Yury Korolev

Yury Korolev
University of Cambridge | Cam · Department of Applied Mathematics and Theoretical Physics

PhD

About

33
Publications
4,793
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94
Citations
Introduction
I am interested in applied analysis, in particular inverse problems, variational methods, imaging and theoretical machine learning.
Additional affiliations
April 2018 - present
University of Cambridge
Position
  • PostDoc Position
October 2017 - March 2018
University of Münster
Position
  • Fellow
January 2017 - September 2017
Universität zu Lübeck
Position
  • PhD Student
Education
August 2012 - December 2012
University of Montana
Field of study
  • Mathematics
March 2010 - April 2013
Lomonosov Moscow State University
Field of study
  • Mathematical Physics
September 2004 - January 2010

Publications

Publications (33)
Article
Full-text available
We present and analyse an approach to image reconstruction problems with imperfect forward models based on partially ordered spaces - Banach lattices. In this approach, errors in the data and in the forward models are described using order intervals. The method can be characterised as the lattice analogue of the residual method, where the feasible...
Article
Full-text available
Errors in the data and the forward operator of an inverse problem can be handily modelled using partial order in Banach lattices. We present some existing results of the theory of regularisation in this novel framework, where errors are represented as bounds by means of the appropriate partial order. We apply the theory to Diffusion Tensor Imaging,...
Article
Mathematical formulations of applied inverse problems often involve operator equations in normed functional spaces. In many cases, these spaces can, in addition, be endowed with a partial order relation, which turns them into Banach lattices. The fact that two tools, such as a partial order relation and a monotone (with respect to this partial orde...
Article
Full-text available
In many applications, the concepts of inequality and comparison play an essential role, and the nature of the objects under consideration is better described by means of partial order relations. To reflect this nature, the conventional problem statements in normed spaces have to be modified. There is a need to enrich the structure of the functional...
Article
Full-text available
We study the problem of deconvolution for light-sheet microscopy, where the data is corrupted by spatially varying blur and a combination of Poisson and Gaussian noise. The spatial variation of the point spread function of a light-sheet microscope is determined by the interaction between the excitation sheet and the detection objective PSF. We intr...
Preprint
The total variation (TV) flow generates a scale-space representation of an image based on the TV functional. This gradient flow observes desirable features for images such as sharp edges and enables spectral, scale, and texture analysis. The standard numerical approach for TV flow requires solving multiple non-smooth optimisation problems. Even wit...
Preprint
Full-text available
We study Gaussian random fields on certain Banach spaces and investigate conditions for their existence. Our results apply inter alia to spaces of Radon measures and H\"older functions. In the former case, we are able to define Gaussian white noise on the space of measures directly, avoiding, e.g., an embedding into a negative-order Sobolev space....
Preprint
Full-text available
We study the problem of deconvolution for light-sheet microscopy, where the data is corrupted by spatially varying blur and a combination of Poisson and Gaussian noise. The spatial variation of the point spread function (PSF) of a light-sheet microscope is determined by the interaction between the excitation sheet and the detection objective PSF. F...
Preprint
Full-text available
In this article we characterize the L∞ eigenvalue problem associated to the Rayleigh quotient ∥∇u∥∞/∥u∥∞ and relate it to a divergence-form PDE, similarly to what is known for Lp eigenvalue problems and the p-Laplacian for p<∞. Contrary to existing methods, which study L∞-problems as limits of Lp-problems for p→∞, we develop a novel framework for a...
Preprint
Full-text available
We study two-layer neural networks whose domain and range are Banach spaces with separable preduals. In addition, we assume that the image space is equipped with a partial order, i.e. it is a Riesz space. As the nonlinearity we choose the lattice operation of taking the positive part; in case of $\mathbb R^d$-valued neural networks this corresponds...
Preprint
Full-text available
We study the problem of regularization of inverse problems adopting a purely data driven approach, by using the similarity to the method of regularization by projection. We provide an application of a projection algorithm, utilized and applied in frames theory, as a data driven reconstruction procedure in inverse problems, generalizing the algorith...
Article
Full-text available
We study variational regularisation methods for inverse problems with imperfect forward operators whose errors can be modelled by order intervals in a partial order of a Banach lattice. We carry out analysis with respect to existence and convex duality for general data fidelity terms and regularisation functionals. Both for a priori and a posterior...
Preprint
Full-text available
Non-linear spectral decompositions of images based on one-homogeneous functionals such as total variation have gained considerable attention in the last few years. Due to their ability to extract spectral components corresponding to objects of different size and contrast, such decompositions enable filtering, feature transfer, image fusion and othe...
Preprint
Full-text available
We study variational regularisation methods for inverse problems with imperfect forward operators whose errors can be modelled by order intervals in a partial order of a Banach lattice. We carry out analysis with respect to existence and convex duality for general data fidelity terms and regularisation functionals. Both for a-priori and a-posterior...
Preprint
Full-text available
In this work we analyse the functional ${\cal J}(u)=\|\nabla u\|_\infty$ defined on Lipschitz functions with homogeneous Dirichlet boundary conditions. Our analysis is performed directly on the functional without the need to approximate with smooth $p$-norms. We prove that its ground states coincide with multiples of the distance function to the bo...
Preprint
Full-text available
We introduce a first order Total Variation type regulariser that decomposes a function into a part with a given Lipschitz constant (which is also allowed to vary spatially) and a jump part. The kernel of this regulariser contains all functions whose Lipschitz constant does not exceed a given value, hence by locally adjusting this value one can dete...
Preprint
Full-text available
We demonstrate that regularisation by projection and variational regularisation can be formulated in a purely data driven setting when the forward operator is given only through training data. We study convergence and stability of the regularised solutions. Our results also demonstrate that the role of the amount of training data is twofold. In reg...
Chapter
Full-text available
We introduce a new regularizer in the total variation family that promotes reconstructions with a given Lipschitz constant (which can also vary spatially). We prove regularizing properties of this functional and investigate its connections to total variation and infimal convolution type regularizers \({{\,\mathrm{{{\,\mathrm{TVL}\,}}^p}\,}}\) and,...
Preprint
Full-text available
We introduce a new regularizer in the total variation family that promotes reconstructions with a given Lipschitz constant (which can also vary spatially). We prove regularizing properties of this functional and investigate its connections to total variation and infimal convolution type regularizers TVLp and, in particular, establish topological eq...
Chapter
The paper presents an HPC implementation of the Multipoint Approximation Method (MAM) applied to problems with uncertainty in design variables as well as in additional environmental variables. The approach relies on approximations built in the combined space of design variables and environmental variables, and subsequent application of a risk measu...
Article
Full-text available
The goal of this paper is to further develop an approach to inverse problems with imperfect forward operators that is based on partially ordered spaces. Studying the dual problem yields useful insights into the convergence of the regularised solutions and allow us to obtain convergence rates in terms of Bregman distances—as usual in inverse problem...
Preprint
Full-text available
The goal of this paper is to further develop an approach to inverse problems with imperfect forward operators that is based on partially ordered spaces. Studying the dual problem yields useful insights into the convergence of the regularised solutions and allow us to obtain convergence rates in terms of Bregman distances - as usual in inverse probl...
Conference Paper
Full-text available
An automaitic optimization technique has been applied for solving the inverse problem in order to determine seven calibratioin parameters of a simple fast-turn-around-time RANS-based acoustic model based on the Goldstein generalized acoustic analogy to predict jet mixing noise from axi-symmetric nozzles. Examples of sound spectra predictions obtain...
Conference Paper
Full-text available
Abstract. An adaptation of the Multipoint Approximation Method (MAM) to a highperformance computing (HPC) environment is presented and demonstrated by high-fidelity CFDbased design optimization of a highly-loaded transonic rotor, significantly improving its efficiency in a much reduced design time. MAM is incorporated into the Rolls-Royce SOPHY des...
Chapter
We consider ill-posed inverse problems for linear operator equations Az=u with an operator A acting between two normed spaces. It is well known that, in general, no error estimate can be provided for approximate solution of an ill-posed problem. But in some special cases, when we are aware of some a priori information about the unknown exact soluti...
Article
Full-text available
We consider ill-posed inverse problems for linear operator equations Az=u in Banach lattices with a priori information that the exact solution belongs to a compact set. We provide an error estimate for an approximate solution to the ill-posed problem. We also show the existence of a supremum and infimum of the set of approximate solutions and their...
Article
We consider an inverse problem of parameter identification for a parabolic equation. The underlying practical example is the reconstruction of the unknown drift in the extended Black-Scholes option pricing model. Using a priori information about the unknown solution (i.e. its Lipschitz constant), we provide a solution to this non-linear ill-posed p...
Chapter
We consider an inverse problem for an operator equation Az = u. The exact operator A and the exact right-hand side u are unknown. Only their upper and lower estimations are available. We provide techniques of calculating upper and lower estimations for the exact solution belonging to a compact set in this case, as well as a posteriori error estimat...

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Projects

Projects (7)
Project
Studying neural network approximations of nonlinear operators acting between infinite-dimensional Banach spaces, with an eye on applications in imaging and inverse problems.
Archived project
Developing optimisation techniques for computationally expensive and noisy functions using surrogate modelling, and applying them in aerospace engineering.
Project
Developing a rigorous regularisation theory for inverse problems with learned operators.