46 research outputs found
Uniqueness, stability and global convergence for a discrete inverse elliptic Robin transmission problem
We derive a simple criterion that ensures uniqueness, Lipschitz stability and
global convergence of Newton's method for the finite dimensional zero-finding
problem of a continuously differentiable, pointwise convex and monotonic
function. Our criterion merely requires to evaluate the directional derivative
of the forward function at finitely many evaluation points and for finitely
many directions.
We then demonstrate that this result can be used to prove uniqueness,
stability and global convergence for an inverse coefficient problem with
finitely many measurements. We consider the problem of determining an unknown
inverse Robin transmission coefficient in an elliptic PDE. Using a relation to
monotonicity and localized potentials techniques, we show that a
piecewise-constant coefficient on an a-priori known partition with a-priori
known bounds is uniquely determined by finitely many boundary measurements and
that it can be uniquely and stably reconstructed by a globally convergent
Newton iteration. We derive a constructive method to identify these boundary
measurements, calculate the stability constant and give a numerical example
Monotonicity and local uniqueness for the Helmholtz equation
This work extends monotonicity-based methods in inverse problems to the case
of the Helmholtz (or stationary Schr\"odinger) equation in a bounded domain for fixed non-resonance frequency and real-valued
scattering coefficient function . We show a monotonicity relation between
the scattering coefficient and the local Neumann-Dirichlet operator that
holds up to finitely many eigenvalues. Combining this with the method of
localized potentials, or Runge approximation, adapted to the case where
finitely many constraints are present, we derive a constructive
monotonicity-based characterization of scatterers from partial boundary data.
We also obtain the local uniqueness result that two coefficient functions
and can be distinguished by partial boundary data if there is a
neighborhood of the boundary where and
Solving an inverse elliptic coefficient problem by convex non-linear semidefinite programming
Several applications in medical imaging and non-destructive 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 non-linear ill-posed 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 non-linear 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 optimization-based
approaches for inverse coefficient problems