# Anatoly YagolaLomonosov Moscow State University, Faculty of Physics · Mathematics

Anatoly Yagola

Doctor of Sciences (Physics-Mathematics)

## About

255

Publications

8,435

Reads

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4,270

Citations

Introduction

Topics: ill-posed problems: theory, numerical methods and applications.
More than 400 scientific publications, including about 30 books.

Additional affiliations

April 1971 - present

Education

September 1963 - February 1968

**Lomonosov Moscow State University**

Field of study

- Physics

## Publications

Publications (255)

One of the possible approaches to reconstructing the map of the distribution of magnetization parameters in the Mercury’s crust from the data of the Messenger orbital mission is considered. Possible ways of increasing the accuracy of reconstructing the magnetic image of Mercury are discussed.

The reverse engineering problem of determining the layer thicknesses of deposited optical coatings from on-line monochromatic measurements is considered. To solve this inverse problem, non-local algorithms are proposed that use all the data accumulated during the deposition process. For the proposed algorithms, the accuracy of solving the inverse p...

One of the possible approaches to reconstructing the map of the distribution of magnetization parameters in the crust of Mars from the data of the Mars MAVEN orbiter mission is considered. Possible ways of increasing the accuracy of reconstruction of the magnetic image of Mars are discussed.

A fast algorithm for calculating the gradient of the Tikhonov functional is proposed for solving inverse coefficient problems for linear partial differential equations of a general form by the regularization method. The algorithm is designed for problems with discretized differential operators that linearly depend on the desired coefficients. When...

A new technique for three-dimensional surface reconstruction of relatively smooth surface topography using the scanning electron microscopy with backscattered electron detector is considered. Experiments show high effectiveness of the method.

On-line optical monitoring of multilayer coating production requires solving inverse identification problems of determining the thicknesses of coating layers. Regardless of the algorithm used to solve inverse problems, the errors in the thicknesses of the deposited layers are correlated by the monitoring procedure. Studying the correlation of thick...

We study the three-dimensional inverse problem of elastography, that is finding the Young's modulus of a biological tissue from known values of its vertical displacements. In this way, one can find inclusions with Young's modulus several times higher than its known background value. Such inclusions are interpreted as tumours. A quasistatic statemen...

This article presents a computational approach for comparing various broadband monitoring strategies, taking into account the positive and negative effects associated with the correlation of thickness errors caused by the monitoring procedure. The approach is based on statistical estimates of the strength of the error self-compensation effect and t...

The paper presents the solution of a special three-dimensional inverse elastography problem: given a quasistatic model of a linear-elastic isotropic body subject to surface forces, to find the Young’s modulus distribution in the biological tissues under study using the known values of vertical displacements of these tissues. This study is aimed at...

Full tensor magnetic gradient measurements are available nowadays. These are essential for determining magnetization parameters in deep layers. Using full or partial tensor magnetic gradient measurements to determine the subsurface properties , e.g., magnetic susceptibility, is an inverse problem. Inversion using total magnetic intensity data is a...

This review describes the specifics of the application of the approximation approach in solving the linear and nonlinear inverse problems of geophysics, geodesy, and geomorphology. Within the paradigm proposed by V.N. Strakhov, almost all the geophysical problems can be reduced to solving systems of linear (and, in some cases, nonlinear) algebraic...

The basic algorithms for determining the thicknesses of layers of deposited multilayer optical coatings are discussed and compared. Using a series of model numerical experiments, the advantage of one of these algorithms—the modified T-algorithm—is demonstrated; this algorithm reduces the influence of the effect of error accumulation in the determin...

Asymptotic analysis of a singularly perturbed reaction—diffusion—advection equation, which is called a Burgers-type equation in applications and has a solution with a sharp transition layer, is applied to solve the coefficient inverse problem of determining the coefficient of linear amplification from known information on the observed solution of t...

The key tool of this paper is a new Carleman estimate for an arbitrary parabolic operator of the second order for the case of reversed time data. This estimate works on an arbitrary time interval. On the other hand, the previously known Carleman estimate for the reversed time case works only on a sufficiently small time interval. First, a stability...

An inverse microtomography problem is under consideration in a class of functions with bounded V H variation. An algorithm for solving this problem is proposed based on Tikhonov's regularization with a special regularizer. The algorithm ensures piecewise uniform convergence of approximate solutions to exact solution of the inverse problem. In addit...

This paper introduces an estimate for the strength of the correlation of layer thickness errors in the process of coating deposition with broadband optical monitoring. It is shown that a strong effect of self-compensation of deposition errors is observed in the case of a strong correlation of layer thickness errors. An estimate for the strength of...

Computational manufacturing experiments are used to detect the types of optical coatings that are showing the presence of a strong error self-compensation effect in the coating production with direct broad band monitoring. It is shown that predictions made on the basis of these experiments coincide with the predictions of the previously developed r...

Algorithms for the online determination of thicknesses of already-deposited layers are important for the reliable control of optical coating production. Possible ways of constructing such algorithms in the case of coating production with direct broadband monitoring are discussed. A modified triangular algorithm is proposed. In contrast to the well-...

A method for improving the accuracy of the broad-band monitoring of the process of depositing optical coatings is proposed. The method is based on determining the actual set of thicknesses of deposited layers in the deposition process. The effectiveness of the proposed approach is demonstrated in a series of model numerical experiments using a simu...

Production of the modern advanced multi-layer optical coatings requires on-line monitoring of the growing layer thickness. We present regularizing algorithms for the continuous on-line determination of the deposited layer thickness that can be used in the coating production with broadband optical monitoring. These algorithms are based on minimizati...

The inverse problem of optical coatings control during their deposition is an important practical and industrial problem. The paper presents the mathematical investigation of the layer thickness errors self-compensation effect that is quite important for the quality production of modern types of thin film optical coatings. It studies the mechanism...

The main theoretical results related to the investigation of the error self-compensation mechanism associated with direct broad band monitoring of optical coating production are presented. The presented results are illustrated using the production of Brewster angle polarizer where this effect is especially strong. Specific properties of the design...

A new algorithm for determining the optical parameters of deposited multilayer optical coatings based on comparing the positions of the extrema of the optical characteristics of multilayer optical coatings is proposed. Two versions of this algorithm are compared. Using a series of numerical simulation experiments, the advantage of one of these vers...

An error self-compensation mechanism is investigated for use during the deposition of optical coatings with broadband optical monitoring. The correlation of thickness errors caused by monitoring procedure is mathematically described. It is shown that this correlation of errors may result in the effect of selfcompensation of errors.

This article presents the solution of a special inverse elastography problem: knowing vertical displacements of compressed biological tissue to find a piecewise constant distribution of Young’s modulus in an investigated specimen. Our goal is to detect homogeneous inclusions in the tissue, which can be interpreted as oncological. To this end, we co...

Regularizing algorithms developed for joint treatment of gas-phase electron diffraction and vibrational spectroscopic data and extended to include systems with large-amplitude oscillatory motion are discussed. The treatment is augmented by the inclusion of microwave rotational constants. The analysis of data from experimental sources is guided by q...

We solve numerically the side Cauchy problem for a 1-D parabolic equation. The initial condition is unknown. This is an ill-posed problem. The main difference with previous results is that our equation is quasilinear, whereas known publications on this topic work only with linear PDEs. The key idea is to minimize a weighted Tikhonov functional with...

The article deals with one of inverse problems of elastography: knowing displacement of compressed tissue finds the distribution of Young’s modulus in the investigated specimen. The direct problem is approximated and solved by the finite element method. The inverse problem can be stated in different ways depending on whether the solution to be foun...

Recovery of magnetic target parameters from magnetic sensor measurements has attracted wide interests and found many practical applications. However, difficulties present in identifying the magnetization due to the complications of magnetization distributions over investigated object, errors and noises of measurement data, degrade the accuracy and...

Inverse problems of molecular force field calculation arising as a result of data processing in vibrational spectroscopy belong to the class of nonlinear ill-posed problems. In this paper we discuss the main mathematical results obtained within the theory of regularization for solving these problems. Different algorithms on the basis of regularizin...

This article is devoted to a Lagrange principle application to an inverse problem of a two-dimensional integral equation of the first kind with a positive kernel. To tackle the ill-posedness of this problem, a new numerical method is developed. The optimal and regularization properties of this method are proved. Moreover, a pseudo-optimal error of...

Iterative re-weighted least square (IRLS) algorithms for -minimization problems require to select proper value of regularization parameter, for which Stein’s unbiased risk estimate (SURE) – an unbiased estimate of prediction error – is often used as a criterion for this selection. In this paper, we propose a recursive SURE to estimate the predictio...

In this article, we consider an inverse problem for the integral equation of the convolution type in a multidimensional case. This problem is severely ill-posed. To deal with this problem, using a priori information (sourcewise representation) based on optimal recovery theory we propose a new method. The regularization and optimization properties o...

The book is written based on the course of lectures given to the Lomonosov Moscow State University Physical Department students. Inverse problems of geophysics are considered as the main applications.
The mathematical apparatus described in the first chapter has been successfully applied to the solution of inverse problems of astrophysics, image p...

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...

In this article we discuss the determination of the aerosol particle size distribution function using the particle spectrum extinction equation. This is an ill-posed integral equation of the first kind, since we are often faced with limited or insufficient observations in remote sensing and the observations are contaminated. To overcome the ill-pos...

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...

This paper presents mathematical background of data processing in vibrational spectroscopy. Regularizing algorithms of molecular force fields calculation based on the joint treatment of experimental and quantum mechanical data have been proposed within the frame of theory of regularization of nonlinear ill-posed problems. Different models of molecu...

Рассматриваются линейные некорректно поставленные задачи при наличии априорной инфор-
мации о решении. С помощью метода расширяющихся компактов, принципа Лагранжа и теории
оптимального восстановления функционала строится оптимальный регуляризирующий алго-
ритм для решения линейных некорректно поставленных задач с истокопредставимым решением
и вычис...

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...

The inverse problem of reconstructing the true spectrum of electrons backscattered from massive and layered targets with allowance for the spread function of the toroidal sector energy analyzer and for the response function of the spectrometer’s electron detector is solved. We present the results from studying the energy spectra of a number of homo...

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...

This monograph reports recent advances of inversion theory and recent developments with practical applications in frontiers of sciences, especially inverse design and novel computational methods for inverse problems. Readers who do research in applied mathematics, engineering, geophysics, biomedicine, image processing, remote sensing, and environme...

Рассматривается решение важной практической задачи восстановления функции распределения размеров частиц аэрозоля в атмосфере по измеренным значениям коэффициента поглощения различных длин волн излучения. Эта задача сводится к интегральному уравнению Фредгольма первого рода, для решения которого применяется алгоритм, основанный на минимизации функци...

A new method for increasing spatial resolution in the detection of backscattered electrons and induced current in scanning electron microscopy (SEM) is proposed in terms of regularized Fourier transform. The real size of an electron probe and its blurring in a solid target sample are considered in forming the algorithm. The experiments reveal an al...

Due to the fast evolution of micro- and nanotechnology, diagnostic techniques and research methods devoted to study micro objects are rapidly developed. One of such methods is the tomography in the backscattered electron mode. The spatial resolution of electronic microscope signals can be improved by solving the inverse problem of reconstructing a...

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...

"Optimization and Regularization for Computational Inverse Problems and Applications" focuses on advances in inversion theory and recent developments with practical applications, particularly emphasizing the combination of optimization and regularization for solving inverse problems. This book covers both the methods, including standard regularizat...

A method of ring artefact suppression in X-ray computerised tomography (CT) reconstructions is proposed. The method is based on the assumption that a sinogram is a smooth function along the horizontal spatial coordinate. Methods based on the theory of ill-posed problems are applied to find a regularised solution. An analytical formula for the solut...

Recovery of magnetic target parameters from magnetic sensor measurements has attracted wide interests and found many practical applications. However, difficulties present in identifying the permanent magnetization due to the complications of magnetization distributions over the ship body, and errors and noises of measurement data degrade the accura...

A ring artefact suppression algorithm in x-ray tomography is proposed and allows one to process input data in real time. The
algorithm is based on methods of the theory of inverse and ill-posed problems. Its numerical implementation uses minimisation
of the smoothing Tikhonov’s functional with the conjugate gradient method.
Key wordsill-posed prob...

The basic conceptions of the theory of ill-posed problems and numerical methods for their solving under different a priori information are described. Hadamard’s definition of well-posedness and examples of ill-posed problems are given. Tikhonov’s definition of a regularizing algorithm and classification of mathematical problems are described. The m...

The gravitational lensing phenomenon can provide us with the information about luminous and dark matter in our Universe. But robust and effective tools are needed to extract that valuable information from observations. In this chapter, two inverse problems arising in gravitational lensing research are considered. Both problems are ill-posed, so reg...

Energy distribution spectra of backscattered electrons in the range 5–25 keV are obtained experimentally. An inverse problem
of the reconstruction of the true electron spectrum is solved taking into account the instrument response function of the
spectrometer; on the basis of the obtained solution, we specify functions of the real energy distribut...

In many real applications such as remote sensing and space surveillance, traditional method based on ideal imaging has already been well-established. It is widely applied to analyze point-target detection performance of electro-optical imaging system, including signal-to-noise ratio (SNR) and noise equivalent temperature difference (NETD). However,...