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In this paper, we propose two parameter choice rules for the discretizing Tikhonov regularization via multiscale Galerkin projection for solving linear ill-posed integral equations. In contrast to previous theoretical analyses, we introduce a new concept called the projection noise level to obtain error estimates for the approximate solutions. This concept allows us to assess how noise levels change during projection. The balance principle and Hanke–Raus rule are modified by incorporating the error estimates of the projection noise level. We demonstrate the convergence rate of these two modified parameter choice rules through rigorous proof. In addition, we find that the error between the approximate solution and the exact solution improves as the noise frequency increases. Finally, numerical experiments are provided to illustrate the theoretical findings presented in this paper. [ABSTRACT FROM AUTHOR]

In 2021, Z. Fu, Y. Chen and B. Han introduced an inexact Newton regularization (REGINN-IT) using an idea involving the non-stationary iterated Tikhonov regularization scheme for solving nonlinear ill-posed operator equations. In this paper, we suggest a simplified version of the REGINN-IT scheme by using the Bregman distance, duality mapping and a suitable parameter choice strategy to produce an approximate solution. The method is comprised of inner and outer iteration steps. The outer iterates are stopped by a Morozov-type stopping rule, while the inner iterate is executed by making use of the non-stationary iterated Tikhonov scheme. We have studied convergence of the proposed method under some standard assumptions and utilizing tools from convex analysis. The novelty of the method is that it requires computation of the Fréchet derivative only at an initial guess of an exact solution and hence can be identified as more efficient compared to the method given by Z. Fu, Y. Chen and B. Han. Further, in the last section of the paper, we discuss test examples to inspect the proficiency of the method. [ABSTRACT FROM AUTHOR]

This paper concentrates on the recovery of block-sparse signals, which are not only sparse but also nonzero elements are arrayed into some blocks (clusters) rather than being arbitrary distributed all over the vector, from linear measurements. We establish high-order sufficient conditions based on block RIP, which could ensure the exact recovery of every block s-sparse signal in the noiseless case via mixed l 2 / l p minimization method, and the stable and robust recovery in the case that signals are not accurately block-sparse in the presence of noise. Additionally, a lower bound on necessary number of random Gaussian measurements is gained for the condition to be true with overwhelming probability. Furthermore, a series of numerical experiments are conducted to demonstrate the performance of the mixed l 2 / l p minimization. To the best of the authors' knowledge, the recovery guarantees established in this paper are superior to those currently available. [ABSTRACT FROM AUTHOR]

Image denoising and edge extraction are two main tasks in image processing. In this paper, a game model is proposed to solve the image denoising and edge extraction, which combines an adaptive improved total variation (AdITV) model for image denoising and a global sparse gradient (GSG) model for edge extraction. The AdITV model is a forward-and-backward diffusion model. In fact, forward diffusion is applied to the homogeneous region to denoise, and backward diffusion is applied to the edge region to enhance the edge. A unified explicit discrete scheme is established in this paper to solve the AdITV model, which is compatible to forward diffusion and backward diffusion. The stability of the scheme is proved. On the other hand, GSG is a functional model based on sparse representation, which is robust to extract edges under the influence of noise. AdITV and GSG are chosen as two components of the game model. The alternate iteration method is used to solve the game problem. The convergence of the algorithm is proved and numerical experiments show the effectiveness of the model. [ABSTRACT FROM AUTHOR]

Overdetermined boundary value problems and the minimal operators generated by them are extremely important in the description of regular boundary value problems for differential equations, and are also widely used in the study of local properties of solutions. In addition, for inverse problems of mathematical physics arising from applications, when determining unknown data, it is necessary to study problems with overdetermined boundary conditions, which is reflected in the study of problems, including for hyperbolic equations and systems, arising in physics, geophysics, seismic tomography, geoelectrics, electrodynamics, medicine, ecology, economics and many other practical areas. Thus, the study of overdetermined boundary value problems is of both theoretical and applied interest. In this paper, a criterion for the regular solvability of the overdetermined Cauchy problem for the Gellerstedt equation and the minimal differential operator generated by it in a hyperbolic domain is established, as which both the case of a characteristic triangle and the case of a more general domain with fairly general assumptions about the boundary of the domain are considered. Due to overdetermined boundary conditions, the problem under consideration will be ill-posed in the general case, therefore, for its regular solvability, additional conditions must be imposed on the initial data. In other words, we have considered the inverse problem: to determine what requirements the initial data of the problem, in particular the right part of the Gellerstedt equation, should meet, in question, so that the overdetermined Cauchy problem is regularly solvable. The proof is based on the Gellerstedt potential, the properties of solutions of the Goursat problem in the characteristic triangle, and the properties of special functions. [ABSTRACT FROM AUTHOR]

The probe and singular sources methods are two well-known classical direct reconstruction methods in inverse obstacle problems governed by partial differential equations. In this paper, by considering an inverse obstacle problem governed by the Laplace equation in a bounded domain as a prototype case, an integrated theory of the probe and singular sources methods is proposed. The theory consists of three parts: (i) introducing the singular sources method combined with the notion of the probe method; (ii) finding a third indicator function whose two ways decomposition yields the indicator functions in the probe and singular sources methods; (iii) finding the completely integrated version of the probe and singular sources methods. [ABSTRACT FROM AUTHOR]

At present, computationally intensive numerical weather prediction systems based on physics equations are widely used for short-term weather forecasting. In this paper, we investigate the potential of accelerating the Weather Research and Forecasting (WRF-ARW) model using machine learning techniques. Two main approaches are considered. First, we assess the viability of complete replacing the numerical weather model with deep learning models, capable of predicting the full range forecast directly from basic initial data. Second, we consider a “super-resolution” technique involving low-resolution WRF computation and a machine learning based downscaling using coarse-grid forecast for conditioning. The process of downscaling is intrinsically an ill-posed problem. In both categories, several prominent and promising machine learning methods are evaluated and compared on real data from a variety of sources. for the Moscow region Namely, in addition to the ground truth WRF forecasts that were utilized for training, we compare the model predictions against ERA5 reanalysis and measurements from local weather stations. We show that deep learning approaches can be successfully applied to accelerate a numerical model and even produce more realistic forecasts in other aspects. As a practical outcome, this study offers empirically validated guidance for the selection and application of deep learning methods to accelerate the computation of detailed short-term atmospheric forecasts tailored to specific needs. [ABSTRACT FROM AUTHOR]

In this paper, we formulate the modified iteratively regularized Landweber iteration method in Banach spaces to solve the inverse problems for which the forward operator may be smooth or non-smooth. We study the convergence analysis of the modified method for both the perturbed as well as unperturbed data by utilizing the Hölder stability estimates. In the presence of perturbed data, we terminate the method via a discrepancy principle and show that it is in fact a convergence regularization method that terminates within a few iterations. In the presence of unperturbed data, we show that the iterates converge to the exact solution. Additionally, we deduce the convergence rates in the presence of perturbed as well as unperturbed data. Finally, we discuss two inverse problems on which the method is applicable. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to investigate a newly proposed hybrid and hybrid inclusion problem consisting of fractional differential problems involving two different fractional derivatives of order μ, Caputo and Liouville–Riemann operators, with multi-order mixed Riemann–Liouville integro-derivative conditions. Although α is between one and two, we need three boundary value conditions to find the integral equation. The study investigates the results of existence for hybrid, hybrid inclusion, and non-hybrid inclusion problems by employing several analytical approaches, including Dhage’s technique, α - ψ {\alpha-\psi} -contractive mappings, fixed points, and endpoints of the product operators. To further illustrate our findings, we present three examples. [ABSTRACT FROM AUTHOR]

Generalized Abel equations have been employed in the recent literature to invert Radon transforms which arise in a number of important imaging applications, including Compton Scatter Tomography (CST), Ultrasound Reflection Tomography (URT), and X-ray CT. In this paper, we present novel injectivity results and inversion methods for generalized Abel operators. We apply our theory to a new Radon transform, R j , of interest in URT, which integrates a square integrable function of compact support, 푓, over ellipsoid and hyperboloid surfaces with centers on a plane. Using our newly established theory on generalized Abel equations, we show that R j is injective and provide an inversion method based on Neumann series. In addition, using algebraic methods, we present image phantom reconstructions from R j ⁢ f data with added pseudo-random noise. [ABSTRACT FROM AUTHOR]

Identification of physical properties of materials is very important because they are in general unknown. Furthermore, their direct experimental measurement could be costly and inaccurate. In such a situation, a cheap and efficient alternative is to mathematically formulate an inverse, but difficult, problem that can be solved, in general, numerically; the challenge being that the problem is, in general, nonlinear and ill-posed. In this paper, the reconstruction of a lower-order unknown time-dependent coefficient in a Cahn–Hilliard-type fourth-order equation from an additional integral observation, which has application to characterizing the nonlinear saturation of the collisional trapped-ion mode in a tokamak, is investigated. The local existence and uniqueness of the solution to such inverse problem is established by utilizing the Rothe method. Moreover, the continuous dependence of the unknown coefficient upon the measured data is derived. Next, the Tikhonov regularization method is applied to recover the unknown coefficient from noisy measurements. The stability estimate of the minimizer is derived by investigating an auxiliary linear fourth-order inverse source problem. Henceforth, the variational source condition can be verified. Then the convergence rate is obtained under such source condition. [ABSTRACT FROM AUTHOR]

Discrete inverse problems correspond to solving a system of equations in a stable way with respect to noise in the data. A typical approach to select a meaningful solution is to introduce a regularizer. While for most applications the regularizer is convex, in many cases it is neither smooth nor strongly convex. In this paper, we propose and study two new iterative regularization methods, based on a primal-dual algorithm, to regularize inverse problems efficiently. Our analysis, in the noise free case, provides convergence rates for the Lagrangian and the feasibility gap. In the noisy case, it provides stability bounds and early stopping rules with theoretical guarantees. The main novelty of our work is the exploitation of some a priori knowledge about the solution set: we show that the linear equations determined by the data can be used more than once along the iterations. We discuss various approaches to reuse linear equations that are at the same time consistent with our assumptions and flexible in the implementation. Finally, we illustrate our theoretical findings with numerical simulations for robust sparse recovery and image reconstruction. We confirm the efficiency of the proposed regularization approaches, comparing the results with state-of-the-art methods. [ABSTRACT FROM AUTHOR]

In this paper, we consider a Bayesian method for nonlinear elastic inverse problems. As a working model, we are interested in the inverse problem of restoring elastic properties from measured tissue displacement. In order to reduce the computational cost, we will use the following multi-fidelity model approach. First, we construct a surrogate low-fidelity DNNs-based model in the prior distribution, then use a certain number of simulations of high fidelity model associated with an adaptive strategy online to update the low-fidelity model locally. Numerical examples show that the proposed method can solve nonlinear elastic inverse problems efficiently and accurately. [ABSTRACT FROM AUTHOR]

Extracting the useful information has been used almost everywhere in many fields of mathematics and applied mathematics. It is a classical ill-posed problem due to the unstable dependence of approximations on small perturbation of the data. The traditional regularization methods depend on the choice of the regularization parameter, which are closely related to an available accurate upper bound of noise level; thus it is not appropriate for the randomly distributed noise with big or unknown variance. In this paper, a purely data driven statistical regularization method is proposed, effectively extracting the information from randomly noisy observations. The rigorous upper bound estimation of confidence interval of the error in L 2 norm is established, and some numerical examples are provided to illustrate the effectiveness and computational performance of the method. [ABSTRACT FROM AUTHOR]

In this paper, free surface problems of Stefan type for the parabolic heat equation are considered. The analytical solutions of the problems are based on the method of heat polynomials and integral error function in the form of series. Convergence of the series solution is considered and proved. Both one-and two-phase Stefan-type problems are investigated. Numerical results for one-phase inverse Stefan problem are presented and discussed. [ABSTRACT FROM AUTHOR]

This paper presents classification and analysis of the mathematical models of the spread of COVID-19 in different groups of population such as family, school, office (3–100 people), town (100–5000 people), city, region (0.5–15 million people), country, continent, and the world. The classification covers major types of models (time-series, differential, imitation ones, neural networks models and their combinations). The time-series models are based on analysis of time series using filtration, regression and network methods. The differential models are those derived from systems of ordinary and stochastic differential equations as well as partial differential equations. The imitation models include cellular automata and agent-based models. The fourth group in the classification consists of combinations of nonlinear Markov chains and optimal control theory, derived by methods of the mean-field game theory. COVID-19 is a novel and complicated disease, and the parameters of most models are, as a rule, unknown and estimated by solving inverse problems. The paper contains an analysis of major algorithms of solving inverse problems: stochastic optimization, nature-inspired algorithms (genetic, differential evolution, particle swarm, etc.), assimilation methods, big-data analysis, and machine learning. [ABSTRACT FROM AUTHOR]

Finding the history of a groundwater contaminant plume from final measurements is an ill-posed problem and, consequently, its solution is extremely sensitive to errors in the input data. In this paper, we study this problem mathematically. So, firstly, existence and uniqueness theorems of a quasi-solution in an appropriate class of admissible initial data are given. Secondly, in order to overcome the ill-posedness of the problem and also approximate the quasi-solution, two approaches (computational and iterative algorithms) are provided. In the computational algorithm, the finite element method and TSVD regularization are applied. This method is tested by two numerical examples. The results reveal the efficiency and applicability of the proposed method. Also, in order to construct the iterative methods, an explicit formula for the gradient of the cost functional J is given. This result helps us to construct two iterative methods, i.e., the conjugate gradient algorithm and Landweber iteration algorithm. We prove the Lipschitz continuity of the gradient of the cost functional, monotonicity and convergence of the iterative methods. At the end of the paper, a numerical example is given to show the validation of the iterative algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we study the well-posedness of the solution of an optimal control problem related to a multi-parameters identification problem in degenerate parabolic equations. Problems of this type have important applications in several fields of applied science. Unlike other inverse coefficient problems for classical parabolic equations, the mathematical model discussed in the paper is degenerate on both lateral boundaries of the domain. Moreover, the status of the two unknown coefficients are different, namely that the reconstruction of the source term is mildly ill-posed, while the inverse initial value problem is severely ill-posed. On the basis of optimal control framework, the problem is transformed into an optimization problem. The existence of the minimizer is proved and the necessary conditions which must be satisfied by the minimizer are also established. Due to the difference between ill-posedness degrees of the two unknown coefficients, the extensively used conjugate theory for parabolic equations cannot be directly applied for our problem. By carefully analyzing the necessary conditions and the direct problem, the uniqueness, stability and convergence of the minimizer are obtained. The results obtained in the paper are interesting and useful, and can be extended to more general parabolic equations with degenerate coefficients. [ABSTRACT FROM AUTHOR]

In this paper, we consider the simplified Levenberg–Marquardt method for nonlinear ill-posed inverse problems in Hilbert spaces for obtaining stable approximations of solutions to the ill-posed nonlinear equations of the form F ⁢ ( u ) = y {F(u)=y} , where F : 풟 ⁢ ( F ) ⊂ 햴 → 햸 {F:\mathcal{D}(F)\subset\mathsf{U}\to\mathsf{Y}} is a nonlinear operator between Hilbert spaces 햴 {\mathsf{U}} and 햸 {\mathsf{Y}} . The method is defined as follows: u n + 1 δ = u n δ - ( T 0 ∗ ⁢ T 0 + α n ⁢ I ) - 1 ⁢ T 0 ∗ ⁢ ( F ⁢ ( u n δ ) - y δ ) , u_{n+1}^{\delta}=u_{n}^{\delta}-(T_{0}^{\ast}T_{0}+\alpha_{n}I)^{-1}T_{0}^{% \ast}(F(u_{n}^{\delta})-y^{\delta}), where T 0 = F ′ ⁢ ( u 0 ) {T_{0}=F^{\prime}(u_{0})} and T 0 ∗ = F ′ ⁢ ( u 0 ) ∗ {T_{0}^{\ast}=F^{\prime}(u_{0})^{\ast}} . Here F ′ ⁢ ( u 0 ) {F^{\prime}(u_{0})} denotes the Frèchet derivative of F at an initial guess u 0 ∈ 풟 ⁢ ( F ) {u_{0}\in\mathcal{D}(F)} for the exact solution u † {u^{\dagger}} , F ′ ⁢ ( u 0 ) ∗ {F^{\prime}(u_{0})^{\ast}} is the adjoint of F ′ ⁢ ( u 0 ) {F^{\prime}(u_{0})} and { α n } {\{\alpha_{n}\}} is an a priori chosen sequence of non-negative real numbers satisfying suitable properties. We use Morozov-type stopping rule to terminate the iterations. Under suitable non-linearity conditions on operator F, we show convergence of the method and also obtain a convergence rate result under a Hölder-type source condition on the element u 0 - u † {u_{0}-u^{\dagger}} . Furthermore, we derive convergence of the method for the case when no source conditions are used and the study concludes with numerical examples which validate the theoretical conclusions. [ABSTRACT FROM AUTHOR]

A controlled system of differential equations under the action of an unknown disturbance is considered. The problem discussed in the paper consists in constructing algorithms for forming a control that provides the realization of a prescribed motion for any admissible disturbance. Namely these algorithms should provide the closeness in the metric of the space of differentiable functions of a phase trajectory of a given controlled system and some etalon trajectory of an analogous system functioning when any outer actions are absent. As the space of admissible disturbances, we take the space of measurable square integrable (with respect to the Euclidean norm) functions. The cases of inaccurate measurements of phase trajectories of both systems at all times and at discrete times are under study. Two computer oriented algorithms for solving the problem are designed. The algorithms are based on the (well-known in the theory of guaranteed control) method of extremal shift. In the process, its local (at each time of control correction) regularization is performed by the method of smoothing functional (the Tikhonov method). In addition, estimates for algorithm’s convergence rate are presented. [ABSTRACT FROM AUTHOR]

The non-convex α ⁢ ∥ ⋅ ∥ ℓ 1 − β ⁢ ∥ ⋅ ∥ ℓ 2 ( α ≥ β ≥ 0 ) regularization is a new approach for sparse recovery. A minimizer of the α ⁢ ∥ ⋅ ∥ ℓ 1 − β ⁢ ∥ ⋅ ∥ ℓ 2 regularized function can be computed by applying the ST-( α ⁢ ℓ 1 − β ⁢ ℓ 2 ) algorithm which is similar to the classical iterative soft thresholding algorithm (ISTA). Unfortunately, It is known that ISTA converges quite slowly, and a faster alternative to ISTA is the projected gradient (PG) method. Nevertheless, the current applicability of the PG method is limited to linear inverse problems. In this paper, we extend the PG method based on a surrogate function approach to nonlinear inverse problems with the α ⁢ ∥ ⋅ ∥ ℓ 1 − β ⁢ ∥ ⋅ ∥ ℓ 2 ( α ≥ β ≥ 0 ) regularization in the finite-dimensional space R n . It is shown that the presented algorithm converges subsequentially to a stationary point of a constrained Tikhonov-type functional for sparsity regularization. Numerical experiments are given in the context of a nonlinear compressive sensing problem to illustrate the efficiency of the proposed approach. [ABSTRACT FROM AUTHOR]

We study inverse spectral problems for Dirac-type functional-differential operators with two constant delays greater than two fifths the length of the interval, under Dirichlet boundary conditions. The inverse problem of recovering operators from four spectra has been studied. We consider cases when delays are greater or less than half the length of the interval. The main result of the paper refers to the proof that in both cases operators can be recovered uniquely from four spectra. [ABSTRACT FROM AUTHOR]

The paper is devoted to the regularization of ill-posed stochastic Cauchy problems in Hilbert spaces: (0.1) d ⁢ u ⁢ (t) = A ⁢ u ⁢ (t) ⁢ d ⁢ t + B ⁢ d ⁢ W ⁢ (t) , t > 0 , u ⁢ (0) = ξ . The need for regularization is connected with the fact that in the general case the operator A is not supposed to generate a strongly continuous semigroup and with the divergence of the series defining the infinite-dimensional Wiener process { W ⁢ (t) : t ≥ 0 } . The construction of regularizing operators uses the technique of Dunford–Schwartz operators, regularized semigroups, generalized Fourier transform and infinite-dimensional Q-Wiener processes. [ABSTRACT FROM AUTHOR]

In this paper, we study the convergence analysis of the inexact Newton–Landweber iteration method (INLIM) with frozen derivative in Hilbert as well as Banach spaces. To study the convergence analysis, we incorporate the Hölder stability of the inverse mapping and Lipschitz continuity of the Fréchet derivative of the forward mapping. Moreover, we derive the convergence rates of INLIM in Hilbert as well as Banach spaces without using any extra smoothness condition. Finally, we compare our convergence rates results with that of several other frozen methods proposed in the literature to solve inverse problems. [ABSTRACT FROM AUTHOR]

The paper is devoted to a problem of acquiring elastic properties of a composite material from the vibration testing data with a simplified experimental acquisition scheme. The specimen is considered to abide by the linear elasticity laws and subject to viscoelastic damping. The boundary value problem for transverse movement of such a specimen in the frequency domain is formulated and solved with finite-element method. The correction method is suggested for the finite element matrices to account for the mass of the accelerometer. The problem of acquiring the elastic parameters is then formulated as a nonlinear least-square optimization problem. The usage of the automatic differentiation technique for stable and efficient computation of the gradient and hessian allows to use well-studied first and second order local optimization methods. We also explore the possibility of generating initial guesses for local minimization by heuristic global methods. The results of the numerical experiments on simulated data are analyzed in order to provide insights for the following real life experiments. [ABSTRACT FROM AUTHOR]

A fast numerical algorithm for solving the Cauchy problem for elliptic equations with variable coefficients in standard calculation domains (rectangles, circles, or rings) is proposed. The algorithm is designed to calculate the heat flux at the inaccessible boundary. It is based on the separation of variables method. This approach employs a finite difference approximation and allows obtaining a solution to a discrete problem in arithmetic operations of the order of N ⁢ ln ⁡ N , where 푁 is the number of grid points. As a rule, iterative procedures are needed to solve the Cauchy problem for elliptic equations. The currently available direct algorithms for solving the Cauchy problem have been developed only for (Laplace, Helmholtz) operators with constant coefficients and for use of analytical solutions for problems with such operators. A novel feature of the results of the present paper is that the direct algorithm can be used for an elliptic operator with variable coefficients (of a special form). It is important that in this case no analytical solution to the problem can be obtained. The algorithm significantly increases the range of problems that can be solved. It can be used to create devices for determining in real time heat fluxes on the parts of inhomogeneous constructions that cannot be measured. For example, to determine the heat flux on the inner radius of a pipe made of different materials. [ABSTRACT FROM AUTHOR]

The paper considers the scattering problem for the first-order system of hyperbolic equations on the half-axis with a nonhomogeneous boundary condition. This problem models the phnomennon of wave propagation in a nonstationary medium where an incoming wave unaffected by a potential field. The scattering operator on the half-axis with a nonzero boundary condition is defined and the uniqueness of the inverse scattering problem (the problem of finding the potential with respect to scattering operator) is studied. [ABSTRACT FROM AUTHOR]

The article features Russian academician and mathematician M. M. Lavrentiev and his papers that were published during the period of 1953 to 1965 which are said to have contributed to the development of the theory of inverse and ill-posed problems. His first paper focused on increasing the accuracy of the solution of a system of linear algebraic equations, while the second paper dealt with improving the accuracy of the solution to a system of linear algebraic equations.

This paper deals with differential pencils possessing a term depending on the unknown function with a fixed argument. We deduce the so called main equation together with its fine structure for the spectral problem. Then, according to the boundary conditions and the position of argument, we describe two cases: degenerate and non-degenerate. For these two cases, the uniqueness of inverse spectral problem is studied and a constructive procedure for reconstructing the potentials along with necessary and sufficient conditions of the inverse problem solvability are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we present a new methodology, based on the inverse problem approach, for the determination of an unknown shear force acting on the inaccessible tip of the microcantilever, which is a key component of transverse dynamic force microscopy (TDFM). The mathematical modelling of this phenomenon leads to the inverse problem of determining the shear force g ⁢ (t) acting on the inaccessible boundary x = ℓ in a system governed by the variable coefficient Euler–Bernoulli equation ρ A ⁢ (x) ⁢ u t ⁢ t + μ ⁢ (x) ⁢ u t + (r ⁢ (x) ⁢ u x ⁢ x + κ ⁢ (x) ⁢ u x ⁢ x ⁢ t ) x ⁢ x = 0 , (x , t) ∈ (0 , ℓ) × (0 , T) , subject to the homogeneous initial conditions and the boundary conditions u ⁢ (0 , t) = u 0 ⁢ (t) , u x ⁢ (0 , t) = 0 , ( u x ⁢ x ⁢ (x , t) + κ ⁢ (x) ⁢ u x ⁢ x ⁢ t ) x = ℓ = 0 , (- (r ⁢ (x) ⁢ u x ⁢ x + κ ⁢ (x) ⁢ u x ⁢ x ⁢ t ) x) x = ℓ = g ⁢ (t) , from the final time measured output (displacement) u T ⁢ (x) := u ⁢ (x , T) . We introduce the input-output map (Φ ⁢ g) ⁢ (x) := u ⁢ (x , T ; g) , g ∈ 풢 , and prove that it is a compact and Lipschitz continuous linear operator. Then we introduce the Tikhonov functional J ⁢ (F) = 1 2 ⁢ ∥ Φ ⁢ g - u T ∥ L 2 ⁢ (0 , ℓ) 2 and prove the existence of a quasi-solution of the inverse problem. We derive a gradient formula for the Fréchet gradient of the Tikhonov functional through the corresponding adjoint problem solution and prove that it is a Lipschitz continuous functional. The results of the numerical experiments clearly illustrate the effectiveness and feasibility of the proposed approach. [ABSTRACT FROM AUTHOR]

This paper is concerned with the inverse scattering of the rough surfaces with multi-frequency phase and phaseless measurements. We present a high-order recursive iteration method based on the homotopy iteration technique to reconstruct the rough surfaces. The convergence for the multi-frequency homotopy iteration method is obtained under some conditions. Some numerical experiments show the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we address stability results in determining the time-dependent scalar and vector potentials appearing in the convection-diffusion equation from the knowledge of the Cauchy data set. We prove Hölder-type stability estimates. The key tool used in this work is the geometric optics solution. [ABSTRACT FROM AUTHOR]

In this paper, we consider the inverse scattering problem of identifying a two-layered cavity by internal acoustic measurements under the condition that the interior interface has a mixed transmission boundary condition. We focus on the mathematical analysis of recovering the shape of the interior interface by using the linear sampling method, including reconstructing the surface conductivity by the same measurements. [ABSTRACT FROM AUTHOR]

This paper investigates the inverse problem for the non-stationary radiation transfer equation, which involves finding the attenuation coefficient using the data of serial irradiation of the medium with pulses of various durations. In the framework of single and double scattering approximations, we obtain asymptotic estimates of the scattered radiation flux density for a short duration of the probing pulse. We propose extrapolation procedures for the ballistic component of the radiation transfer equation solution using the data of multiple irradiations of the medium by pulsed radiation sources, which allows us to obtain approximate formulas for finding the attenuation coefficient. The results of numerical experiments with a well-known digital phantom confirm the effectiveness of the extrapolation algorithm for improving the quality of tomographic images of scattering media. [ABSTRACT FROM AUTHOR]

In this paper, we study the inverse problems of determining the unknown transverse shear force g ⁢ (t) in a system governed by the damped Euler–Bernoulli equation ρ ⁢ (x) ⁢ u t ⁢ t + μ ⁢ (x) ⁢ u t + (r ⁢ (x) ⁢ u x ⁢ x ) x ⁢ x + (κ ⁢ (x) ⁢ u x ⁢ x ⁢ t ) x ⁢ x = 0 , (x , t) ∈ (0 , ℓ) × (0 , T ] , subject to the boundary conditions u ⁢ (0 , t) = 0 , u x ⁢ (0 , t) = 0 , [ r ⁢ (x) ⁢ u x ⁢ x + κ ⁢ (x) ⁢ u x ⁢ x ⁢ t ] x = ℓ = 0 , - [ (r ⁢ (x) ⁢ u x ⁢ x + κ ⁢ (x) ⁢ u x ⁢ x ⁢ t ) x ] x = ℓ = g ⁢ (t) , for t ∈ [ 0 , T ] , from the measured deflection ν ⁢ (t) := u ⁢ (ℓ , t) , t ∈ [ 0 , T ] , and from the bending moment ω ⁢ (t) := - (r ⁢ (0) ⁢ u x ⁢ x ⁢ (0 , t) + κ ⁢ (0) ⁢ u x ⁢ x ⁢ t ⁢ (0 , t) ) , t ∈ [ 0 , T ] , where the terms (κ ⁢ (x) ⁢ u x ⁢ x ⁢ t ) x ⁢ x and μ ⁢ (x) ⁢ u t account for the Kelvin–Voigt damping and external damping, respectively. The main purpose of this study is to analyze the Kelvin–Voigt damping effect on determining the unknown transverse shear force (boundary input) through the given boundary measurements. The inverse problems are transformed into minimization problems for Tikhonov functionals, and it is shown that the regularized functionals admit unique solutions for the inverse problems. By suitable regularity on the admissible class of shear force g ⁢ (t) , we prove that these functionals are Fréchet differentiable, and the derivatives are expressed through the solutions of corresponding adjoint problems posed with measured data as boundary data associated with the direct problem. The solvability of these adjoint problems is obtained under the minimal regularity of the boundary data g ⁢ (t) , which turns out to be the regularizing effect of the Kelvin–Voigt damping in the direct problem. Furthermore, using the Fréchet derivative of the more regularized Tikhonov functionals, we obtain remarkable Lipschitz stability estimates for the transverse shear force in terms of the given measurement by a feasible condition only on the Kelvin–Voigt damping coefficient. [ABSTRACT FROM AUTHOR]

This paper is concerned with the inverse scattering of acoustic waves by an unbounded periodic elastic medium in the three-dimensional case. A novel uniqueness theorem is proved for the inverse problem of recovering a bi-periodic interface between acoustic and elastic waves using the near-field data measured only from the acoustic side of the interface, corresponding to a countably infinite number of quasi-periodic incident acoustic waves. The proposed method depends only on a fundamental a priori estimate established for the acoustic and elastic wave fields and a new mixed-reciprocity relation established in this paper for the solutions of the fluid-solid interaction scattering problem. [ABSTRACT FROM AUTHOR]

This paper is concerned with discontinuous inverse problem generated by complex-valued weight Sturm–Liouville differential operator with λ-dependent boundary conditions. We establish some properties of spectral characteristic and prove that the potential on the whole interval can be uniquely determined by the Weyl-type function or two spectra. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a projected homotopy perturbation method based on sequential Bregman projections for nonlinear inverse problems in Banach spaces. To expedite convergence, the approach uses two search directions given by homotopy perturbation iteration, and the new iteration is calculated as the projection of the current iteration onto the intersection of stripes decided by above directions. The method allows to use L 1 -like penalty terms, which is significant to reconstruct sparsity solutions. Under reasonable conditions, we establish the convergence and regularization properties of the method. Finally, two parameter identification problems are presented to indicate the effectiveness of capturing the property of the sparsity solutions and the acceleration effect of the proposed method. [ABSTRACT FROM AUTHOR]

The modified transmission eigenvalue (MTE) problem was introduced in [S. Cogar, D. Colton, S. Meng and P. Monk, Modified transmission eigenvalues in inverse scattering theory, Inverse Problems 33 2017, 12, Article ID 125002] and used as a target signature for nondestructive testing. In this paper, we study the inverse spectral problem to reconstruct the modified transmission eigenvalues using Cauchy data. We propose a reciprocity gap functional method and show that the MTEs can be determined by solving some linear ill-posed integral equations. Numerical examples for both absorbing and non-absorbing media are presented to validate the effectiveness and robustness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper is devoted to the inverse problem of determining the spatially dependent source in a time fractional diffusion-wave equation, with the aid of extra measurement data at a subboundary. A uniqueness result is obtained by using the analyticity and the newly established unique continuation principle provided that the coefficients are all temporally independent. We also derive a Lipschitz stability of our inverse source problem under a suitable topology whose norm is given via the adjoint system of the fractional diffusion-wave equation. [ABSTRACT FROM AUTHOR]

This work presents an application of tensor field tomography for non-destructive reconstructions of axially symmetric residual stresses in a graded-index YAG single crystal for the case of beam deflection. The axis of the cylinder coincides with the crystallographic axis [001] of the single crystal and it has an axially symmetric refractive index distribution. The transformation of the polarization of light is measured in a plane orthogonal to the axis of the cylinder. Stresses are determined within the framework of the Maxwell piezo-optic law (linear dependence of the permittivity tensor on stresses) and small rotation of quasi principal stress axes. This paper generalizes the method of integrated photoelasticity for the case of ray deflection. [ABSTRACT FROM AUTHOR]

In this paper, the existence and uniqueness of a strong solution of the inverse problem of determining a coefficient of right-hand side of the integro-differential Kelvin–Voigt equation are investigated. The unknown coefficient that we search defends on space variables. Additional information on a solution of the inverse problem is given here as an integral overdetermination condition. The original inverse problem is reduced to study an equivalent inverse problem with homogeneous initial condition. Then the equivalences of the last inverse problem to an operator equation of second kind is proved. We establish the sufficient conditions for the unique solvability of the operator equation of second kind. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the stability of the system of partial differential equations modelling scalar image velocimetry. We first revisit a successful numerical technique to reconstruct velocity vectors u from images of a passive scalar field ψ by minimizing a cost functional that penalizes the difference between the reconstructed scalar field ϕ and the measured scalar field ψ, under the constraint that ϕ is advected by the reconstructed velocity field u , which again is governed by the Navier–Stokes equations. We investigate the stability of the reconstruction by applying this method to synthetic scalar fields in two-dimensional turbulence that are generated by numerical simulation. Then we present a mathematical analysis of the nonlinear coupled problem and prove that, in the two-dimensional case, smooth solutions of the Navier–Stokes equations are uniquely determined by the measured scalar field. We also prove a conditional stability estimate showing that the map from the measured scalar field ψ to the reconstructed velocity field u, on any interior subset, is Hölder continuous. [ABSTRACT FROM AUTHOR]

In this paper, we consider the iteratively regularized Gauss–Newton method with frozen derivative and formulate its convergence rates in the settings of Banach spaces. The convergence rates of iteratively regularized Gauss–Newton method with frozen derivative are well studied via generalized source conditions. We utilize the recently developed concept of conditional stability of the inverse mapping to derive the convergence rates. Also, in order to show the practicality of this paper, we show that our results are applicable on an ill-posed inverse problem. Finally, we compare the convergence rates derived in this paper with the existing convergence rates in the literature. [ABSTRACT FROM AUTHOR]

In this article, we consider an inverse problem for determining simultaneously a fractional order and a time-dependent source term in a multi-dimensional time-fractional diffusion-wave equation by a nonlocal condition. Based on a uniformly bounded estimate of the Mittag-Leffler function given in this paper, we prove the uniqueness of the inverse problem and the Lipschitz continuity properties for the direct problem. Then we employ the Levenberg–Marquardt method to recover simultaneously the fractional order and the time source term, and establish a finite-dimensional approximation algorithm to find a regularized numerical solution. Moreover, a fast tensor method for solving the direct problem in the three-dimensional case is provided. Some numerical results in one and multidimensional spaces are presented for showing the robustness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

The theory of inverse problems is an actively studied area of modern differential equation theory. This paper studies the solvability of the inverse problem for a linearized system of Navier–Stokes equations in a cylindrical domain with a final overdetermination condition. Our approach is to reduce the inverse problem to a direct problem for a loaded equation. In contrast to the well-known works in this field, our approach is to find an equation for a loaded term whose solvability condition provides the solvability of the original inverse problem. At the same time, the classical theory of spectral decomposition of unbounded self-adjoint operators is actively used. Concrete examples demonstrate that the assertions of our theorems naturally develop and complement the known results on inverse problems. Various cases are considered when the known coefficient on the right-hand side of the equation depends only on time or both on time and a spatial variable. Theorems establishing new sufficient conditions for the unique solvability of the inverse problem under consideration are proved. [ABSTRACT FROM AUTHOR]

In this paper, we consider the numerical method for an optimal control problem governed by an obstacle problem. An approximate optimization problem is proposed by regularizing the original non-differentiable constrained problem with a simple method. The connection between the two formulations is established through some convergence results. A sufficient condition is derived to decide whether a solution of the first-order optimality system is a global minimum. The method with a second-order in time dissipative system is developed to solve the optimality system numerically. Several numerical examples are reported to show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper we study a partial inverse spectral problem for non-self-adjoint Sturm–Liouville operators with a constant delay and show that subspectra of two boundary value problems with one common boundary condition are sufficient to determine the complex potential. We developed the Horváth's method in [M. Horváth, On the inverse spectral theory of Schrödinger and Dirac operators, Trans. Amer. Math. Soc. 353 2001, 10, 4155–4171] for the self-adjoint Sturm–Liouville operator without delay into the non-self-adjoint Sturm–Liouville differential operator with a constant delay. [ABSTRACT FROM AUTHOR]

In this paper, we consider Carleman estimates and inverse problems for the coupled quantitative thermoacoustic equations. In part I, we establish Carleman estimates for the coupled quantitative thermoacoustic equations by assuming that the coefficients satisfy suitable conditions and taking the usual weight function φ ⁢ (x , t) = e λ ⁢ ψ ⁢ (x , t) , ψ ⁢ (x , t) = | x - x 0 | 2 - β ⁢ | t - t 0 | 2 + β ⁢ t 0 2 for x in a bounded domain in ℝ n with C 3 -boundary and t ∈ (0 , T) , where t 0 = T 2 . We will discuss applications of the Carleman estimates to some inverse problems for the coupled quantitative thermoacoustic equations in the succeeding part II paper [M. Cristofol, S. Li and Y. Shang, Carleman estimates and inverse problems for the coupled quantitative thermoacoustic equations. Part II: Inverse problems, preprint 2020, https://hal.archives-ouvertes.fr/hal-02863385]. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the role of the damping term μ ⁢ u t in unique determination of unknown spatial load F ⁢ (x) in a damped Euler–Bernoulli beam equation u t ⁢ t + μ ⁢ u t + (r ⁢ (x) ⁢ u x ⁢ x ) x ⁢ x = F ⁢ (x) ⁢ G ⁢ (t) from the measured final time displacement u T ⁢ (x) := u ⁢ (x , T) or velocity ν T ⁢ (x) := u t ⁢ (x , T) . In [A. Hasanov Hasanoğlu and V. G. Romanov, Introduction to Inverse Problems for Differential Equations, Springer, Cham, 2017] it was shown that the unique determination of the spatial load F ⁢ (x) in the undamped wave equation u t ⁢ t - (k ⁢ (x) ⁢ u x) x = F ⁢ (x) ⁢ G ⁢ (t) from final data (displacement or velocity) is not possible. This result is also valid for the undamped beam equation u t ⁢ t + (r ⁢ (x) ⁢ u x ⁢ x ) x ⁢ x = F ⁢ (x) ⁢ G ⁢ (t) . We show that in the presence of the damping term, the spatial load can be uniquely determined from the final time output under some acceptable conditions with respect to the final time T > 0 and the damping coefficient μ > 0 . The cases of pure spatial load G ⁢ (t) ≡ 1 , and the exponentially decaying load G ⁢ (t) = e - η ⁢ t are analyzed separately, leading also to sufficient conditions for the uniqueness of the inverse problem solution. The results presented in this paper clearly demonstrate the key role of the damping term in the unique determination of the unknown spatial load F ⁢ (x) in the damped Euler–Bernoulli beam equation. [ABSTRACT FROM AUTHOR]