Theory Of Ill-posed Problems Research Articles

On stable solutions of the dynamical reconstruction and tracking control problems for a coupled ordinary differential equation–heat equation

A dynamical reconstruction problem for a system described by a coupled ordinary differential equation—heat equation is considered. The problem consists in reconstructing an unknown varying in time right-hand part of this system on the basis of inaccurate measurements of its solution. A dynamical algorithm for solving this problem is designed. An estimate for convergence rate is presented. The algorithm is stable with respect to informational noises and computational errors. It is based on the combination of the feedback control method and the method of smoothing functional well-known in the theory of ill-posed problems. The algorithm suggested in the paper is applied to solve tracking control problems. Results of a numerical experiment are discussed.

Microwave Near-Field Dynamical Tomography of Thorax at Pulmonary and Cardiovascular Activity.

The developed near-field microwave diagnostics of dynamical lung tomography provide information about variations of air and blood content depth structure in the processes of breathing and heart beating that are unattainable for other available methods. The method of dynamical pulse 1D tomography (profiling) is based on solving the corresponding nonlinear ill-posed inverse problem in the extremely complicated case of the strongly absorbing frequency-dispersive layered medium with the dual regularization method-a new Lagrange approach in the theory of ill-posed problems. This method has been realized experimentally by data of bistatic measurements with two electrically small bow-tie antennas that provide a subwavelength resolution. The proposed methods of 3D lung tomography based on the multisensory pulse, multifrequency, or multi-base measurements are based on solving the corresponding integral equations in the Born approximation. The experimental 3D tomography of lung air content was obtained by the results of the multiple 1D pulse profiling by pulse measurements in several grid points over the planar square region of the thorax. Additionally, the possible applicability of multifrequency measurements of scattered harmonic signals in the monitoring of lungs was demonstrated by four-frequency measurements in the process of breathing. The results demonstrated the feasibility of the proposed control in the diagnosis of some lung diseases.

University Learning Outcomes: Statement of the Issue Within the Theory of Ill-Posed Problems

The article discusses the problem of determining the learning outcomes for educational programs of the university, including at the program level and the level of individual disciplines included in the program. The goal is to substantiate the understanding of the definition of learning outcomes at a university as an inverse ill-posed problem within the framework of the theory of ill-posed problems, which will allow further reengineering of the educational process at a university based on models of inverse and ill-posed problems. The article identifies the types of uncertainty inherent in education, which affect the possibility of an incorrect decision. The publication substantiates a new understanding of the educational process focused on learning outcomes at a university, as a process of finding a solution to an inverse incorrect problem. In this case, the learning outcomes formulated for a specific educational program are considered as an available solution to the inverse problem. The research carried out by the authors made it possible to construct a mathematical model of the educational process at the university and identify its components. The authors demonstrate that the optimal way to design the educational process is to design it as an inverse problem, and the list of disciplines, their content and the corresponding learning outcomes are determined and developed at subsequent stages, focusing on the goals of the program.

О некорректных задачах, экстремалях функционала Тихонова и регуляризованных принципах Лагранжа

The problem of finding a normal solution to an operator equation of the first kind on a pair of Hilbert spaces is classical in the theory of ill-posed problems. In accordance with the theory of regularization, its solutions are approximated by the extremals of the Tikhonov functional. From the point of view of the theory of problems for constrained extremum, the problem of minimizing a functional, equal to the square of the norm of an element, with an operator equality constraint (that is, given by an operator with an infinite-dimensional image) is equivalent to the classical ill-posed problem. The paper discusses the possibility of regularizing the Lagrange principle (LP) in the specified constrained extremum problem. This regularization is a transformation of the LP that turns it into a universal tool of stable solving illposed problems in terms of generalized minimizing sequences (GMS) and preserves its “general structural arrangement” based on the constructions of the classical Lagrange function. The transformed LP “contains” the classical analogue as its limiting variant when the numbers of the GMS elements tend to infinity. Both non-iterative and iterative variants of the regularization of the LP are discussed. Each of them leads to stable generation of the GMS in the original constrained extremum problem from the extremals of the regular Lagrange functional taken at the values of the dual variable generated by the corresponding procedure for the regularization of the dual problem. In conclusion, the article discusses the relationship between the extremals of the Tikhonov and Lagrange functionals in the considered classical ill-posed problem.

On dynamical input reconstruction in a distributed second order equation

Abstract A second order nonlinear differential equation is considered. An algorithm for reconstructing an input from inaccurate measurements of the solution at discrete times is designed. The algorithm based on the constructions of feedback control theory and theory of ill-posed problems is stable with respect to informational noises and computational errors.

Approximate calculation of the Caputo-type fractional derivative from inaccurate data. Dynamical approach

Abstract A specific formulation of the “classical” problem of mathematical analysis is considered. This is the problem of calculating the derivative of a function. The purpose of this work is to construct an algorithm for the approximate calculation of the Caputo-type fractional derivative based on the methods of control theory. The input data of the algorithm is represented by inaccurate measured function values at discrete, frequently enough, times. The proposed algorithm is based on two aspects: a local modification of the Tikhonov regularization method from the theory of ill-posed problems and the Krasovskii extremal shift method from the guaranteed control theory, both of which ensure the stability to informational noises and computational errors. Numerical experiments were carried out to illustrate the operation of the algorithm.

Market demand: a holistic theory and its verification

The paper presents a revision of the contemporary reductionistic demand theory, replacing the studying object, i.e. an individual, with a fuzzy collection of market buyers, regarded as a “statistical ensemble of consumers”. The new holistic market demand theory formally retains the neoclassical demand theory with reconsidering the utility function as a collective one. A nonparametric verification method is presented, which uses the economic (Konus) and formula Fisher indexes. The method has a variational character based on the theory of ill-posed problems. Verification is carried out simultaneously with the construction of Konus indexes. The method is demonstrated on an example of the new Giffen demand model.

The Parabolic Transform and Some Singular Integral Evolution Equations

Some singular integral evolution equations with wide class of closed operators are studied in Banach space. The considered integral equations are investigated without the existence of the resolvent of the closed operators. Also, some non-linear singular evolution equations are studied. An abstract parabolic transform is constructed to study the solutions of the considered ill-posed problems. Applications to fractional evolution equations and Hilfer fractional evolution equations are given. All the results can be applied to general singular integro-differential equations. The Fourier Transform plays an important role in constructing solutions of the Cauchy problems for parabolic and hyperbolic partial differential equations. This means that the Fourier transform is suitable but under conditions on the characteristic forms of the partial differential operators. Also, the Laplace transform plays an important role in studying the Cauchy problem for abstract differential equations in Banach space. But in this case, we need the existence of the resolvent of the considered abstract operators. This note is devoted to exploring the Cauchy problem for general singular integro-partial differential equations without conditions on the characteristic forms and also to study general singular integral evolution equations. Our approach is based on applying the new parabolic transform. This transform generalizes the methods developed within the regularization theory of ill-posed problems.

Extremal Shift in a Problem of Tracking a Solution of an Operator Differential Equation

A control problem for an operator differential equation in a Hilbert space is considered. The problem consists in constructing an algorithm generating a feedback control and guaranteeing that the solution of the equation follows a solution of another equation, which is subject to an unknown disturbance. We assume that both equations are given on an infinite time interval and the unknown disturbance is an element of the space of square integrable functions; i.e., the perturbation may be unbounded. We construct two algorithms based on elements of the theory of ill-posed problems and the extremal shift method known in the theory of positional differential games. The algorithms are stable with respect to information noises and calculation errors. The first and second algorithms can be used in the cases of continuous and discrete measurement of solutions, respectively.

Numerical solution of direct and inverse problems of heat transfer in oil reservoirs exploiting with horizontal well

In this paper, a mathematical model is constructed and its grid analog for modeling the thermohydrodynamic processes occurring in the oil reservoir and horizontal wellbore after its launch. On the basis of the proposed model and the theory of ill-posed problems, a numerical method is proposed for solving the inverse coefficient problem for determining the filtration properties of an oil reservoir.

Numerical methods for solving inverse problems of fluid filtration to a horizontal well

In this paper, a three-dimensional mathematical model of thermohydrodynamic processes occurring in an oil reservoir exposed by a horizontal well after its launch is constructed. Based on the proposed model and the theory of ill-posed problems, a numerical method is proposed for solving the inverse coefficient problem for determining the filtration properties of an oil reservoir. As the initial information, temperature change curves taken simultaneously by several deep autonomous measuring instruments installed in various sections of the horizontal well bore are used. According to the proposed method, the heterogeneity of the reservoir is evaluated along the horizontal part of the well bore.

On a Control Problem for a Linear System with Measurements of a Part of Phase Coordinates

We consider a control problem for a system of linear ordinary differential equations. It is required to design a feedback control procedure under which the rate of change of a part of the phase coordinates of the system would track the rate of change of a part of the phase coordinates of another system subject to an unknown disturbance. It is assumed that a part of phase coordinates of each of the systems is measured with error at discrete times. We propose a solution algorithm that is stable with respect to informational noises and computation errors. The algorithm is based on the extremal shift method known in the theory of guaranteed control. Since it is impossible to apply the “classical” extremal shift due to the incompleteness of the information on the phase coordinates, we propose a modification of this method that employs elements of the dynamic inversion theory. The latter is based on constructions from the theory of ill-posed problems. In the concluding section of the paper, we specify a class of systems nonlinear in the phase coordinates for which the algorithm is applicable.

The stability of images taken in computed tomography by back-projection method

Background: The back-projection method is quite widespread in the modern computed tomography and there are not principled critical comments addressed to it in the scientific literature. But there are reasons for such comments. The main one of them, in our opinion, is that the method does not use the theory of ill-posed problems in any way, despite the fact that the problem of the reconstruction is ill-posed. Objectives: The aim of the work is a development of a method of the reconstruction of tomograms, which is a modification of the back-projection method taking into account the theory of ill-posed problems. Materials and methods: In the back-projection method, the value of the filter parameter is chosen practically arbitrarily. In the method proposed in the article, this choice receives a justification: the filter parameter is identified with the regularization parameter that allows us to use the theory of ill-posed problems for its determination, and thereby to ensure the stability of the reconstructed image. Results: The dependence of the quality of the reconstruction on the selected filter width was obtained. The value of the filter at the given error level of the initial data and the given geometry corresponding to the minimum error of the reconstructed image was found. The value of the filter width depends on the scanning settings and the noise level on the projections, so the result is not the specific value of the filter but it’s the way to select the optimal value. Conclusions: In this paper we show that it is possible without completely abandon the back-projection method with all its positive sides to modify this method supplementing it with the approaches used in the theory of ill-posed problems. Such an approach must ensure the stability of the reconstructed image. This can be, for example, the identification of the filter parameter with the regularization parameter that formed the basis for the method of image reconstruction proposed in this article. It’s possible there are other ways of using the theory of ill-posed problems in the of back-projection method.

Optimizing performance of the deconvolution model reduction for large ODE systems

We investigate the numerical performance of the regularized deconvolution closure introduced recently by the authors. The purpose of the closure is to furnish constitutive equations for Irving-Kirkwood-Noll procedure, a well known method for deriving continuum balance equations from the Newton’s equations of particle dynamics. A version of this procedure used in the paper relies on spatial averaging developed by Hardy, and independently by Murdoch and Bedeaux. The constitutive equations for the stress are given as a sum of nonlinear operator terms acting on the mesoscale average density and velocity. Each term is a “convolution sandwich” containing the deconvolution operator, a composition or a product operator, and the convolution (averaging) operator. Deconvolution is constructed using filtered regularization methods from the theory of ill-posed problems. The purpose of regularization is to ensure numerical stability. The particular technique used for numerical experiments is truncated singular value decomposition (SVD). The accuracy of the constitutive equations depends on several parameters: the choice of the averaging window function, the value of the mesoscale resolution parameter, scale separation, the level of truncation of singular values, and the level of spectral filtering of the averages. We conduct numerical experiments to determine the effect of each parameter on the accuracy and efficiency of the method. Partial error estimates are also obtained.

Higher-order total variations for functions of several variables and their application in the theory of ill-posed problems

In the space of functions of two variables with Hardy-Krause property, new notions of higher-order total variations and Banach spaces of functions of two variables with bounded higher variations are introduced. The connection of these spaces with the Sobolev spaces W 1 , m ∈ ℕ, is studied. In the Sobolev spaces, a wide class of integral functionals with the weak regularization properties and the H-property is defined. It is proved that the application of these functionals in the Tikhonov variational scheme generates for m ≥ 3 the convergence of approximate solutions with respect to the total variation of order m − 3. The results are naturally extended to the case of functions of N variables.

On instabilities in data assimilation algorithms

Data assimilation algorithms are a crucial part of operational systems in numerical weather prediction, hydrology and climate science, but are also important for dynamical reconstruction in medical applications and quality control for manufacturing processes. Usually, a variety of diverse measurement data are employed to determine the state of the atmosphere or to a wider system including land and oceans. Modern data assimilation systems use more and more remote sensing data, in particular radiances measured by satellites, radar data and integrated water vapor measurements via GPS/GNSS signals. The inversion of some of these measurements are ill-posed in the classical sense, i.e. the inverse of the operator H which maps the state onto the data is unbounded. In this case, the use of such data can lead to significant instabilities of data assimilation algorithms. The goal of this work is to provide a rigorous mathematical analysis of the instability of well-known data assimilation methods. Here, we will restrict our attention to particular linear systems, in which the instability can be explicitly analyzed. We investigate the three-dimensional variational assimilation and four-dimensional variational assimilation. A theory for the instability is developed using the classical theory of ill-posed problems in a Banach space framework. Further, we demonstrate by numerical examples that instabilities can and will occur, including an example from dynamic magnetic tomography.

On the reconstruction of inputs in linear parabolic equations

The problem of reconstructing distributed inputs in linear parabolic equations is investigated. The algorithm proposed for solving this problem is stable with respect to informational disturbances and computational errors. It is based on the combination of methods from the theory of ill-posed problems and from the theory of positional control. The process of reconstructing unknown inputs implemented by the algorithm employs inaccurate measurements of phase coordinates of the system at discrete sufficiently frequent times. In the case when the input is a function of bounded variation, an upper estimate is established for the convergence rate.

Some algorithms for the dynamic reconstruction of inputs

For some classes of systems described by ordinary differential equations, a survey of algorithms for the dynamic reconstruction of inputs is presented. The algorithms described in the paper are stable with respect to information noises and computation errors; they are based on methods from the theory of ill-posed problems as well as on appropriate modifications of N.N. Krasovskii’s principle of extremal aiming, which is known in the theory of guaranteed control.

An analytical formula for ring artefact suppression in X-ray tomography

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 solution is proposed allowing fast ring artefact suppression. Its performance is demonstrated for parallel beam synchrotron X-ray tomography.

Iterative processes of Fejér type in ill-posed problems with a priori information

In the latter thirty years, the solution of ill-posed problems with a priori information formed a separate field of research in the theory of ill-posed problems. We mean the class of problems, where along with the basic equation one has some additional data on the desired solution. Namely, one states some relations and constraints which contain important information on the object under consideration. As a rule, taking into account these data in a solution algorithm, one can essentially increase its accuracy for solving ill-posed (unstable) problems. It is especially important in the solution of applied problems in the case when a solution is not unique, because this approach allows one to choose a solution that meets the reality. In this paper we survey the methods for solving such problems. We briefly describe all relevant approaches (known to us), but we pay the main attention to the method proposed by us. This technique is based on the application of iterative processes of Fejer type which admit a flexible and effective realization for a wide class of a priori constraints.