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This paper is concerned with recovering the solution of a final value problem associated with a parabolic equation involving a non linear source and a non-local term, which to the best of our knowledge has not been studied earlier. It is…
Singular source terms expressed as weighted summations of Dirac-delta functions are regularized through approximation theory with convolution operators. We consider the numerical solution of scalar and one-dimensional hyperbolic…
The Tikhonov-Phillips method is widely used for regularizing ill-posed inverse problems mainly due to the simplicity of its formulation as an optimization problem. The use of different penalizers in the functionals associated to the…
By singular value decomposition (SVD) of a numerically singular Hessian matrix and a numerically singular system of linear equations for the experimental data (accumulated in the respective ${\chi ^2}$ function) and constraints, least…
Recovering a function from integrals over conical surfaces recently got significant interest. It is relevant for emission tomography with Compton cameras and other imaging applications. In this paper, we consider the weighted conical Radon…
This paper focuses on the development of a space-variant regularization model for solving an under-determined linear inverse problem. The case study is a medical image reconstruction from few-view tomographic noisy data. The primary…
We develop new techniques for proving lower bounds on the least singular value of random matrices with limited randomness. The matrices we consider have entries that are given by polynomials of a few underlying base random variables. This…
Regularization plays a pivotal role when facing the challenge of solving ill-posed inverse problems, where the number of observations is smaller than the ambient dimension of the object to be estimated. A line of recent work has studied…
We introduce a fully-corrective generalized conditional gradient method for convex minimization problems involving total variation regularization on multidimensional domains. It relies on alternatively updating an active set of subsets of…
We deal with the shape reconstruction of inclusions in elastic bodies. For solving this inverse problem in practice, data fitting functionals are used. Those work better than the rigorous monotonicity methods from [5], but have no…
Inverse problems are characterized by their inherent non-uniqueness and sensitivity with respect to data perturbations. Their stable solution requires the application of regularization methods including variational and iterative…
Recently, three numerical methods for the computation of eigenvalues of singular matrix pencils, based on a rank-completing perturbation, a rank-projection, or an augmentation were developed. We show that all three approaches can be…
Inverse problems are fundamental in fields like medical imaging, geophysics, and computerized tomography, aiming to recover unknown quantities from observed data. However, these problems often lack stability due to noise and…
In recent years, a series of convergence rates conditions for regularization methods has been developed. Mainly, the motivations for developing novel conditions came from the desire to carry over convergence rates results from the Hilbert…
The total generalized variation (TGV) is a popular regularizer in inverse problems and imaging combining discontinuous solutions and higher order smoothing. In particular, empirical observations suggest that its order two version strongly…
These lecture notes evolve around mathematical concepts arising in inverse problems. We start by introducing inverse problems through examples such as differentiation, deconvolution, computed tomography and phase retrieval. This then leads…
Singular Value Decomposition (SVD) is a powerful tool in linear algebra.We propose an extension of SVD for both the qualitative detection and quantitative determination of nonlinearity in a time series. The paper illustrates nonlinear SVD…
We present a general variational framework for the training of freeform nonlinearities in layered computational architectures subject to some slope constraints. The regularization that we add to the traditional training loss penalizes the…
Besides perturbation theory (which clearly requires the knowledge of the exact unperturbed solution), variational techniques represent the main tool for any investigation of the eigenvalue problem of some semibounded operator H in quantum…
In representation learning (RL), how to make the learned representations easy to interpret and less overfitted to training data are two important but challenging issues. To address these problems, we study a new type of regulariza- tion…