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The problems of making inferences about the natural world from noisy observations and imperfect theories occur in almost all scientific disciplines. This 2006 book addresses these problems using examples taken from geophysical fluid dynamics. It focuses on discrete formulations, both static and time-varying, known variously as inverse, state estimation or data assimilation problems. Starting with fundamental algebraic and statistical ideas, the book guides the reader through a range of inference tools including the singular value decomposition, Gauss-Markov and minimum variance estimates, Kalman filters and related smoothers, and adjoint (Lagrange multiplier) methods. The final chapters discuss a variety of practical applications to geophysical flow problems. Discrete Inverse and State Estimation Problems is an ideal introduction to the topic for graduate students and researchers in oceanography, meteorology, climate dynamics, and geophysical fluid dynamics. It is also accessible to a wider scientific audience; the only prerequisite is an understanding of linear algebra.

Oceanography --- Estimation theory --- Geophysics --- Mathematical models --- Fluid models --- Problèmes inverses (équations différentielles) --- Inverse problems (Differential equations) --- Estimation theory. --- Fluid models in geophysics --- Rotating dishpan --- Estimating techniques --- Least squares --- Mathematical statistics --- Stochastic processes --- Mathematical models. --- Fluid models. --- Fluides, Mécanique des --- Fluid mechanics --- Géophysique --- Mécanique des fluides. --- Oceanography - Mathematical models --- Geophysics - Fluid models

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The book develops the statistical approach to inverse problems with an emphasis on modeling and computations. The framework is the Bayesian paradigm, where all variables are modeled as random variables, the randomness reflecting the degree of belief of their values, and the solution of the inverse problem is expressed in terms of probability densities. The book discusses in detail the construction of prior models, the measurement noise modeling and Bayesian estimation. Markov Chain Monte Carlo-methods as well as optimization methods are employed to explore the probability distributions. The results and techniques are clarified with classroom examples that are often non-trivial but easy to follow. Besides the simple examples, the book contains previously unpublished research material, where the statistical approach is developed further to treat such problems as discretization errors, and statistical model reduction. Furthermore, the techniques are then applied to a number of real world applications such as limited angle tomography, image deblurring, electrical impedance tomography and biomagnetic inverse problems. The book is intended to researchers and advanced students in applied mathematics, computational physics and engineering. The first part of the book can be used as a text book on advanced inverse problems courses. The authors Jari Kaipio and Erkki Somersalo are Professors in the Applied Physics Department of the University of Kuopio, Finland and the Mathematics Department at the Helsinki University of Technology, Finland, respectively.

Inverse problems (Differential equations) --- Problèmes inversés (Equations différentielles) --- Numerical solutions. --- Solutions numériques --- Inverse problems (Differential equations) -- Numerical solutions. --- Mathematics - General --- Mathematical Theory --- Calculus --- Mathematics --- Physical Sciences & Mathematics --- Numerical solutions --- Inverse problems (Differential equations). --- Problèmes inversés (Equations différentielles) --- Solutions numériques --- EPUB-LIV-FT LIVMATHE SPRINGER-B --- Mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Computer mathematics. --- Probabilities. --- Physics. --- Complexity, Computational. --- Biomedical engineering. --- Probability Theory and Stochastic Processes. --- Analysis. --- Computational Mathematics and Numerical Analysis. --- Theoretical, Mathematical and Computational Physics. --- Complexity. --- Biomedical Engineering. --- Clinical engineering --- Medical engineering --- Bioengineering --- Biophysics --- Engineering --- Medicine --- Complexity, Computational --- Electronic data processing --- Machine theory --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Computer mathematics --- Discrete mathematics --- 517.1 Mathematical analysis --- Mathematical analysis --- Math --- Science --- Numerical analysis --- Differential equations --- Distribution (Probability theory. --- Global analysis (Mathematics). --- Computer science --- Engineering. --- Biomedical Engineering and Bioengineering. --- Construction --- Industrial arts --- Technology --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Mathematical physics. --- Computational complexity. --- Physical mathematics --- Physics --- Inverse problems (Differential equations) - Numerical solutions.