Inverse problems are often formulated as constrained optimization problems. For example, one may wish to find the parameter distribution that provides the best match to the measured data. The Lagrange multiplier method is usually used in the formulation of such problems, resulting in a two-field variational formulation that is often called a mixed formulation. Typical straightforward discretizations of this problem, for example classical Galerkin finite element method, are frequently non-convergent and can cause severe problems. These include oscillations, locking, singular matrices, and spurious nonphysical solutions. This is the case even when precisely the same method for the forward problem gives optimally accurate discrete solutions and shows optimal convergence. We show in the context of inverse elasticity how to address such diffculties by residual based stabilization. We formulated the inverse problem to reconstruct the elastic modulus distribution of soft tissue as a constrained minimization problem for which we seek the stationary point of a Lagrangian. We discretized the resulting equations using the Galerkin approximation of the unite element method. Stabilization was considered here by adding to the weak form a perturbation term based on the residual form of the Euler-Lagrange equation. In this way stability was enhanced without upsetting consistency. Preliminary results obtained when solving the inverse problem for various cases show that the stabilization terms employed in this formulation enable us to deal with the spurious solutions that appear in the absence of stabilization. The purpose of this study is to find the best stabilization terms in order to solve accurately the inverse problem.


Poster presented at the 2006 Thrust R1A Nonlinear and Dual Wave Probes Conference


Elastic Modulus Reconstruction, two-field variational formulation

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Lagrange problem




Bernard M. Gordon Center for Subsurface Sensing and Imaging Systems (Gordon-CenSSIS)

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Bernard M. Gordon Center for Subsurface Sensing and Imaging Systems (Gordon-CenSSIS)

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