Blockage detection in networks: the area reconstruction method. (English) Zbl 1437.65122
Summary: In this note we present a reconstructive algorithm for solving the cross-sectional pipe area from boundary measurements in a tree network with one inaccessible end. This is equivalent to reconstructing the first order perturbation to a wave equation on a quantum graph from boundary measurements at all network ends except one. The method presented here is based on a time reversal boundary control method originally presented by M. M. Sondhi and B. Gopinath [“Determination of vocal tract shape from impulse response at the lips”, J. Acoust. Soc. Am. 49, 1867–1873 (1971; doi:10.1121/1.1912593)] for one dimensional problems and later by L. Oksanen [Inverse Probl. Imaging 5, No. 3, 731–744 (2011; Zbl 1230.35145)] to higher dimensional manifolds. The algorithm is local, so is applicable to complicated networks if we are interested only in a part isomorphic to a tree. Moreover the numerical implementation requires only one matrix inversion or least squares minimization per discretization point in the physical network. We present a theoretical solution existence proof, a step-by-step algorithm, and a numerical implementation applied to two numerical experiments.
MSC:
| 65M32 | Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65K10 | Numerical optimization and variational techniques |
| 76M21 | Inverse problems in fluid mechanics |
| 76N20 | Boundary-layer theory for compressible fluids and gas dynamics |
| 35Q35 | PDEs in connection with fluid mechanics |
| 35R02 | PDEs on graphs and networks (ramified or polygonal spaces) |
| 35R30 | Inverse problems for PDEs |
Keywords:
impulse response; blockage detection; transient flow; area reconstruction; network; boundary control; reconstruction algorithmCitations:
Zbl 1230.35145References:
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