Related papers: A regularization method for ill-posed bilevel opti…

Solving a bilevel optimization problem is at the core of several machine learning problems such as hyperparameter tuning, data denoising, meta- and few-shot learning, and training-data poisoning. Different from simultaneous or…

For the solution of full-rank ill-posed linear systems a new approach based on the Arnoldi algorithm is presented. Working with regularized systems, the method theoretically reconstructs the true solution by means of the computation of a…

The numerical solution of linear discrete ill-posed problems typically requires regularization, i.e., replacement of the available ill-conditioned problem by a nearby better conditioned one. The most popular regularization methods for…

We investigate the ill-posed inverse problem of recovering unknown spatially dependent parameters in nonlinear evolution PDEs. We propose a bi-level Landweber scheme, where the upper-level parameter reconstruction embeds a lower-level state…

Several generalizations of the traditional Tikhonov-Phillips regularization method have been proposed during the last two decades. Many of these generalizations are based upon inducing stability throughout the use of different penalizers…

Optimization problems with composite functions consist of an objective function which is the sum of a smooth and a (convex) nonsmooth term. This particular structure is exploited by the class of proximal gradient methods and some of their…

The widely used nuclear norm heuristic for rank minimization problems introduces a regularization parameter which is difficult to tune. We have recently proposed a method to approximate the regularization path, i.e., the optimal solution as…

The Golub-Kahan-Tikhonov method is a popular solution technique for large linear discrete ill-posed problems. This method first applies partial Golub-Kahan bidiagonalization to reduce the size of the given problem and then uses Tikhonov…

We consider a bilevel program involving a linear lower level problem with left-hand-side perturbation. We then consider the Karush-Kuhn-Tucker reformulation of the problem and subsequently build a tractable optimization problem with linear…

Parameter identification problems typically consist of a model equation, e.g. a (system of) ordinary or partial differential equation(s), and the observation equation. In the conventional reduced setting, the model equation is eliminated…

In this paper, we consider bilevel optimization problem where the lower-level has coupled constraints, i.e. the constraints depend both on the upper- and lower-level variables. In particular, we consider two settings for the lower-level…

In this paper we consider new regularization methods for linear inverse problems of dynamic type. These methods are based on dynamic programming techniques for linear quadratic optimal control problems. Two different approaches are…

In this paper we consider the iteratively regularized Gauss-Newton method for solving nonlinear ill-posed inverse problems. Under merely Lipschitz condition, we prove that this method together with an a posteriori stopping rule defines an…

In this work, we analyze the regularizing property of the stochastic gradient descent for the efficient numerical solution of a class of nonlinear ill-posed inverse problems in Hilbert spaces. At each step of the iteration, the method…

In this paper, we deal with nonlinear ill-posed problems involving monotone operators and consider Lavrentiev's regularization method. This approach, in contrast to Tikhonov's regularization method, does not make use of the adjoint of the…

This manuscript is designed to introduce students in applied mathematics and data science to the concept of regularization for ill-posed inverse problems. Construct a mathematical model that describes how an image gets blurred. Convert a…

Regularization is a powerful technique for extracting useful information from noisy data. Typically, it is implemented by adding some sort of norm constraint to an objective function and then exactly optimizing the modified objective…

A simple bilevel variational problem where the lower level is a variational inequality while the upper level is an optimization problem is studied. We consider an inexact version of the lower problem, which guarantees enough regularity to…

Subspace recycling techniques have been used quite successfully for the acceleration of iterative methods for solving large-scale linear systems. These methods often work by augmenting a solution subspace generated iteratively by a known…

This paper focuses on regularisation methods using models up to the third order to search for up to second-order critical points of a finite-sum minimisation problem. The variant presented belongs to the framework of [3]: it employs random…