in App Sciences

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The inverse problem of recovery of a potential on a quantum tree graph from the Weyl matrix given at a number of points is considered. A method for its numerical solution is proposed. The overall approach is based on the leaf peeling method combined with Neumann series of Bessel functions (NSBF) representations for solutions of Sturm–Liouville equations. In each step, the solution of the arising inverse problems reduces to dealing with the NSBF coefficients. The leaf peeling method allows one to localize the general inverse problem to local problems on sheaves, while the approach based on the NSBF representations leads to splitting the local problems into two‐spectrum inverse problems on separate edges and reduces them to systems of linear algebraic equations for the NSBF coefficients. Moreover, the potential on each edge is recovered from the very first NSBF coefficient. The proposed method leads to an efficient numerical algorithm that is illustrated by numerical tests.

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Section: Introductionmentioning

confidence: 99%

Reconstruction techniques for quantum trees

Avdonin,

Khmelnytskaya,

Kravchenko

2024

Math Methods in App Sciences

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Inverse problems for Sturm–Liouville operators with interior discontinuities and boundary conditions dependent on the spectral parameter

Wang

2012

Math Methods in App Sciences

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Determining a distributed parameter in a neural cable model via a boundary control method

Avdonin

Bell

2012

J. Math. Biol.

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Dendrites of nerve cells have membranes with spatially distributed densities of ionic channels and hence non-uniform conductances. These conductances are usually represented as constant parameters in neural models because of the difficulty in experimentally estimating them locally. In this paper we investigate the inverse problem of recovering a single spatially distributed conductance parameter in a one-dimensional diffusion (cable) equation through a new use of a boundary control method. We also outline how our methodology can be extended to cable theory on finite tree graphs. The reconstruction is unique.

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Inverse problem on the semi-axis: local approach

Cited by 4 publications

References 24 publications

Reconstruction techniques for quantum trees

Reconstruction techniques for quantum trees

Inverse problems for Sturm–Liouville operators with interior discontinuities and boundary conditions dependent on the spectral parameter

Determining a distributed parameter in a neural cable model via a boundary control method

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