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Fall 2007 Schedule

Friday, October 5, 2007
Inverse Problems Seminar
Speaker: Professor Isaac Herari
Professor from Tel Aviv University, Currently visiting Duke University
Title: Direct Computation of Inverse Potential Problems with Interior Data
Location: Amos Eaton 402
Time: 12:00PM - 1:00PM

Friday, November 9, 2007
Inverse Problems Seminar
Speaker: Venky Krishnan
From Tufts University, Medford MA
Title: Support Theorems for the Geodesic Ray Transform on Riemannian Manifolds
Location: Amos Eaton 411
Time: 12:00PM - 1:00PM
Host: Birsen Yacizi

Abstract:  
   The inverse problem of heat conduction provides a model for many applications governed by diffusion equations, for which interior data are available.  We wish to find the single, unknown, thermal conductivity field by direct (i.e. non-iterative) computation. This approach requires at least two interior temperature fields. The strong form of the problem is governed by two partial differential equations of pure advective transport.  The given temperature fields must satisfy a compatibility condition for the problem to have a solution.  The standard weak (variational) form does not provide a suitable basis for finite element computation.  A conventional least-squares variational approach yields a well-posed problem, but does not perform well numerically.
   We introduce a novel variational formulation, the Adjoint Weighted Equation (AWE), for solving the two-field problem.  In this case, the gradients of two given temperature fields must be linearly independent in the entire domain, a weaker condition than the compatibility required by the strong form.  The solution of the AWE formulation is equivalent to that of the strong form when both are well posed.  The Galerkin discretization of the AWE formulation converges with optimal rates. Computational examples confirm these optimal rates, and demonstrate superior performance to conventional numerical methods on problems with both smooth and rough coefficients and solutions.
   This work was done in collaboration with Assad Oberai of RPI and Paul Barbone of BU.  We are currently extending these ideas to problems of incompressible elasticity.

 

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