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This book is devoted to the mathematical theory of regularization methods and gives an account of the currently available results about regularization methods for linear and nonlinear ill-posed problems. Both continuous and iterative regularization methods are considered in detail with special emphasis on the development of parameter choice and stopping rules which lead to optimal convergence rates.
This book provides a concise description of the current status of a fascinating scientific problem — the inverse variational problem in classical mechanics. The essence of this problem is as follows: one is given a set of equations of motion describing a certain classical mechanical system, and the question to be answered is: Do these equations of motion correspond to some Lagrange function as its Euler-Lagrange equations? In general, not for every system of equations of motion does a Lagrange function exist; it can, however, happen that one may modify the given equations of motion in such a way that they yield the same set of solutions as the original ones and they correspond already to a Lagrange function. Moreover, there can even be infinitely many such Lagrange functions, the relations among which are not trivial. The book deals with this scope of problems. No advanced mathematical methods, such as, contemporary differential geometry, are used. The intention is to meet the standard educational level of a broad group of physicists and mathematicians. The book is well suited for use as lecture notes in a university course for physicists.
Inverse Problems is a monograph which contains a self-contained presentation of the theory of several major inverse problems and the closely related results from the theory of ill-posed problems. The book is aimed at a large audience which include graduate students and researchers in mathematical, physical, and engineering sciences and in the area of numerical analysis.
The 100+ Series, Geometry, offers in-depth practice and review for challenging middle school math topics such as rotations, reflections, and transformations; congruence and similarity; and sine and cosine functions. Bonus activities on each page help extend the learning and activities, making these books perfect for daily review in the classroom or at home. Common Core State Standards have raised expectations for math learning, and many students in grades 6Ð8 are studying more accelerated math at younger ages. The 100+ Series provides the solution with titles that include over 100 targeted practice activities for learning algebra, geometry, and other advanced math topics. It also features over 100 reproducible, subject specific practice pages to support standards-based instruction.
In inverse problems one wants to find some parameter of interest which is not directly observable by indirect measurement. These measurements are usually noisy while the mapping of measurement to parameter is typically illposed (that is unstable). Therefore one applies regularization techniques that balance these two factors to find a stable approximation of the sought for parameter. However, in order to bound the reconstruction error, one needs additional information on the true parameter, which is nowadays typically formulated in terms of variational source conditions. In this thesis, we develop a general strategy to verify these conditions based on smoothness of the true parameter and the illposedness of the problem; the latter will be characterized by exploiting structural similarities to stability estimates. Following this, we apply our strategy to verify variational source conditions for parameter identification problems, inverse scattering and electrical impedance tomography.