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The article deals with gradient-like iterative methods for solving nonlinear operator equations on Hilbert and Banach spaces. The authors formulate a general principle of studying such methods. This principle allows to formulate simple…
We propose a neural network-based algorithm for solving forward and inverse problems for partial differential equations in unsupervised fashion. The solution is approximated by a deep neural network which is the minimizer of a cost…
We consider solving a probably infinite dimensional operator equation, where the operator is not modeled by physical laws but is specified indirectly via training pairs of the input-output relation of the operator. Neural operators have…
In this work, we study the problem of non-blind image deconvolution and propose a novel recurrent network architecture that leads to very competitive restoration results of high image quality. Motivated by the computational efficiency and…
Further development of the method of computational experiments for solving ill-posed problems is given. The effective (unoverstated) estimate for solution error of the first-kind equation is obtained using the truncating singular numbers…
This article presents an approach to the two-dimensional Schr\"odinger equation based on automatic learning methods with neural networks. It is intended to determine the ground state of a particle confined in any two-dimensional potential,…
This paper presents some algorithms in linear algebraic groups. These algorithms solve the word problem and compute the spinor norm for orthogonal groups. This gives us an algorithmic definition of the spinor norm. We compute the double…
A new method for solving inverse spectral problems on quantum star graphs is proposed. The method is based on Neumann series of Bessel functions representations for solutions of Sturm-Liouville equations. The representations admit estimates…
We analyse an operator arising in the description of singular solutions to the two-dimensional Keller-Segel problem. It corresponds to the linearised operator in parabolic self-similar variables, close to a concentrated stationary state.…
Various applications in signal processing and machine learning give rise to highly structured spectral optimization problems characterized by low-rank solutions. Two important examples that motivate this work are optimization problems from…
One of the grand challenges of Mathematics instruction is to provide students with problems that are both accessible and have a reasonably elegant solution. Instructors commonly resort to resources like course textbooks, online-learning…
This paper concerns a spectral estimation problem in which we want to find a spectral density function that is consistent with estimated second-order statistics. It is an inverse problem admitting multiple solutions, and selection of a…
Convolution-type integral equations commonly occur in signal processing and image processing. Discretizing these equations yields large and ill-conditioned linear systems. While the classic multigrid method is effective for solving linear…
Solving inverse problems requires the knowledge of the forward operator, but accurate models can be computationally expensive and hence cheaper variants that do not compromise the reconstruction quality are desired. This chapter reviews…
This paper proposes a new way of regularizing an inverse problem in imaging (e.g., deblurring or inpainting) by means of a deep generative neural network. Compared to end-to-end models, such approaches seem particularly interesting since…
Motivated by a certain molecular reconstruction methodology in cryo-electron microscopy, we consider the problem of solving a linear system with two unknown orthogonal matrices, which is a generalization of the well-known orthogonal…
An unsteady problem is considered for a space-fractional equation in a bounded domain. A first-order evolutionary equation involves the square root of an elliptic operator of second order. Finite element approximation in space is employed.…
In this work, we consider methods for solving large-scale optimization problems with a possibly nonsmooth objective function. The key idea is to first specify a class of optimization algorithms using a generic iterative scheme involving…
Many current neural networks for medical imaging generalise poorly to data unseen during training. Such behaviour can be caused by networks overfitting easy-to-learn, or statistically dominant, features while disregarding other potentially…
Spectral Photon-Counting Computed Tomography (SPCCT) is a promising technology that has shown a number of advantages over conventional X-ray Computed Tomography (CT) in the form of material separation, artefact removal and enhanced image…