Class Of Ill-posed Problems Research Articles

Extract the information via multiple repeated observations under randomly distributed noise

Abstract 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 L^{2} norm is established, and some numerical examples are provided to illustrate the effectiveness and computational performance of the method.

PU-GAN: A One-Step 2-D InSAR Phase Unwrapping Based on Conditional Generative Adversarial Network

Two-dimensional phase unwrapping (PU) is a classical ill-posed problem in synthetic aperture radar interferometry (InSAR). The traditional algorithmic model-based 2-D PU methods are limited by the Itoh condition, which is from the PU researchers’ experience and has critical challenges under strong phase noises or violent phase changes. Recently, advanced learning-based 2-D PU methods could break through the limitation of the Itoh condition owing to their data-driven frameworks, offering promising results in terms of both the speed and accuracy. The one-step learning-based PU method, as one of the representatives, retrieves the unwrapped phase directly from the wrapped phase through regression. However, the main disadvantage of one-step learning-based PU is that it usually blurs the output unwrapped phase due to its <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L_{2}$ </tex-math></inline-formula> loss, that is, it cannot guarantee the congruency between the rewrapped interferometric fringes of the PU solution and the input interferogram. To solve this problem, we propose a one-step 2-D PU method based on the conditional generative adversarial network (referred to as PU-GAN), which treats 2-D PU as an image-to-image translation problem. The generator in PU-GAN can be trained to generate the unwrapped phase through minimizing a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L_{1}$ </tex-math></inline-formula> -norm loss based on a U-Net architecture, while simultaneously the corresponding discriminator can learn an adversarial loss by a structure of Patch-GAN that tries to classify if the output unwrapped phase image is real or fake. Both a theoretical analysis and the experimental results show that the proposed method outperforms the representative algorithmic model-based and learning-based 2-D PU methods.

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.

Inverse, ill-posed problems, essentially different solutions and explicit formulas for solutions

A request for an inverse problem, as well as for an incorrect problem produces tens of millions of answers in the Internet. In the past few decades, hundreds of international conferences on these topics have been held annually. Problems of this kind are quite involved, and their numerical analysis requires the development of special methods and numerical algorithms. Explicit formulas provide the main tool for testing these methods and numerical algorithms. The Cauchy problem for an elliptic equation is a classical ill-posed problem, which serves as a model for many inverse and incorrect problems. In the present paper we give a numerically realizable explicit formula for solving the Cauchy problem in a two-dimensional domain for a general second-order linear elliptic equation with analytic coefficients and the Cauchy analytic data on the analytic boundary.

Simplification method for nonlinear equations of monotone type in Banach space

Abstract. In a Banach space, we study an operator equation with a monotone operator T. The operator is an operator from a Banach space to its conjugate, and T=AC, where A and C are operators of some classes. The considered problem belongs to the class of ill-posed problems. For this reason, an operator regularization method is proposed to solve it. This method is constructed using not the operator T of the original equation, but a more simple operator A, which is B-monotone, B=C−1. The existence of the operator B is assumed. In addition, when constructing the operator regularization method, we use a dual mapping with some gauge function. In this case, the operators of the equation and the right-hand side of the given equation are assumed to be perturbed. The requirements on the geometry of the Banach space and on the agreement conditions for the perturbation levels of the data and of the regularization parameter are established, which provide a strong convergence of the constructed approximations to some solution of the original equation. An example of a problem in Lebesgue space is given for which the proposed method is applicable.

Sharp minimax distribution estimation for current status censoring with or without missing

Nonparametric estimation of the cumulative distribution function and the probability density of a lifetime X modified by a current status censoring (CSC), including cases of right and left missing data, is a classical ill-posed problem with biased data. The biased nature of CSC data may preclude us from consistent estimation unless the biasing function is known or may be estimated, and its ill-posed nature slows down rates of convergence. Under a traditionally studied CSC, we observe a sample from $(Z,\Delta )$ where a continuous monitoring time $Z$ is independent of $X$, $\Delta :=I(X\leq Z)$ is the status, and the bias of observations is created by the density of $Z$ which is estimable. In presence of right or left missing, we observe corresponding samples from $(\Delta Z,\Delta )$ or $((1-\Delta )Z,\Delta )$; the data are again biased but now the density of $Z$ cannot be estimated from the data. As a result, to solve the estimation problem, either the density of $Z$ must be known (like in a controlled study) or an extra cross-sectional sampling of $Z$, which is typically simpler than an underlying CSC study, be conducted. The main aim of the paper is to develop for this biased and ill-posed problem the theory of efficient (sharp-minimax) estimation which is inspired by known results for the case of directly observed $X$. Among interesting aspects of the developed theory: (i) While sharp-minimax analysis of missing CSC may follow the classical Pinsker’s methodology, analysis of CSC requires a more complicated estimation procedure based on a special smoothing in both frequency and time domains; (ii) Efficient estimation requires solving an old-standing problem of approximating aperiodic Sobolev functions; (iii) If smoothness of the cdf of $X$ is known, then its rate-minimax estimation is possible even if the density of $Z$ is rougher. Real and simulated examples, as well as extensions of the core models to dependent $X$ and Z and case-control CSC, are presented.

Energy Activation Spectrum of Low-Temperature Acoustic Relaxation in High-Purity Iron Single Crystal. Solution of the Inverse Problem of Mechanical Spectroscopy by the Tikhonov Regularization Method

When studying the temperature dependences of the acoustic absorption and the modulus of elasticity, absorption peaks are often observed, which correspond to the characteristic step on the temperature dependence of the modulus of elasticity. Such features are called relaxation resonances. It is believed that the occurrence of such relaxation resonances is due to the presence in the structure of the material of elementary microscopic relaxors that interact with the studied vibrational mode of mechanical vibrations of the sample. In a sufficiently perfect material, such a process is characterized by a relaxation time τ, and in a real defective material by a relaxation time spectrum P(τ). Most often such relaxation processes have a thermally activated character and the relaxation time τ(T) is determined by the Arrhenius ratio τ(T)=τ0exp(U0/kT), and the characteristics of the process will be U0 - activation energy, τ0 - period of attempts, Δ0 - characteristic elementary contribution of a single relaxator to the dynamic response of the material and their spectra. In the low temperatures region the statistical distribution of parameters τ0 and Δ0 can be neglected with exponential accuracy, and the relaxation contribution to the temperature dependences of absorption and the dynamic elasticity modulus of the material will be determined only by the activation energy spectrum P(U) of microscopic relaxors. The main task of mechanical spectroscopy in the analysis of such relaxation resonances is to determine U0, τ0, Δ0 and P(U). It is shown, that the problem of recovering of spectral function P(U) of acoustic relaxation of a real crystal can be reduced to the solving of the Fredholm integral equation of the first kind with an approximately known right part and concerns to a class of ill-posed problems. The method based on Tikhonov regularizing algorithm for recovering P(U) from experimental temperature dependences of absorption or elasticity module is offered. It is established, that acoustic relaxation in high-purity iron single crystal in the temperature range 5-100 K is characterized by two-modes spectral function P(U) with maxima at 0.037 eV and 0.015 eV, which correspond to the a-peak and its a' satellite.

COMBINED PATCH-WISE MINIMAL-MAXIMAL PIXELS REGULARIZATION FOR DEBLURRING

Abstract. Deblurring is a vital image pre-processing procedure to improve the quality of images. It is a classical ill-posed problem. A new blind deblurring method based on image sparsity prior is proposed here. The proposed image sparsity prior combines patch-wise minimal and maximal pixels of latent image, and improves gradually the image sparsity during deblurring. An algorithm that is different with half quadratics splitting algorithm is applied under the maximum a posterior (MAP) framework. Experiment results demonstrate that the proposed method can keep more subtle texture and sharpened edges, reduce the artefacts in visual, and the corresponding evaluated indexes perform favourably against it of the state-of-the-art methods on synthesized, natural and remote sensing images (RSI) quantitatively.

On quasi-reversibility solutions to the Cauchy problem for the Laplace equation: regularity and error estimates

We are interested in the classical ill-posed Cauchy problem for the Laplace equation. One method to approximate the solution associated with compatible data consists in considering a family of regularized well-posed problems depending on a small parameterε &gt; 0. In this context, in order to prove convergence of finite elements methods, it is necessary to get regularity results of the solutions to these regularized problems which hold uniformly inε. In the present work, we obtain these results in smooth domains and in 2D polygonal geometries. In presence of corners, due to the particular structure of the regularized problems, classical techniquesà laGrisvard do not work and instead, we apply the Kondratiev approach. We describe the procedure in detail to keep track of the dependence inεin all the estimates. The main originality of this study lies in the fact that the limit problem is ill-posed in any framework.

Estimates of efficiency for two methods of stable numerical summation of smooth functions

We consider a classical ill-posed problem of reconstruction of continuous functions from their noisy Fourier coefficients. We study the case of functions of two variables that has been much less investigated. The smoothness of reconstructed functions is measured in terms of the Sobolev classes as well as the classes of functions with dominated mixed derivatives. We investigate two summation methods, that are based on ideas of the rectangle and the hyperbolic cross, respectively. For both of these methods we establish the estimates of the accuracy on the classes that are considered as well the estimates of computational costs. Moreover, we made the comparison of their efficiency based on obtained estimates. A somehow surprising outcome of our study is that for both types of the considered smoothness classes one should employ hyperbolic cross approximation that is not typical for the functions under consideration.

Construction and investigation of a method for measuring the non-stationary pressure using a wavelet transform

Automated control systems badly need measurements of fast-changing non-stationary physical quantities in real time, or close to that. In this area, there is a separate group of tasks on measuring the non-stationary pressure in liquids and gases. This paper demonstrates that measuring the non-stationary pressure in real time, or close to that, represents a problem on restoring an input signal, which, in terms of mathematics, belongs to the class of ill-posed problems (according to J. Hadamard). We have derived a solution to the inverse problem of measurement that is based on a mathematical model for measuring transformation enabled by a pressure sensor. Based on this solution, we have constructed a measuring method, which implies the wavelet processing of the sensor's output signal. In this case, we suggest that such basic functions of wavelet transformation should be selected that are the modification of the pulse transition function of the sensor. This paper reports an experimental study into the feasibility of the developed method, based on the measurement of the simulated pressure pulse. A pressure pulse is simulated by dropping a ball of the calibrated mass onto the sensor's membrane. We have proposed a measurement scheme for determining the duration of touch between the ball and the membrane. Testing the accuracy of the method implies comparing the actual mass of the ball with that derived from the sensor's output signal. The proposed method has demonstrated high accuracy because the maximum relative error in determining the mass of the falling ball was only 0.65 %. The proposed method for measuring the non-stationary pressure could be used in control systems that require the high-speed dynamic correction of a measurement error. Specifically, these include control system in aerospace engineering, testing complexes, military technology, scientific research

On an asymmetric backward heat problem with the space and time-dependent heat source on a disk

Abstract In this article, we investigate the backward heat problem (BHP) which is a classical ill-posed problem. Although there are many papers relating to the BHP in many domains, considering this problem in polar coordinates is still scarce. Therefore, we wish to deal with this problem associated with a space and time-dependent heat source in polar coordinates. By modifying the quasi-boundary value method, we propose the stable solution for the problem. Furthermore, under some initial assumptions, we get the Hölder type of error estimates between the exact solution and the approximated solution. Eventually, a numerical experiment is provided to prove the effectiveness and feasibility of our method.

One Approach to the Comparison of Error Bounds at a Point and on a Set in the Solution of Ill-Posed Problems

The approximate solution of ill-posed problems by the regularization method always involves the issue of estimating the error. It is a common practice to use uniform bounds on the whole class of well-posedness in terms of the modulus of continuity of the inverse operator on this class. Local error bounds, which are also called error bounds at a point, have been studied much less. Since the solution of a real-life ill-posed problem is unique, an error bound obtained on the whole class of well-posedness roughens to a great extent the true error bound. In the present paper, we study the difference between error bounds on the class of well-posedness and error bounds at a point for a special class of ill-posed problems. Assuming that the exact solution is a piecewise smooth function, we prove that an error bound at a point is infinitely smaller than the exact bound on the class of well-posedness.

Stable iterative Lagrange principle in convex programming as a tool for solving unstable problems

A convex programming problem in a Hilbert space with an operator equality constraint and a finite number of functional inequality constraints is considered. All constraints involve parameters. The close relation of the instability of this problem and, hence, the instability of the classical Lagrange principle for it to its regularity properties and the subdifferentiability of the value function in the problem is discussed. An iterative nondifferential Lagrange principle with a stopping rule is proved for the indicated problem. The principle is stable with respect to errors in the initial data and covers the normal, regular, and abnormal cases of the problem and the case where the classical Lagrange principle does not hold. The possibility of using the stable sequential Lagrange principle for directly solving unstable optimization problems is discussed. The capabilities of this principle are illustrated by numerically solving the classical ill-posed problem of finding the normal solution of a Fredholm integral equation of the first kind.

A wavelet method for numerical fractional derivative with noisy data

Numerical fractional differentiation is a classical ill-posed problem in the sense that a small perturbation in the data can cause a large change in the fractional derivative. In this paper, we consider a wavelet regularization method for solving a reconstruction problem for numerical fractional derivative with noise. A Meyer wavelet projection regularization method is given, and the Hölder-type stability estimates under both apriori and aposteriori regularization parameter choice rules are obtained. Some numerical examples show that the method works well.

Recovery of function from noisy Fourier coefficients with respect to the Haar system

We consider the classical ill-posed problem of recovering of a function from noisy Fourier coefficients with respect to the Haar system. For the classes of functions given in terms of generalized smoothness, we present a priori and a posteriori regularization parameter choice realizing an order-optimal error bound.

The Generalized Tikhonov Regularization Method forHigh Order Numerical Derivatives

Numerical differentiation is a classical ill-posed problem. The generalized Tikhonov regularization method is proposed to solve this problem. The error estimates are obtained for a priori and a posteriori parameter choice rules, respectively. Numerical examples are presented to illustrate the validity and effectiveness of this method.

Spectral method for ill-posed problems based on the balancing principle

In this article, we consider a class of ill-posed problems in two-dimensional setting which can be formulated as the problem of solving ill-posed operator equation. The spectral method is a simple and effective regularization method for such a class of ill-posed problems. For the spectral method, we obtain the error estimates with optimal order under an adaptive a posteriori parameter choice rule called balancing principle. Two ill-posed examples including an inverse diffraction problem and a deblurring problem are provided.

REALISTIC AND CORRECT MODELS OF IMPRESSED SOURCES FOR TIME-HARMONIC ELECTROMAGNETIC BOUNDARY VALUE PROBLEMS INVOLVING METAMATERIALS

The aim of this work is to analyze the role of the impressed sources in determining the well or ill-posedness of time harmonic electromagnetic boundary value problems involving isotropic effective media. It is shown, in particular, that, even if all interfaces are regular, the class of ill-posed problems can be very large in the presence of general square-integrable impressed sources. However, when a simple and realistic constraint is enforced on these sources, requiring that the support of the sources does not include any interface between a traditional medium and a metamaterial, among the problems here considered just those involving an interface between complementary materials remain ill-posed. These considerations have a very significant impact also on the approximability of the solution of well-posed problems since the numerical noise can introduce small fictitious sources even where the sources to be simulated are not present. These effects on finite element simulators are fully analyzed. Finally, we propose an algorithm that allows to obtain much better approximations of the solutions of the most critical well-posed problems.

Numerical differentiation by a Tikhonov regularization method based on the discrete cosine transform

In this article, we discuss a classical ill-posed problem on numerical differentiation by a Tikhonov regularization method based on the discrete cosine transform (DCT). After implementing an eigenvalue decomposition of the second-order difference matrix in the penalty we obtain an explicit minimizer of the Tikhonov functional in a linear combination of DCT-2 components. The choice of the regularization parameter thus can be properly chosen after calling a single variable bounded nonlinear function minimization. Moreover, we propose two approaches to weaken the Gibbs phenomena appearing near the boundary of the reconstructed derivatives. Several numerical examples are provided to show the computational efficiency of our proposed algorithm. Finally, the extension to the multidimensional numerical differentiation leads the approach to a wider scope.