Stanford 50: State of the Art and Future Directions of Computational Mathematics and Numerical Computing


  • March 30, 2007
  • 9:50 am - 10:15 am

    Model reduction and electrical impedance tomography

    Liliana Borcea (Rice University)

    We present a novel inversion algorithm for electrical impedance tomography in two dimensions, based on a model reduction approach. The reduced models are resistor networks that arise in five-point stencil discretizations of the elliptic partial differential equation satisfied by the electric potential, on adaptive grids that are computed as part of the problem. We prove the unique solvability of the model reduction problem for a broad class of measurements of the Dirichlet to Neumann map. The size of the networks (reduced models) is limited by the precision of the measurements. The resulting grids are naturally refined near the boundary, where we make the measurements and where we expect better resolution of the images. To determine the unknown conductivity, we use the resistor networks to define a nonlinear mapping of the data that behaves as an approximate inverse of the forward map. Then we propose an efficient Newton-type iteration for finding the conductivity, using this map. We also show how to incorporate a priori information about the conductivity in the inversion scheme.

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