| Author | Mohit Tyagi, Paul E. Barbone, Assad A. Oberai |
|---|---|
| Title | Residual-based stabilized formulation for the solution of inverse elliptic PDE |
| Year | 2016 |
| Journal | Computers andMathematics in Applications |
| Volume | To be submitted |
| Abstract | In several fields of science and engineering, such as thermometry, elasticity imaging, and geophysics, the solution to an inverse problem with interior data is sought, wherein the forward model is in the form of an elliptic partial differential equation. A common approach to solving these problems is to pose them as a constrained minimization problem, where the difference between the measured and a predicted response is minimized under the constraint that the predicted response satisfy the forward elliptic model. The optimization parameters represent the spatial distribution of the material properties in the forward model, and the data mismatch is measured in the L2 norm. In this manuscript we consider an instantiation of this problem, where the forward problem is that of linear plane stress elasticity, or equivalently that of linear heat/hydraulic conduction. We demonstrate that the linearized version of the saddle point problem obtained from the minimization problem inherits some stability from the forward elliptic problem. In particular, it is stable for the response variable and the Lagrange multiplier, but not for the material property field. This lack of stability implies that we are unable to prove optimal convergence with mesh refinement for the overall problem. We overcome this difficulty by adding to the saddle point problem a residual-based term that provides sufficient stability, and prove optimal convergence in an energy-like norm. We verify these estimates through simple numerical examples. We note that while we have considered a specific model for an inverse elliptic problem in this manuscript, similar ideas could be developed for a broad class of inverse elliptic problems. |
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