
An introduction to finite element methods for inverse coefficient problems in elliptic PDEs
Several novel imaging and nondestructive testing technologies are based...
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On stable invertibility and global Newton convergence for convex monotonic functions
We derive a simple criterion that ensures uniqueness, Lipschitz stabilit...
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Finite element analysis for identifying the reaction coefficient in PDE from boundary observations
This work is devoted to the nonlinear inverse problem of identifying the...
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Error analysis of an unfitted HDG method for a class of nonlinear elliptic problems
We study Hibridizable Discontinuous Galerkin (HDG) discretizations for a...
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A Matrix Factorization Approach for Learning SemidefiniteRepresentable Regularizers
Regularization techniques are widely employed in optimizationbased appr...
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Applications of the BackusGilbert method to linear and some non linear equations
We investigate the use of a functional analytical version of the Backus...
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Multifrequency inverse obstacle scattering with unknown impedance boundary conditions using recursive linearization
We consider the reconstruction of the shape and the impedance function o...
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Solving an inverse elliptic coefficient problem by convex nonlinear semidefinite programming
Several applications in medical imaging and nondestructive material testing lead to inverse elliptic coefficient problems, where an unknown coefficient function in an elliptic PDE is to be determined from partial knowledge of its solutions. This is usually a highly nonlinear illposed inverse problem, for which unique reconstructability results, stability estimates and global convergence of numerical methods are very hard to achieve. The aim of this note is to point out a new connection between inverse coefficient problems and semidefinite programming that may help addressing these challenges. We show that an inverse elliptic Robin transmission problem with finitely many measurements can be equivalently rewritten as a uniquely solvable convex nonlinear semidefinite optimization problem. This allows to explicitly estimate the number of measurements that is required to achieve a desired resolution, to derive an error estimate for noisy data, and to overcome the problem of local minima that usually appears in optimizationbased approaches for inverse coefficient problems.
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