Advances in the development of novel cobalt Fischer-Tropsch catalysts for synthesis of long-chain hydrocarbons and clean fuels.

Atlas of Clinical Fungi

Lectures on Cauchy’s Problem in Linear Partial Differential Equations. By J. Hadamard. Pp. viii+316. 15s.net. 1923. (Per Oxford University Press.)

The goal of this final chapter is to show how the boundary value problems of mathematical physics can be solved by the methods of the preceding chapters. This will be done by solving a variety of specific problems that illustrate the principal types of problems that were formulated in Chapter 7. Additional applications are developed in the Exercises. The primary solution method is Fourier’s method of separation of variables and the associated Sturm-Liouville theory of Chapter 8.

Boundary Value Problems of Mathematical Physics

Linear Inverse Problems.- Large-Scale Inverse Problems in Imaging.- Regularization Methods for Ill-Posed Problems.- Distance Measures and Applications to Multi-Modal Variational Imaging.- Energy Minimization Methods.- Compressive Sensing.- Duality and Convex Programming.- EM Algorithms.- Iterative Solution Methods.- Level Set Methods for Structural Inversion and Image Reconstructions.- Expansion Methods.- Sampling Methods.- Inverse Scattering.- Electrical Impedance Tomography.- Synthetic Aperture Radar Imaging.- Tomography.- Optical Imaging.- Photoacoustic and Thermoacoustic Tomography: Image Formation Principles.- Mathematics of Photoacoustic and Thermoacoustic Tomography.- Wave Phenomena.- Statistical Methods in Imaging.- Supervised Learning by Support Vector Machines.- Total Variation in Imaging.- Numerical Methods and Applications in Total Variation Image Restoration.- Mumford and Shah Model and its Applications in Total Variation Image Restoration.- Local Smoothing Neighbourhood Filters.- Neighbourhood Filters and the Recovery of 3D Information.- Splines and Multiresolution Analysis.- Gabor Analysis for Imaging.- Shaper Spaces.- Variational Methods in Shape Analysis.- Manifold Intrinsic Similarity.- Image Segmentation with Shape Priors: Explicit Versus Implicit Representations.- Starlet Transform in Astronomical Data Processing.- Differential Methods for Multi-Dimensional Visual Data Analysis.- Wave fronts in Imaging, Quinto.- Ultrasound Tomography, Natterer.- Optical Flow, Schnoerr.- Morphology, Petros.- Maragos.- PDEs, Weickert. - Registration, Modersitzki. - Discrete Geometry in Imaging, Bobenko, Pottmann.-Visualization, Hege.- Fast Marching and Level Sets, Osher.- Couple Physics Imaging, Arridge.- Imaging in Random Media, Borcea.- Conformal Methods, Gu.- Texture, Peyre.- Graph Cuts, Darbon.- Imaging in Physics with Fourier Transform (i.e. Phase Retrieval e.g Dark field imaging), J. R. Fienup.- Electron Microscopy, Oktem Ozan.- Mathematical Imaging OCT (this is also FFT based), Mark E. Brezinski.- Spect, PET, Faukas, Louis.

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Handbook of mathematical methods in imaging

Abstract The terms “inverse problems” and “ill-posed problems” have been steadily and surely gaining popularity in modern science since the middle of the 20th century. A little more than fifty years of studying problems of this kind have shown that a great number of problems from various branches of classical mathematics (computational algebra, differential and integral equations, partial differential equations, functional analysis) can be classified as inverse or ill-posed, and they are among the most complicated ones (since they are unstable and usually nonlinear). At the same time, inverse and ill-posed problems began to be studied and applied systematically in physics, geophysics, medicine, astronomy, and all other areas of knowledge where mathematical methods are used. The reason is that solutions to inverse problems describe important properties of media under study, such as density and velocity of wave propagation, elasticity parameters, conductivity, dielectric permittivity and magnetic permeability, and properties and location of inhomogeneities in inaccessible areas, etc. In this paper we consider definitions and classification of inverse and ill-posed problems and describe some approaches which have been proposed by outstanding Russian mathematicians A. N. Tikhonov, V. K. Ivanov and M. M. Lavrentiev.

Definitions and examples of inverse and ill-posed problems

The theory of ill-posed problems originated in an unusual way. As a rule, a new concept is a subject in which its creator takes a keen interest. The concept of ill-posed problems was introduced by Hadamard with the comment that these problems are physically meaningless and not worthy of the attention of serious researchers. Despite Hadamard's pessimistic forecasts, however, his unloved "child" has turned into a powerful theory whose results are used in many fields of pure and applied mathematics. What is the secret of its success? The answer is clear. Ill-posed problems occur everywhere and it is unreasonable to ignore them. Unlike ill-posed problems, inverse problems have no strict mathematical definition. In general, they can be described as the task of recovering a part of the data of a corresponding direct (well-posed) problem from information about its solution. Inverse problems were first encountered in practice and are mostly ill-posed. The urgent need for their solution, especially in geological exploration and medical diagnostics, has given powerful impetus to the development of the theory of ill-posed problems. Nowadays, the terms "inverse problem" and "ill-posed problem" are inextricably linked to each other. Inverse and ill-posed problems are currently attracting great interest. A vast literature is devoted to these problems, making it necessary to systematize the accumulated material. This book is the first small step in that direction. We propose a classification of inverse problems according to the type of equation, unknowns and additional information. We consider specific problems from a single position and indicate relationships between them. The problems relate to different areas of mathematics, such as linear algebra, theory of integral equations, integral geometry, spectral theory and mathematical physics. We give examples of applied problems that can be studied using the techniques we describe. This book was conceived as a textbook on the foundations of the theory of inverse and ill-posed problems for university students.The author'sintention was to explain this complex material in the most accessible way possible. The monograph is aimed primarily at those who are just beginning to get to grips with inverse and ill-posed problems but we hope that it will be useful to anyone who is interested in the subject.

Inverse and Ill-Posed Problems: Theory and Applications

Inverse and Ill-posed Problems

Main definitions and notations Introduction Chapter 1. Finite-difference scheme inversion (FDSI) 1.1. Introduction 1.2. Volterra operator equations 1.3. Definitions and examples 1.4. Convergence of FDSI 1.5. Numerical examples Chapter 2. Linearized multidimensional inverse problem for the wave equation 2.1. Introduction 2.2. Problem formulation 2.3. Linearization 2.4 Analyzing the structure of the solution to one-dimensional direct problem 2.5. Existence theorem for the direct problem 2.6. Uniqueness of solutions to the inverse problem and regularization 2.7. Numerical examples vi Direct Methods of Solving Multidimensional Inverse Problems Chapter 3. Methods of I.M. Gel'fand, B.M. Levitan and M. G. Krein 3.1. Introduction 3.2. Gel'fand-Levitan-Krein (GLK) equation for one-dimensional inverse problem

Direct Methods of Solving Multidimensional Inverse Hyperbolic Problems

Part 1 Cauchy problem for Maxwell's equations: Maxwell's equations as a hyperbolic symmetric system structure of the Cauchy problem solution in case of the current located on the media interface. Part 2 One-dimensional inverse problems: structure of the Fourier-image of the Cauchy problem solution for one-dimensional medium in case of the current located at a point the problem of determining the medium permittivity the problem of determining the conductivity co-efficient the problem of determining all the co-efficients of Maxwell's equations. Part 3 Multi-dimensional inverse problems: linearization method applied to the inverse problems investigation of the linearized problem of determining the permittivity co-efficient unique solvability theorem for a two-dimensional problem of determining the conductivity co-efficient analytic in one variable on the uniqueness of the solution of three-dimensional inverse problems. Part 4 Inverse problems in the case of source periodic in time: one-dimensional inverse problems linear one-dimensional inverse problem linearized three-dimensional inverse problems. Part 5 Inverse problems for quasi-stationary Maxwell's equations: on correspondence between the solutions of quasi-stationary and wave Maxwell's equations a one-dimensional inverse problem of determining the conductivity and permeability co-efficients the one-dimensional inverse problem for wave-quasi-stationary system of equations. Part 6 The inverse problems for the simplest anisotropic media: on the uniqueness of determination of permittivity and permeability in anisotropic media on the problem of determining permittivity and conductivity tensors. Part 7 Numerical methods. Part 8 Convergence results. Part 9 Examples (Part contents)

Sergey Kabanikhin

Papers

Definitions and examples of inverse and ill-posed problems

Inverse and Ill-Posed Problems: Theory and Applications

Inverse and Ill-posed Problems

Direct Methods of Solving Multidimensional Inverse Hyperbolic Problems

Inverse Problems for Maxwell's Equations