Analysis of generalized iteratively regularized Landweber iterations driven by data
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Andrea Aspri
Abstract
We investigate generalized versions of the iteratively regularized Landweber method to address linear and nonlinear ill-posed problems. Our approach draws inspiration from a data-driven perspective, particularly when dealing with unlabeled data. Specifically, in the iteratively regularized Landweber iteration, we replace the damping term with either the average or the geometric mean of the unlabeled data. We provide a rigorous analysis establishing convergence and stability results and present numerical outcomes for linear operators, with the Radon transform serving as a prototype.
Abstract
We investigate generalized versions of the iteratively regularized Landweber method to address linear and nonlinear ill-posed problems. Our approach draws inspiration from a data-driven perspective, particularly when dealing with unlabeled data. Specifically, in the iteratively regularized Landweber iteration, we replace the damping term with either the average or the geometric mean of the unlabeled data. We provide a rigorous analysis establishing convergence and stability results and present numerical outcomes for linear operators, with the Radon transform serving as a prototype.
Chapters in this book
- Frontmatter I
- Preface V
- Contents VII
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Part I: Mathematical aspects of data-driven methods in inverse problems
- On optimal regularization parameters via bilevel learning 1
- Learned regularization for inverse problems 39
- Inverse problems with learned forward operators 73
- Unsupervised approaches based on optimal transport and convex analysis for inverse problems in imaging 107
- Learned reconstruction methods for inverse problems: sample error estimates 163
- Statistical inverse learning problems with random observations 201
- General regularization in covariate shift adaptation 245
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Part II: Applications of data-driven methods in inverse problems
- Analysis of generalized iteratively regularized Landweber iterations driven by data 273
- Integration of model- and learning-based methods in image restoration 303
- Dynamic computerized tomography using inexact models and motion estimation 331
- Deep Bayesian inversion 359
- Utilizing uncertainty quantification variational autoencoders in inverse problems with applications in photoacoustic tomography 413
- Electrical impedance tomography: a fair comparative study on deep learning and analytic-based approaches 437
- Classification with neural networks with quadratic decision functions 471
- Index 495
Chapters in this book
- Frontmatter I
- Preface V
- Contents VII
-
Part I: Mathematical aspects of data-driven methods in inverse problems
- On optimal regularization parameters via bilevel learning 1
- Learned regularization for inverse problems 39
- Inverse problems with learned forward operators 73
- Unsupervised approaches based on optimal transport and convex analysis for inverse problems in imaging 107
- Learned reconstruction methods for inverse problems: sample error estimates 163
- Statistical inverse learning problems with random observations 201
- General regularization in covariate shift adaptation 245
-
Part II: Applications of data-driven methods in inverse problems
- Analysis of generalized iteratively regularized Landweber iterations driven by data 273
- Integration of model- and learning-based methods in image restoration 303
- Dynamic computerized tomography using inexact models and motion estimation 331
- Deep Bayesian inversion 359
- Utilizing uncertainty quantification variational autoencoders in inverse problems with applications in photoacoustic tomography 413
- Electrical impedance tomography: a fair comparative study on deep learning and analytic-based approaches 437
- Classification with neural networks with quadratic decision functions 471
- Index 495
