1987, ISBN 9780444427656, xvi, 613

Book

Foundations of Computational Mathematics, ISSN 1615-3375, 12/2012, Volume 12, Issue 6, pp. 805 - 849

In applications throughout science and engineering one is often faced with the challenge of solving an ill-posed inverse problem, where the number of available measurements is smaller...

60D05 | Semidefinite programming | Economics general | 52A41 | Linear and Multilinear Algebras, Matrix Theory | Mathematics | Real algebraic geometry | 41A45 | Convex optimization | 90C25 | Numerical Analysis | 90C22 | Atomic norms | Math Applications in Computer Science | Applications of Mathematics | Computer Science, general | Gaussian width | Symmetry | MATHEMATICS, APPLIED | CUT | APPROXIMATION | ALGORITHM | EQUATIONS | RANK | SPACE | MATHEMATICS | RECOVERY | MINIMIZATION | NORM | COMPUTER SCIENCE, THEORY & METHODS | Geometry | Computational mathematics | Algebra | Optimization | Inverse problems | Mathematical analysis | Norms | Programming | Mathematical models | Matrices | Atomic structure | Matrix methods

60D05 | Semidefinite programming | Economics general | 52A41 | Linear and Multilinear Algebras, Matrix Theory | Mathematics | Real algebraic geometry | 41A45 | Convex optimization | 90C25 | Numerical Analysis | 90C22 | Atomic norms | Math Applications in Computer Science | Applications of Mathematics | Computer Science, general | Gaussian width | Symmetry | MATHEMATICS, APPLIED | CUT | APPROXIMATION | ALGORITHM | EQUATIONS | RANK | SPACE | MATHEMATICS | RECOVERY | MINIMIZATION | NORM | COMPUTER SCIENCE, THEORY & METHODS | Geometry | Computational mathematics | Algebra | Optimization | Inverse problems | Mathematical analysis | Norms | Programming | Mathematical models | Matrices | Atomic structure | Matrix methods

Journal Article

Annals of Statistics, ISSN 0090-5364, 2018, Volume 46, Issue 6B, pp. 3569 - 3602

.... a dictionary U from observations Y in an inverse regression model Y = T f + xi with linear operator T and general random error xi...

Deconvolution | Super-resolution | Ill-posed problem | Scan statistic | Multiscale analysis | Gumbel extreme value limit | APPROXIMATION | super-resolution | RESOLUTION | STATISTICS & PROBABILITY | ADAPTIVE ESTIMATION | ill-posed problem | scan statistic | DENSITY | deconvolution | SIGNAL-DETECTION | FLUORESCENCE MICROSCOPY | LINEAR FUNCTIONALS | REGULARIZATION | CONVERGENCE-RATES

Deconvolution | Super-resolution | Ill-posed problem | Scan statistic | Multiscale analysis | Gumbel extreme value limit | APPROXIMATION | super-resolution | RESOLUTION | STATISTICS & PROBABILITY | ADAPTIVE ESTIMATION | ill-posed problem | scan statistic | DENSITY | deconvolution | SIGNAL-DETECTION | FLUORESCENCE MICROSCOPY | LINEAR FUNCTIONALS | REGULARIZATION | CONVERGENCE-RATES

Journal Article

Inverse Problems, ISSN 0266-5611, 09/2015, Volume 31, Issue 9, pp. 93001 - 93021

This paper is concerned with computational approaches and mathematical analysis for solving inverse scattering problems in the frequency domain...

diffraction limit | multiple frequency | inverse scattering | LOCATION | NONUNIQUENESS | MATHEMATICS, APPLIED | RECONSTRUCTION | STABILITY | ALGORITHM | PHYSICS, MATHEMATICAL | NUMERICAL-SOLUTION | HELMHOLTZ-EQUATION | OBSTACLE | MAP | LINEAR SAMPLING METHOD | Reconstruction | Algorithms | Inverse problems | Diffraction | Inverse scattering | Mathematical analysis | Mathematical models | Inverse | Analysis of PDEs | Mathematics

diffraction limit | multiple frequency | inverse scattering | LOCATION | NONUNIQUENESS | MATHEMATICS, APPLIED | RECONSTRUCTION | STABILITY | ALGORITHM | PHYSICS, MATHEMATICAL | NUMERICAL-SOLUTION | HELMHOLTZ-EQUATION | OBSTACLE | MAP | LINEAR SAMPLING METHOD | Reconstruction | Algorithms | Inverse problems | Diffraction | Inverse scattering | Mathematical analysis | Mathematical models | Inverse | Analysis of PDEs | Mathematics

Journal Article

Mathematical problems in engineering, ISSN 1024-123X, 7/2013, Volume 2013, pp. 1 - 19

.... To do this we employ acoustic, electromagnetic, and elastic waves for presenting different types of inverse problems...

MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | ENGINEERING, MULTIDISCIPLINARY | BOUNDARY-VALUE PROBLEM | SCATTERING PROBLEM | ENSEMBLE KALMAN FILTER | NEWTON METHOD | SHAPE RECONSTRUCTION | DATA ASSIMILATION | NO-RESPONSE TEST | HARMONIC ACOUSTIC-WAVES | INTEGRAL-EQUATION METHODS | LINEAR SAMPLING METHOD | Mathematical research | Functions, Inverse | Research | Studies | Algorithms | Medical imaging | Inverse problems | Nondestructive testing | Research & development--R&D | Theory | Weather forecasting | Superconductivity | Artificial intelligence | Remote sensing | Mathematical analysis | Geophysics | Differential equations | Acoustics | Boundaries | Data assimilation

MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | ENGINEERING, MULTIDISCIPLINARY | BOUNDARY-VALUE PROBLEM | SCATTERING PROBLEM | ENSEMBLE KALMAN FILTER | NEWTON METHOD | SHAPE RECONSTRUCTION | DATA ASSIMILATION | NO-RESPONSE TEST | HARMONIC ACOUSTIC-WAVES | INTEGRAL-EQUATION METHODS | LINEAR SAMPLING METHOD | Mathematical research | Functions, Inverse | Research | Studies | Algorithms | Medical imaging | Inverse problems | Nondestructive testing | Research & development--R&D | Theory | Weather forecasting | Superconductivity | Artificial intelligence | Remote sensing | Mathematical analysis | Geophysics | Differential equations | Acoustics | Boundaries | Data assimilation

Journal Article

SIAM Journal on Imaging Sciences, ISSN 1936-4954, 2009, Volume 2, Issue 1, pp. 183 - 202

We consider the class of iterative shrinkage-thresholding algorithms (ISTA) for solving linear inverse problems arising in signal/image processing...

Deconvolution | Optimal gradient method | Image deblurring | Global rate of convergence | Two-step iterative algorithms | Least squares and l | Linear inverse problem | Iterative shrinkage-thresholding algorithm | regularization problems | MATHEMATICS, APPLIED | global rate of convergence | iterative shrinkage-thresholding algorithm | image deblurring | IMAGING SCIENCE & PHOTOGRAPHIC TECHNOLOGY | linear inverse problem | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | COMPUTER SCIENCE, SOFTWARE ENGINEERING | deconvolution | two-step iterative algorithms | IMAGE | optimal gradient method | least squares and l regularization problems | CONVERGENCE | REGULARIZATION | MONOTONE-OPERATORS | Algorithms | Inverse problems | Image processing | Images | Preserves | Signal processing | Iterative methods | Convergence

Deconvolution | Optimal gradient method | Image deblurring | Global rate of convergence | Two-step iterative algorithms | Least squares and l | Linear inverse problem | Iterative shrinkage-thresholding algorithm | regularization problems | MATHEMATICS, APPLIED | global rate of convergence | iterative shrinkage-thresholding algorithm | image deblurring | IMAGING SCIENCE & PHOTOGRAPHIC TECHNOLOGY | linear inverse problem | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | COMPUTER SCIENCE, SOFTWARE ENGINEERING | deconvolution | two-step iterative algorithms | IMAGE | optimal gradient method | least squares and l regularization problems | CONVERGENCE | REGULARIZATION | MONOTONE-OPERATORS | Algorithms | Inverse problems | Image processing | Images | Preserves | Signal processing | Iterative methods | Convergence

Journal Article

2011, ISBN 9789814338776, xxi, 326

Book

IEEE Transactions on Image Processing, ISSN 1057-7149, 05/2012, Volume 21, Issue 5, pp. 2481 - 2499

A general framework for solving image inverse problems with piecewise linear estimations is introduced in this paper...

Deblurring | inverse problem | Dictionaries | piecewise linear estimation | Biological system modeling | Piecewise linear approximation | Estimation | Gaussian distribution | Degradation | interpolation | Inverse problems | super- resolution | Gaussian mixture models | SUPERRESOLUTION | REGRESSION | IMAGE INTERPOLATION | REPRESENTATIONS | super-resolution | SCALE MIXTURES | RECONSTRUCTION | SIGNALS | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | ENGINEERING, ELECTRICAL & ELECTRONIC | RECOVERY | EM ALGORITHM | REGULARIZATION | Artifacts | Reproducibility of Results | Algorithms | Computer Simulation | Image Interpretation, Computer-Assisted - methods | Sensitivity and Specificity | Linear Models | Image Enhancement - methods | Models, Statistical | Normal Distribution | Usage | Image processing | Kernel functions | Mathematical optimization | Innovations | Gaussian processes | Interpolation | Images | Mathematical models | Gaussian | Computational efficiency | Estimators

Deblurring | inverse problem | Dictionaries | piecewise linear estimation | Biological system modeling | Piecewise linear approximation | Estimation | Gaussian distribution | Degradation | interpolation | Inverse problems | super- resolution | Gaussian mixture models | SUPERRESOLUTION | REGRESSION | IMAGE INTERPOLATION | REPRESENTATIONS | super-resolution | SCALE MIXTURES | RECONSTRUCTION | SIGNALS | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | ENGINEERING, ELECTRICAL & ELECTRONIC | RECOVERY | EM ALGORITHM | REGULARIZATION | Artifacts | Reproducibility of Results | Algorithms | Computer Simulation | Image Interpretation, Computer-Assisted - methods | Sensitivity and Specificity | Linear Models | Image Enhancement - methods | Models, Statistical | Normal Distribution | Usage | Image processing | Kernel functions | Mathematical optimization | Innovations | Gaussian processes | Interpolation | Images | Mathematical models | Gaussian | Computational efficiency | Estimators

Journal Article

IEEE Transactions on Information Theory, ISSN 0018-9448, 06/2018, Volume 64, Issue 6, pp. 4129 - 4158

In this paper, we characterize sharp time-data tradeoffs for optimization problems used for solving linear inverse problems...

projected gradient descent | linear inverse problems | Inverse problems | run time prediction of optimization algorithms | Cost function | Prediction algorithms | Complexity theory | Time-data tradeoffs | Noise measurement | Convergence | noncovex and nonsmooth optimization | Economic models | Penalty function | Phase transformations | Transition points | Tradeoffs | Least squares | Phase transitions | Empirical analysis

projected gradient descent | linear inverse problems | Inverse problems | run time prediction of optimization algorithms | Cost function | Prediction algorithms | Complexity theory | Time-data tradeoffs | Noise measurement | Convergence | noncovex and nonsmooth optimization | Economic models | Penalty function | Phase transformations | Transition points | Tradeoffs | Least squares | Phase transitions | Empirical analysis

Journal Article

SIAM Journal on Matrix Analysis and Applications, ISSN 0895-4798, 2017, Volume 38, Issue 2, pp. 458 - 477

In this paper, we consider optimal low-rank regularized inverse matrix approximations and their applications to inverse problems...

Bayes risk | Ill-posed inverse problems | Regularization | Low-rank matrix approximation | SPARSE LINEAR-EQUATIONS | MATHEMATICS, APPLIED | regularization | TOMOSYNTHESIS | LOW-RANK APPROXIMATIONS | ill-posed inverse problems | low-rank matrix approximation | ALGORITHMS | LSQR

Bayes risk | Ill-posed inverse problems | Regularization | Low-rank matrix approximation | SPARSE LINEAR-EQUATIONS | MATHEMATICS, APPLIED | regularization | TOMOSYNTHESIS | LOW-RANK APPROXIMATIONS | ill-posed inverse problems | low-rank matrix approximation | ALGORITHMS | LSQR

Journal Article

The Annals of Statistics, ISSN 0090-5364, 10/2011, Volume 39, Issue 5, pp. 2626 - 2657

The posterior distribution in a nonparametric inverse problem is shown to contract to the true parameter at a rate that depends on the smoothness of the parameter, and the smoothness and scale of the prior...

Minimax | Gaussian distributions | Inverse problems | Covariance | Linear transformations | Eigenvalues | Hilbert spaces | Mathematical independent variables | Coordinate systems | Credible set | Rate of contraction | Gaussian prior | Posterior distribution | STATISTICS & PROBABILITY | PARAMETERS | INFERENCE | rate of contraction | posterior distribution | ASYMPTOTIC NORMALITY | HILBERT SCALES | REGULARIZATION | FUNCTIONALS | POSTERIOR DISTRIBUTIONS | CONVERGENCE-RATES | 62G05 | 62G20 | 62G15

Minimax | Gaussian distributions | Inverse problems | Covariance | Linear transformations | Eigenvalues | Hilbert spaces | Mathematical independent variables | Coordinate systems | Credible set | Rate of contraction | Gaussian prior | Posterior distribution | STATISTICS & PROBABILITY | PARAMETERS | INFERENCE | rate of contraction | posterior distribution | ASYMPTOTIC NORMALITY | HILBERT SCALES | REGULARIZATION | FUNCTIONALS | POSTERIOR DISTRIBUTIONS | CONVERGENCE-RATES | 62G05 | 62G20 | 62G15

Journal Article

Acta numerica, ISSN 0962-4929, 05/2018, Volume 27, pp. 1 - 111

Regularization methods are a key tool in the solution of inverse problems. They are used to introduce prior knowledge and allow a robust approximation of ill-posed (pseudo-) inverses...

POSTERIORI PARAMETER CHOICE | VARIATIONAL IMAGE DECOMPRESSION | MATHEMATICS | LOGARITHMIC CONVERGENCE-RATES | TGV-BASED FRAMEWORK | ILL-POSED PROBLEMS | ITERATED TIKHONOV REGULARIZATION | EM-TV METHODS | GAUSS-NEWTON METHOD | TOTAL VARIATION MINIMIZATION | LINEAR OPERATOR-EQUATIONS | Medical imaging | Inverse problems | Partial differential equations | Image processing | Variational methods | Noise | Applied mathematics | Regularization methods | Learning theory | Ill-posed problems (mathematics) | Banach spaces | Regularization

POSTERIORI PARAMETER CHOICE | VARIATIONAL IMAGE DECOMPRESSION | MATHEMATICS | LOGARITHMIC CONVERGENCE-RATES | TGV-BASED FRAMEWORK | ILL-POSED PROBLEMS | ITERATED TIKHONOV REGULARIZATION | EM-TV METHODS | GAUSS-NEWTON METHOD | TOTAL VARIATION MINIMIZATION | LINEAR OPERATOR-EQUATIONS | Medical imaging | Inverse problems | Partial differential equations | Image processing | Variational methods | Noise | Applied mathematics | Regularization methods | Learning theory | Ill-posed problems (mathematics) | Banach spaces | Regularization

Journal Article