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We study Newton type methods for inverse problems described by nonlinear operator equations $F(u)=g$ in Banach spaces where the Newton equations $F'(u_n;u_{n+1}-u_n) = g-F(u_n)$ are regularized variationally using a general data misfit…
It is a well-known fact that adding noise to the input data often improves network performance. While the dropout technique may be a cause of memory loss, when it is applied to recurrent connections, Tikhonov regularization, which can be…
Convolutional neural network training can suffer from diverse issues like exploding or vanishing gradients, scaling-based weight space symmetry and covariant-shift. In order to address these issues, researchers develop weight regularization…
Tikhonov regularization is one of the most commonly used methods of regularization of ill-posed problems. In the setting of finite element solutions of elliptic partial differential control problems, Tikhonov regularization amounts to…
Classical approach to regularization is to design norms enhancing smoothness or sparsity and then to use this norm or some power of this norm as a regularization function. The choice of the regularization function (for instance a power…
A univariate continuous function can always be decomposed as the sum of a non-increasing function and a non-decreasing one. Based on this property, we propose a non-parametric regression method that combines two spline-fitted monotone…
Machine learning techniques for the solution of inverse problems have become an attractive approach in the last decade, while their theoretical foundations are still in their infancy. In this chapter we want to pursue the study of…
We consider regularization methods of Kaczmarz type in connection with the expectation-maximization (EM) algorithm for solving ill-posed equations. For noisy data, our methods are stabilized extensions of the well established…
Softening material models are known to trigger spurious localizations.This may be shown theoretically by the existence of solutions with zero dissipation when localization occurs and numerically with spurious mesh dependency and…
Regularisation theory in Banach spaces, and non--norm-squared regularisation even in finite dimensions, generally relies upon Bregman divergences to replace norm convergence. This is comparable to the extension of first-order optimisation…
One of the key assumptions in the stability and convergence analysis of variational regularization is the ability of finding global minimizers. However, such an assumption is often not feasible when the regularizer is a black box or…
Regularization is a well studied problem in the context of neural networks. It is usually used to improve the generalization performance when the number of input samples is relatively small or heavily contaminated with noise. The…
Monotone inclusions have a wide range of applications, including minimization, saddle-point, and equilibria problems. We introduce new stochastic algorithms, with or without variance reduction, to estimate a root of the expectation of…
We study in this paper a smoothness regularization method for functional linear regression and provide a unified treatment for both the prediction and estimation problems. By developing a tool on simultaneous diagonalization of two positive…
We consider the monotone inclusion problems in real Hilbert spaces. Proximal splitting algorithms are very popular technique to solve it and generally achieve weak convergence under mild assumptions. Researchers assume the strong conditions…
We present a new approach to convexification of the Tikhonov regularization using a continuation method strategy. We embed the original minimization problem into a one-parameter family of minimization problems. Both the penalty term and the…
The total variation (TV) regularization has phenomenally boosted various variational models for image processing tasks. We propose to combine the backward diffusion process in the earlier literature of image enhancement with the TV…
We consider the efficient minimization of a nonlinear, strictly convex functional with $\ell_1$-penalty term. Such minimization problems appear in a wide range of applications like Tikhonov regularization of (non)linear inverse problems…
We consider the elastic wave scattering problem involving rigid obstacles. This work addresses the inverse problem of reconstructing the position and shape of such obstacles using far-field measurements. A novel monotonicity-based approach…
Regularization-based approaches for injecting constraints in Machine Learning (ML) were introduced to improve a predictive model via expert knowledge. We tackle the issue of finding the right balance between the loss (the accuracy of the…