Why are inverse problems interesting and practically important?
Inverse problems are the problems that consist of finding an unknown property of an object, or a medium, from the observation of a response of this object, or medium, to a probing signal. Thus, the theory of inverse problems yields a theoretical basis for remote sensing and non-destructive evaluation. For example, if an acoustic plane wave is scattered by an obstacle, and one observes the scattered field far from the obstacle, or in some exterior region, then the inverse problem is to find the shape and material properties of the obstacle. Such problems are important in identification of flying objects (airplanes missiles, etc.), objects immersed in water (submarines, paces of fish, etc.), and in many other situations. In geophysics one sends an acoustic wave from the surface of the earth and collects the scattered field on the surface for various positions of the source of the field for a fixed frequency, or for several frequencies. The inverse problem is to find the subsurface inhomogeneities. In technology one measures the eigenfrequencies of a piece of a material, and the inverse problem is to find a defect in this material, for example, a hole in a metal. In geophysics the inhomogeneity can be an oil deposit, a cave, a mine. In medicine it may be a tumor, or some abnormality in a human body. If one is able to find inhomogeneities in a medium by processing the scattered field on the surface, then one does not have to drill a hole in a medium. This, in turn, avoids expensive and destructive evaluation. The practical advantages of remote sensing are what makes the inverse problems important. (Inverse Problems: Mathematical and Analytical Techniques with Applications to Engineering, A. G. Ramm)
Some Books on Inverse Problems
1. An Introduction to the Mathematical Theory of Inverse Problems, A. Kirsch, Springer, 1996.
2. Regularization of Inverse Problems, H. W. Engl, M. Hanke, A. Neubauer, Springer, 1996.
3. Methods for Solving Inverse Problems in Mathematical Physics, A. I. Prilepko, D. G. Orlovsky, I. A. Vasin, Marcel Dekker, 2000.
4. Computational Methods for Inverse Problems, C. R. Vogel, Siam, 2002.
5. Inverse Problem Theory and Methods for Model Parameter Estimation, A. Tarantola, Siam, 2004.
6. Inverse Problems for Partial Differential Equations (Applied Mathematical Sciences), V. Isakov, Springer, 2005.
7. Inverse Problems: Mathematical and Analytical Techniques with Applications to Engineering, A. G. Ramm, , Springer, 2005.
8. Numerical Methods for Solving Inverse Problems of Mathematical Physics , A. A. Samarskii, , de Gruyter, 2007.
9. An Introduction to Inverse Problems with Applications, F. D. M. Neto, A. J. Neto, Springer, 2012.
International SCI Journals on Inverse Problems
1. Inverse Problems
2. Inverse Problems in Science and Engineering
3. Journal of Inverse and ill-Posed Problems
4. Inverse Problems and Imaging
Groups and Societies on Inverse Problems
1. Inverse Problems Group at the Zirve University
2. Inverse Problems Group at the University of Manchester
3. Optimal Control and Inverse Problems Group at the University of Graz
4. Inverse Problems Group at the University of Washington