Lecture notes on nonlinear inversion and tomography: 1. Borehole seismic tomography / Конспекты лекций по нелинейной инверсии и томографии: 1. Скважинная сейсмическая томография

Nonlinear inverse problems are common in science and engineering. In fact the quotation from Feynman shows clearly that the process of discovering physical laws is itself an inverse problem.
What is an inverse problem? Subtraction is the inverse of addition. Division is the inverse of multiplication. Root is the inverse of power. Given the answer (say, the number 4) nd the question (2+2 = ? or 8/2 = ? or p 16 = ?). This last example (commonly seen in the game of Jeopardy) is most important since it is clear that the same answer (i.e., the data) could come from many questions (e.g., models and methods of analysis) | and therefore it is not surprising that a degree of ambiguity (sometimes a very high degree of ambiguity) is an inherent part of most realistic inverse problems. Physical scientists are used to thinking about situations that lead to equations with unique solutions. Because of the traditional training they receive, most scientists are very uncomfortable with mathematical problems that do not have unique solutions. Yet both quantum mechanics and chaos theory provide numerous examples of real experiments (e.g., the double slit experiment and weather) where ambiguities are typically encountered. The subject of inverse problems is another realm where lack of uniqueness commonly occurs. Methods of dealing with the ambiguities therefore play a vital | if not quite central | role in our analysis. <...>