What are the physical motivations and importance of inverse problems?
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Inverse problems arise in various fields of science and technology, including geophysics, medical imaging, and quantum mechanics. These problems involve recovering the internal properties of a medium by making measurements only on the boundary. The importance of inverse problems lies in their ability to provide valuable information about physical systems when the necessary inputs are known. In the context of differential and integral equations, these inputs include initial or boundary data, coefficients, force terms, and the shape and size of the domain. By solving inverse problems, we can compute solutions and predict the behavior of physical systems under different conditions. Inverse problems have applications in areas such as thermodynamics, photonics, elasticity, and network measurement.
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