We discuss the ill-posed Cauchy problem for elliptic equations, which is pervasive in inverse boundary value problems modeled by elliptic equations.We provide essentially optimal stability results, in wide generality and under substantially minimal assumptions.As a general scheme in our arguments, we show that all such stability results can be derived by the use of a single building brick, the three-spheres inequality.
Abstract. We prove that a sound-soft polyhedral scatterer is uniquely determined by the far-field pattern corresponding to an incident plane wave at one given wavenumber and one given incident direction.Lo duca e io per quel cammino ascoso intrammo a ritornar nel chiaro mondo; e sanza cura aver d'alcun riposo, salimmo su, el primo e io secondo, tanto ch'i' vidi de le cose belle che porta'l ciel, per un pertugio tondo; e quindi uscimmo a riveder le stelle.Dante, Inferno, C.XXXIV, 133-139.
We prove an optimal stability estimate for an inverse Robin boundary value problem arising in corrosion detection by electrostatic boundary measurements.
Abstract.We consider the problem of electrical impedance tomography where conductivity distribution in a domain is to be reconstructed from boundary measurements of voltage and currents. It is well-known that this problem is highly illposed. In this work, we propose the use of the Mumford-Shah functional, developed for segmentation and denoising of images, as a regularization. After establishing existence properties of the resulting variational problem, we proceed by demonstrating the approach in several numerical examples. Our results indicate that this is an effective approach for overcoming the illposedness. Moreover, it has the capability of enhancing the reconstruction while at the same time segmenting the conductivity image.
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