Some parameters of a physical system, for example, density or conductivity, may not be known, being inaccessible to direct measurement. To determine the values of these parameters, so that the system is understood as completely as possible, we solve an inverse problem, or seek the cause knowing the effect. Inverse problems arise in geophysics (analysing the interior of the earth, oil field location), medical imaging (MRI, ultrasound), remote sensing, ocean acoustic tomography, nondestructive testing, and astronomy. To stay connected with applications, we model the physical system as a system of PDEs with piecewise smooth coefficients on an anisotropic Riemannian polyhedron, which is constructed of “glued” together pieces of various materials. The inverse problem is then to determine the polyhedron structure, metric and the coefficients of a system of PDEs given partial information of special solutions at accessible points (e.g. on the surface of the earth). The uniqueness problem solved in the book together with introduced techniques are of great importance to mathematicians and might be appealing to anyone interested in modern interdisciplinary research.

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