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Recursive Algorithms Solving Inverse Problems on Quantum Graphs

Presented by: 
S Avdonin [Alaska]
Monday 26th July 2010 - 16:00 to 16:45
INI Seminar Room 1
In this talk we describe a new approach to solving boundary inverse problems on quantum graphs. This approach is based on the Boundary Control method and combines the spectral and dynamical approaches to inverse problems on graphs. It was proposed in [1] for the Schr\"odinger equation with standard matching conditions and was extended in [2] to the two-velocity wave equation. Since the number of edges of graphs arising in applications is typically very big, we propose a recursive procedure which may serve as a base for developing effective numerical algorithms. For trees, this procedure allows recalculating efficiently the inverse data from the original tree to the smaller trees, `removing' leaves step by step up to the rooted edge. Numerical tests for inverse problems are impossible without producing accurate inverse data. This means that we have to have reliable numerical algorithms for solving the direct Problems --- given the coefficients of equations and the graph topology find its spectral (and dynamical) data. Even for the simplest graph --- a finite interval or the semi-axis --- this is a rather difficult problem from the numerical point of view. The surprising fact is that to solve numerically, say, the Gelfand--Levitan equation and find the potential from the given spectral function is much easier than to find the spectral function from the given potential. For graphs with many edges these difficulties increase dramatically. Therefore, at the moment there are no efficient algorithms for, or numerical experiments in, solving inverse problems on graphs. Based on the results of [3], we propose a way to reduces the `direct' problem to solving second kind Volterra integral equations. 1. S. Avdonin and P. Kurasov, Inverse problems for quantum trees,} Inverse Problems and Imaging, {2} (2008), 1--21. 2. S. Avdonin, G. Leugering and V. Mikhaylov, On an inverse problem for tree-like networks of elastic strings}, Zeit. Angew. Math. Mech., {90} (2010), 136--150. 3. S. Avdonin, V. Mikhaylov and A.Rybkin, The boundary control approach to the Titchmarsh-Weyl $m-$function,} Comm. Math. Phys., {275} (2007), 791--803.
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons