A posteriori error estimates for approximate solutions of linear ill-posed inverse problems are under investigation. A brief overview of existing approaches to this problem is given. Special variational a posteriori error estimates are studied for various classes of function spaces. Obtaining all such estimates reduces to solving a certain canonical problem. For the canonical problem, a solution algorithm is given in various forms. This algorithm is several times faster than previously proposed ones. The estimates and algorithm are suitable for various Hilbert and Banach spaces. The obtaining of such estimates is illustrated using the example of a two-dimensional model inverse problem of heat conduction.

We consider the optimization of a neural network previously developed by the authors for the joint inversion of 3D gravitational and magnetic fields in the context of mineral exploration. The distinctive feature of this neural network is that it solves ill-posed (ill-conditioned) inverse problems. The neural network implements a special two-level algorithm. The lower level of the algorithm uses two neural networks with equivalent architectures. The first of them computes the gravitational field sources in a given domain from measurements of this field on a remote surface. The second neural network processes magnetic field measured on the same surface to find magnetic sources in the same domain. The found source distributions are used at the upper level of the algorithm to calculate their structural residual, which determines the degree of difference (closeness) of their geometries. As a result, minimizing this residual, when training a neural network at the upper level, implements a computational algorithm that yields geometrically close source distributions of different fields. The article examines in detail the possibilities of optimizing some elements of the neural networks and the algorithms used (datasets, training process, specific form of loss functions, etc.) Test calculations for model problem demonstrate high quality of joint inversion by our optimized neural networks approach. Calculations were also carried out for the joint processing of real-feald data from gravity and magnetic exploration in Jussara region, Goias State, Brazil. The article also considers the issue of determining in joint field inversion not only the geometric distribution of sources, but also their physical intensities.

We consider the solution of systems of linear algebraic equations (SLAEs) with an ill- conditioned or degenerate exact matrix and an approximate right-hand side. An approach to solving such a problem is proposed and justified, which makes it possible to improve the conditionality of the SLAE matrix and, as a result, obtain an approximate solution that is stable to perturbations of the right hand side with higher accuracy than using other methods. The approach is implemented by an algorithm that uses so-called minimal pseudoinverse matrices. The results of numerical experiments are presented that confirm the theoretical provisions of the article.

For inverse problems of gravimetry and magnetometry, a possible problem formulation is considered, which consists in finding hypothetical point sources at a given depth that correspond to potential fields measured on the Earth’s surface. The uniqueness of solutions to these inverse problems is proved. Their discretized variants are solved numerically using a new algorithm based on improving the condition number of the problem’s matrix with the help of the minimal pseudoinverse matrix method (MPMI algorithm). The algorithm is tested on model problems of gravimetry and magnetometry, which are solved separately. A variant of the MPMI algorithm for the joint solution of these inverse problems is also proposed and tested. Finally, the algorithm is used for separate and joint processing of some well-known gravity and magnetic exploration data, namely, for the Kursk Magnetic Anomaly.

t Theinverse single-frequency problem of scalar acoustics in three-dimensional space is considered. It consists in determining the characteristics of acoustic inhomogeneities lying in a flat layer from the distribution of the complex amplitude of the acoustic field in a flat recording layer. To effectively solve it, a special numerical algorithm is used, previously proposed and justified by the authors. As its component part, the algorithm uses solutions to special one-dimensional Fredholm equations of the 1st kind, and in the case of ambiguity in the solution of the latter, their normal solutions (solutions with a minimum norm) are sought. To regularize this ill-posed inverse problem, Tikhonov regularization and the TSVD method are used. A systematic numerical study is carried out of the influence of various parameters in the data recording scheme on the accuracy of the approximate solutions obtained using the algorithm. In particular, we study the dependence of this accuracy on the position of point sources and composite sources that cause sound vibrations, its dependence on the position of the data recording layer, and on the number of planes in which the recording sensors lie. It is shown that acceptable accuracy of approximate solutions can be obtained even with two such layers. In addition, an approach to the numerical study of the ambiguity of the solution to the inverse problem under consideration is proposed. It is based on special theoretical statements given in the article, and on the same numerical algorithm for solving the inverse problem. The approach is focused on comparing several normal solutions of one-dimensional integral equations from the problem under consideration. These normal solutions are calculated with respect to various elements. An example of using this theory to numerically estimate the non-uniqueness of a solution is given.

The inverse problem of acoustic sounding of a three-dimensional nonstationary medium is considered, based on the Cauchy problem for the wave equation with a sound speed coefficient depending on the spatial coordinates and time. The data in the inverse problem are measurements of time-dependent acoustic pressure in some spatial domain. Using these data, it is necessary to determine the positions of local acoustic inhomogeneities (spatial sound speed distributions), which change over time. A special idealized sounding model is used, in which, in particular, it is assumed that the spatial sound speed distribution changes little in the interval between source time pulses. With such a model, the inverse problem is reduced to solving three-dimensional Fredholm linear integral equations for each sounding time interval. Using these solutions, the spatial sound speed distributions are calculated in each sounding time interval. When a special (plane-layer) geometric scheme for the location of the observation and sounding domains is included in the sounding scheme, the inverse problem can be reduced to solving systems of one-dimensional linear Fredholm integral equations, which are solved by well-known methods for regularizing ill-posed problems. This makes it possible to solve the three-dimensional inverse problem of determining the nonstationary sound speed distribution in the sounded medium on a personal computer of average performance for fairly detailed spatial grids in a few minutes. The efficiency of the corresponding algorithm for solving a three-dimensional nonstationary inverse sounding problem in the case of moving local acoustic inhomogeneities is illustrated by solving a number of model problems.

Рассматриваются типичные трехмерные обратные задачи магниторазведки, а именно: определения векторной плотности магнитных диполей в исследуемой области земной коры по измеряемым на поверхности компонентам вектора (и/или тензора градиента) магнитной индукции. Эти задачи, являясь, как правило, некорректно поставленными, могут быть решены стандартными методами регуляризации. Однако при этом для такого решения на достаточно подробных сетках за время порядка нескольких минут требуются значительные вычислительные ресурсы (вычислительные кластеры, суперкомпьютеры и т.д.). В статье предлагается новый «быстрый» регуляризующий алгоритм решения таких трехмерных задач, позволяющий получать их приближенное решение на персональном компьютере средней производительности за десятки секунд или несколько минут. Кроме того, используемый подход позволяет за сравнимое время вычислить и апостериорную оценку точности найденного решения, и это дает возможность оценить качество решения при интерпретации результатов. Алгоритмы решения обратной задачи и апостериорной оценки точности решения апробируются при решении модельных обратных задач и используются при обработке экспериментальных данных.

The problem of solving systems of linear equations with an ill-conditioned or degenerate exact matrix and an approximate right-hand side is considered. A scheme for solving such a problem is proposed and justified, which allows improving the conditionality of the matrix of the linear system. As a result, an approximate solution that is stable to perturbations of the right-hand side is obtained with a higher accuracy than when using some other methods. The scheme is implemented by an algorithm that uses minimal pseudoinverse matrices. The results of numerical experiments are presented, confirming the theoretical provisions of the article.

We consider direct and inverse problems of three-dimensional quasi-static elastography underlying a cancer diagnosis method. They are based on a model of a tissue exposed to surface compression with deformations obeying linear elasticity laws. The arising three-dimensional displacements of the tissue are described by a boundary value problem for partial differential equations with coefficients determined by a variable Young's modulus and a constant Poisson ratio. The problem contains a small parameter, so it can be solved using the theory of regular perturbations of partial differential equations. This is the direct problem. The inverse problem is to find the Young modulus distribution from given tissue displacements. A significant increase in Young's modulus within a certain tissue domain suggests possible malignancy. Under certain assumptions, simple formulas for solving both direct and inverse problems of three-dimensional quasi-static elastography are derived. Three-dimensional inverse test problems are solved numerically with the help of the proposed formulas. The resulting approximate solutions agree fairly well with the exact model solutions. The computations based on the formulas require only several tens of milliseconds on a moderate-performance personal computer for sufficiently fine grids, so the proposed small-parameter approach can be used in real-time cancer diagnosis.

A three-dimensional multifrequency inverse problem of acoustic sounding of a stationary inhomogeneous medium is considered. This nonlinear inverse problem is reduced to solving an auxiliary three-dimensional linear Fredholm integral equation of the first kind. In the analysis of the uniqueness of the solution to the inverse problem, the connection between the integral equation and determining the source in the Helmholtz equation is indicated. The last problem is ambiguously solvable in the general case. Examples of such ambiguity are given. Questions about detailed data (frequencies, sources) ensuring or not the uniqueness of solutions are considered. A speed-efficient algorithm for solving the inverse problem based on Fourier transforms is proposed. This algorithm makes it possible to calculate uniquely an approximate solution by a stable method under data perturbations. The results of numerical experiments on solving a three-dimensional model inverse problem on fairly detailed grids are presented.

Two modifications with variable coefficients of the well-known SEIR model for epidemic development in the application to the modeling of the infection curves of COVID-19 are considered. The data for these models are information on the number of infections each day obtained from the Johns Hopkins Coronavirus Resource Center database. In our paper, we propose special methods based on Tikhonov regularization for models’ identification on the class of piecewise constant coefficients. In contrast to the model with constant coefficients, which cannot always accurately describe some of infection curves, the first model is able to approximate them for different countries with an accuracy of 2–8%. The second model considered in the article takes into account external sources of infection in the form of an inhomogeneous term in one of the model equations and is able to approximate the data with a slightly better accuracy of 2–4%. For the second model, we also consider the possibility of using other input data, namely the number of infected people per day. Such data are used to model infection curves for several waves of the COVID-19 epidemic, including part of the Omicron wave. Numerical experiments carried out for a number of countries show that the waves of external sources of infection found are ahead of the wave of infection by 10 or more days. At the same time, other piecewise constant coefficients of the model change relatively slowly. These models can be applied fairly reliably to approximate many waves of infection curves with high precision and can be used to identify external and hidden sources of infection. This is the advantage of our models.

A new method for estimating formant frequency tracks of the vocal tract for arbitrary speech segments is proposed. The method uses the ratio of two Fourier transforms of a speech signal with special exponential-type windows depending on some parameter. This ratio is used for specific points in time and is considered as a function of frequency and parameter. By analyzing, for several parameter values, the distribution of minimum points (in terms of frequency) for the phase of this ratio and/or a similar distribution of extreme points for its amplitude, it is possible to estimate formant frequencies from the peaks of these distributions. A mathematical study is presented that substantiates this approach. A series of numerical experiments were carried out on the processing of synthetic and real speech signals, which confirmed the performance capabilities of the proposed formant evaluation method. In particular, in experiments with synthesized vowels, it was found that the error in estimating their resonance frequencies is small and stable with respect to additive noise up to a signal-to-noise ratio of 5 dB. For real speech, the method makes it possible to calculate the formant frequency tracks for both sounds with vocal excitation and for voiceless fricatives, aspirated plosives, and whispered speech.

Mathematical models of the phase function and its parameters in speech-signal analysis problems have been investigated. The phase spectrum of a speech signal has been calculated using the Hilbert transform of signals at the output of a gammatone filterbank. Short- and long-term modulations of the linear phase component and phase derivatives with respect to frequency and time, and mixed derivative have been considered. The method for vowel segmentation using aggregation of the correlation coefficients of the phase parameters is described. Experiments on estimating the formant and pitch frequencies and the glottal opening and closure instants have been performed.

A new algorithm for stable solution of a three-dimensional scalar inverse problem of acoustic sensing of an inhomogeneous medium in a cylindrical domain is proposed. Data for its solution is the complex amplitude of the wave field measured outside the acoustic inhomogeneities in the cylindrical layer. With the help of the Fourier transform and Fourier series, the inverse problem is reduced to a set of one-dimensional Fredholm integral equations of the first kind. Next, the complex amplitude of the wave field is computed in the inhomogeneity region and the desired sonic velocity field is found in this region. When run on a moderate-performance personal computer, the algorithm takes tens of seconds to solve the inverse problem on rather fine three-dimensional grids. The accuracy of the algorithm is analyzed numerically as applied to test inverse problems at different frequencies, and the stability of the algorithm with respect to data perturbations is investigated.

A computational procedure is proposed for refining the position and shape of three-dimensional acoustic inhomogeneities during the sound probing of a medium. The procedure, called a computing magnifier, is based on a high-speed algorithm for solving the inverse problem of acoustic sounding in areas of a special structure (three-dimensional space, cylindrical area, etc.) with a complex wave field amplitude as the data recorded in a thin layer. The algorithm was proposed and studied by the authors in their previous works. The computational magnifying procedure consists of quickly solving the inverse problem using this algorithm on the initial grid in the original 3D region, narrowing the original region to a nested smaller new region containing inhomogeneities, and then solving the inverse problem in this new region on a new grid with the same or even with fewer nodes. By repeating this procedure several times, we can significantly refine the position and shape of the studied inhomogeneities, as if enlarging them. The computational magnifying procedure works much faster than solving the inverse problem on adaptively refined 3D grids in the original area. This makes it easy to implement the procedure on personal computers (PCs) with average performance. A method for the numerical estimation of the quality of refining the position and shape of the studied inhomogeneities based on the use of histograms is proposed. A number of numerical model experiments on a PC on the use of a computing magnifier in a cylindrical region are presented. They include analysis of the quality of the position and shape of the refinement using histograms when solving an inverse problem with accurate and noisy data, the effect of averaging noisy data for determining the position and shape, experiments to assess the resolution of the computing magnifier, etc. The running time of the computing magnifier in each of these three-dimensional numerical experiments is about 10 s.

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 discretizing the problem and calculating the gradient, it is possible to use the finite element method. As an illustration, we consider the solution of two inverse problems of elastography using the finite element method: finding the distribution of Young's modulus in biological tissue from data on its compression and a similar problem of determining the characteristics of local oncological inclusions, which have a special parametric form.

Under consideration is the problem of improving the contrast of the image obtained by processing tomographic projections with phase distortion.The study is based on the wellknown intensity transfer equation.Unlike other works, this equation is solved in a bounded region of variation of the tomographic parameters.In a domain, a boundary value problem is posed for the intensity transfer equation which is then specialized for a three-dimensional parallel tomographic scheme.The case of two-dimensional tomography is also considered, together with the corresponding boundary value problem for the intensity transfer equation.We propose numerical methods for solving the boundary value problems of phase correction.The results are given of the numerical experiments on correction of tomographic projections and reconstruction of the structure of the objects under study (in particular, a slice of a geological sample) by using piecewise uniform regularization.

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 statement of the problem is used in which the fragment of the tissue is considered as a linearly elastic body. It is assumed that the geometry of inclusions is specified parametrically, and the Young's modulus inside and outside the inclusions is constant. The task is reduced to finding the number of inclusions, parameters defining their shape and the Young's modulus inside the inclusions. To solve the problem, a special algorithm is proposed and justified. The results of numerical experiments on solving a three-dimensional model problems are presented. A comparison is made of the solutions to the inverse problem in the three-dimensional domain and two-dimensional inverse problems in selected cross-sections of this domain. It is found that 2D inverse problems do not always allow one to find a true 3D solution. For one of solved inverse problems, a-posteriori error estimates for Young modulus and geometric parameters of inclusions are obtained.