Matti Lassas

Anisotropic conductivities that cannot be detected by EIT

We construct anisotropic conductivities in dimension 3 that give rise to the same voltage and current measurements at the boundary of a body as a homogeneous isotropic conductivity. These conductivities are non-zero, but degenerate close to a surface inside the body.

Inverse boundary spectral problems

© 2001 by Chapman & Hall/CRC. Inverse boundary problems are a rapidly developing area of applied mathematics with applications throughout physics and the engineering sciences. However, the mathematical theory of inverse problems remains incomplete and needs further development to aid in the solution of many important practical problems. Inverse Boundary Spectral Problems develop a rigorous theory for solving several types of inverse problems exactly. In it, the authors consider the following: “Can the unknown coefficients of an elliptic partial differential equation be determined from the eigenvalues and the boundary values of the eigenfunctions?” Along with this problem, many inverse problems for heat and wave equations are solved. The authors approach inverse problems in a coordinate invariant way, that is, by applying ideas drawn from differential geometry. To solve them, they apply methods of Riemannian geometry, modern control theory, and the theory of localized wave packets, also known as Gaussian beams. The treatment includes the relevant background of each of these areas. Although the theory of inverse boundary spectral problems has been in development for at least 10 years, until now the literature has been scattered throughout various journals. This self-contained monograph summarizes the relevant concepts and the techniques useful for dealing with them.

Full-wave invisibility of active devices at all frequencies

There has recently been considerable interest in the possibility, both theoretical and practical, of invisibility (or “cloaking”) from observation by electromagnetic (EM) waves. Here, we prove invisibility with respect to solutions of the Helmholtz and Maxwell’s equations, for several constructions of cloaking devices. The basic idea, as in the papers [GLU2, GLU3, Le, PSS1], is to use a singular transformation that pushes isotropic electromagnetic parameters forward into singular, anisotropic ones. We define the notion of finite energy solutions of the Helmholtz and Maxwell’s equations for such singular electromagnetic parameters, and study the behavior of the solutions on the entire domain, including the cloaked region and its boundary. We show that, neglecting dispersion, the construction of [GLU3, PSS1] cloaks passive objects, i.e., those without internal currents, at all frequencies k. Due to the singularity of the metric, one needs to work with weak solutions. Analyzing the behavior of such solutions inside the cloaked region, we show that, depending on the chosen construction, there appear new “hidden” boundary conditions at the surface separating the cloaked and uncloaked regions. We also consider the effect on invisibility of active devices inside the cloaked region, interpreted as collections of sources and sinks or internal currents. When these conditions are overdetermined, as happens for Maxwell’s equations, generic internal currents prevent the existence of finite energy solutions and invisibility is compromised.We give two basic constructions for cloaking a region D contained in a domain

Cloaking Devices, Electromagnetic Wormholes, and Transformation Optics

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Statistical inversion for medical x-ray tomography with few radiographs: I. General theory

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Invisibility and inverse problems

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