
Solving an inverse elliptic coefficient problem by convex nonlinear semidefinite programming
Several applications in medical imaging and nondestructive material tes...
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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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Series reversion in Calderón's problem
This work derives explicit series reversions for the solution of Calderó...
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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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On the implementation of largescale integral operators with modern HPC solutions – Application to 3D Marchenko imaging by leastsquares inversion
Numerical integral operators of convolution type form the basis of most ...
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Numerical recovery of the piecewise constant leading coefficient of an elliptic equation
We propose a numerical algorithm for the reconstruction of a piecewise c...
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BiParametric Operator Preconditioning
We extend the general operator preconditioning framework [R. Hiptmair, C...
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An introduction to finite element methods for inverse coefficient problems in elliptic PDEs
Several novel imaging and nondestructive testing technologies are based on reconstructing the spatially dependent coefficient in an elliptic partial differential equation from measurements of its solution(s). In practical applications, the unknown coefficient is often assumed to be piecewise constant on a given pixel partition (corresponding to the desired resolution), and only finitely many measurements can be made. This leads to the problem of inverting a finitedimensional nonlinear forward operator ℱ: 𝒟(ℱ)⊆ℝ^n→ℝ^m, where evaluating ℱ requires one or several PDE solutions. Numerical inversion methods require the implementation of this forward operator and its Jacobian. We show how to efficiently implement both using a standard FEM package and prove convergence of the FEM approximations against their truesolution counterparts. We present simple example codes for Comsol with the Matlab Livelink package, and numerically demonstrate the challenges that arise from nonuniqueness, nonlinearity and instability issues. We also discuss monotonicity and convexity properties of the forward operator that arise for symmetric measurement settings.
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