## About

33

Publications

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94

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Introduction

I am interested in applied analysis, in particular inverse problems, variational methods, imaging and theoretical machine learning.

Additional affiliations

April 2018 - present

October 2017 - March 2018

January 2017 - September 2017

Education

August 2012 - December 2012

March 2010 - April 2013

September 2004 - January 2010

## Publications

Publications (33)

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...

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,...

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...

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...

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...

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...

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....

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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,...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

## Projects

Projects (7)

Studying neural network approximations of nonlinear operators acting between infinite-dimensional Banach spaces, with an eye on applications in imaging and inverse problems.

Developing optimisation techniques for computationally expensive and noisy functions using surrogate modelling, and applying them in aerospace engineering.

Developing a rigorous regularisation theory for inverse problems with learned operators.