In this work we study the inverse boundary value problem of determining the refractive index in the acoustic equation. It is known that this inverse problem is ill-posed. Nonetheless, we show that the ill-posedness decreases when we increase the frequency and the stability estimate changes from logarithmic type for low frequencies to a Lipschitz estimate for large frequencies.
We study the inverse problem of determining an electrical inclusion from boundary measurements. We derive a stability estimate for the linearized map with explicit formulae on generic constants that shows that the problem becomes more ill-posed as the inclusion is farther from the boundary. We also show that this estimate is optimal.
We develop a reconstruction algorithm to determine penetrable obstacles inside a domain in the plane from acoustic measurements made on the boundary. This algorithm uses complex geometrical optics solutions to the Helmholtz equation with polynomial-type phase functions.
In this paper we study the local behavior of a solution to the l-th power of the Laplacian with singular coefficients in lower order terms. We obtain a bound on the vanishing order of the nontrivial solution. Our proofs use Carleman estimates with carefully chosen weights. We will derive appropriate three-sphere inequalities and apply them to obtain doubling inequalities and the maximal vanishing order.
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