On 2D Inverse Problems/The inverse problems

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The inverse problems that we discuss in this book are the problems of inferring information about graphs or manifolds from the solutions of difference and differential equations defined on the domains (the measurement data). There're two categories of the inverse problems of finding the local and global properties of networks and manifolds, inverse boundary problems (from the boundary values of the solutions) and inverse spectral problems (from the spectral data of difference or differential operators). In this book we will consider both and also the relationship b/w the continuous inverse problems on manifolds and discrete inverse problems on the embedded networks.

Exercise (*). Restate the problem of finding roots of a polynomial as an inverse problem on the weighted directed graph of the following type:

The weights of the paths are elementary symmetric functions of the weights of individual edges

(Hint). The boundary data consists of values of the elementary symmetric functions of the weights a,b,c,d.