Ill-posed Model Research Articles

An Enhanced Fluid Registration for Image Multi-Frame Super Resolution

The main idea of multi-frame super resolution (SR) algorithms is to recover a single high-resolution image from a sequence of low resolution ones of the same object. The success of the SR approaches is often related to a well registration and restoration steps. Therefore, we propose a new approach based on fluid image registration and we use a second order partial differential equation (PDE) to treat both the registration and restoration steps that guarantees the success of SR algorithms. Since the registration step is usually a variational ill-posed model, a mathematical study is needed to check the existence of the solution to the regularized problem. Thus, we prove the existence and uniqueness of the well posed fluid image registration and assure also the existence of the used second order PDE in the restoration step. The results show that the proposed method is competitive with the existing methods.

An improved PDE-constrained optimization fluid registration for image multi-frame super resolution

The main idea of multi-frame super resolution (SR) algorithms is to recover a single high-resolution image from a sequence of low resolution ones of the same object. The success of the SR approaches is often related to a well registration and restoration steps. Therefore, we propose a new approach based on a partial differential equation (PDE)-constrained optimization fluid image registration and we use a fourth order PDE to treat both the registration and restoration steps that guarantee the success of SR algorithms. Since the registration step is usually a variational ill-posed model, a mathematical study is needed to check the existence of the solution to the regularized problem. Thus, we prove the existence and of the well posed fluid image registration and assure also the existence of the used second order PDE in the restoration step. The results show that the proposed method is competitive with the existing methods.

The hypothesis testing statistics in linear ill-posed models

In the geodetic community, an adjustment framework is established by the four components of model choice, parameter estimation, variance component estimation (VCE), and quality control. For linear ill-posed models, the parameter estimation and VCE have been extensively investigated. However, at least to the best of our knowledge, nothing is known about the quality control of hypothesis testing in ill-posed models although it is indispensable and rather important. In this paper, we extend the theory of hypothesis testing to ill-posed models. As the Tikhonov regularization is typically applied to stabilize the solution of ill-posed models, the solution and its associated residuals are biased. We first derive the statistics of overall-test, w-test and minimal detectable bias for an ill-posed model. Due to the bias influence, both overall-test and w-test statistics do not follow the distributions as those used in well-posed models. Then, we develop the bias-corrected statistics. As a result, the bias-corrected w-test statistic can sufficiently be approximated by a standard normal distribution; while the bias-corrected overall-test statistic can be approximated by two non-central chi-square distributions, respectively. Finally, some numerical experiments with a Fredholm first kind integral equation are carried out to demonstrate the performance of our designed hypothesis testing statistics in an ill-posed model.

Bias Reduction Method for Parameter Inversion of Ill-Posed Surveying Model

AbstractBecause of ill-posed problems, inverted parameters of ill-posed models always have large variances. The regularization method is widely used to solve this problem. Based on the mean square ...

Solution of the Ill-Posed Semiparametric Regression Model Based on Singular Value Modification Restriction

We propose a solution of the ill-posed semi-parametric regression model based on singular value modification restriction, aimed at the ill-posed problem of the normal matrix which may occur in the process of solving the semiparametric regression model. First, the coefficient matrix is decomposed into singular values, and the smaller singular values are selected according to the criterion ∑i=1r1/σi/∑i=1n1/σi≤5% (in the singular value matrix, σ1>σ1>⋯>σr>⋯σn). Second, the relatively smaller singular values are modified by the biased parameter to suppress the magnification of the estimated variance so as to effectively reduce the variance of parameter estimation, reduce the introduction of deviation and obtain more reliable parameter estimation. The results of the numerical experiments show that the improved singular value modification restriction method can not only overcome the effect of the ill-posed normal matrix on the parameter estimation solution but also correctly separate the systematic errors and improve the accuracy of semiparametric regression model calculation results.

Ill posedness in modelling two-dimensional morphodynamic problems: effects of bed slope and secondary flow

A two-dimensional model describing river morphodynamic processes under mixed-size sediment conditions is analysed with respect to its well posedness. Well posedness guarantees the existence of a unique solution continuously depending on the problem data. When a model becomes ill posed, infinitesimal perturbations to a solution grow infinitely fast. Apart from the fact that this behaviour cannot represent a physical process, numerical simulations of an ill-posed model continue to change as the grid is refined. For this reason, ill-posed models cannot be used as predictive tools. One source of ill posedness is due to the simplified description of the processes related to vertical mixing of sediment. The current analysis reveals the existence of two additional mechanisms that lead to model ill posedness: secondary flow due to the flow curvature and the effect of gravity on the sediment transport direction. When parametrising secondary flow, accounting for diffusion in the transport of secondary flow intensity is a requirement for obtaining a well-posed model. When considering the theoretical amount of diffusion, the model predicts instability of perturbations that are incompatible with the shallow water assumption. The effect of gravity on the sediment transport direction is a necessary mechanism to yield a well-posed model, but not all closure relations to account for this mechanism are valid under mixed-size sediment conditions. Numerical simulations of idealised situations confirm the results of the stability analysis and highlight the consequences of ill posedness.

Construction of a Non-negative Solution of the Leontief-Ford Model

The article deals with the Leontief-Ford model, which is allowed to take into account the costs that necessary for the elimination of industrial waste. We consider its non-negative solution for those cases when the errors of the initial data have little influence on the result of the solution and for those cases when the errors of the initial data significantly affect the result of the solution. In the first case, it is called correctly delivered, and in the second incorrectly delivered. We use the iteration method to find a nonnegative solution to a correctly posed model, and also we use the regularization method to find a nonnegative solution to an ill-posed model. The developed technique is brought to a practical algorithm, which is implemented in the program LF_2, it allows you to find a solution to the model, regardless of whether the model is correctly supplied or incorrectly supplied. An example of its application is given, where we enter the initial data of a poorly conditioned balance model into the program and obtain its non-negative solution. The results of this work may be interesting for specialists in economics and mathematical methods and models, as well as economic entities. It is known from practice that in the development of balance models of economic entities there are models with a bad number of conditionality. The application of the developed software product and methods described in this paper will allow economic entities to make quality and quick management decisions on the volume of product output of each sector of the economy, taking into account the costs necessary for the elimination of industrial waste, regardless of the conditionality and dimension of their balance models.

In-Sample Inference and Forecasting in Misspecified Factor Models

This article considers in-sample prediction and out-of-sample forecasting in regressions with many exogenous predictors. We consider four dimension-reduction devices: principal components, ridge, Landweber Fridman, and partial least squares. We derive rates of convergence for two representative models: an ill-posed model and an approximate factor model. The theory is developed for a large cross-section and a large time-series. As all these methods depend on a tuning parameter to be selected, we also propose data-driven selection methods based on cross-validation and establish their optimality. Monte Carlo simulations and an empirical application to forecasting inflation and output growth in the U.S. show that data-reduction methods outperform conventional methods in several relevant settings, and might effectively guard against instabilities in predictors’ forecasting ability.

On the time differential dual-phase-lag thermoelastic model

This paper studies the time differential dual-phase-lag model of a thermoelastic material, where the elastic deformation is accompanied by thermal effects governed by a time differential equation for the heat flux with dual phase lags. This coupling gives rise to a complex differential system requiring a special treatment. Uniqueness and continuous dependence results are established for the solutions of the mixed initial boundary value problems associated with the model of the linear theory of thermoelasticity with dual-phase-lag for an anisotropic and inhomogeneous material. Two methods are developed in this paper, both being based on an identity of Lagrange type and of a conservation law applied to appropriate initial boundary value problems associated with the model in concern. The uniqueness results are established under mild constitutive hypotheses (right like those in the classical linear thermoelasticity), without any restrictions upon the delay times (excepting the class of thermoelastic materials for which the delay time of phase lag of the conductive temperature gradient is vanishing and the delay time in the phase lag of heat flux vector is strictly positive, when an ill-posed model should be expected). The continuous dependence results are established by using a conservation law and a Gronwall inequality, under certain constitutive restrictions upon the thermoelastic coefficients and the delay times.

Simultaneous recovery of the temperature and species concentration from integral equation model

Absorption spectroscopy is an advanced tool for flow diagnostics in measuring multiple parameters of species. Such kinds of problems can be modeled by some integral equations with known kernel, aiming to the determination of the integrands from their integration values along all possible paths of injected lasers. This paper considers the parameters detection problems in combustion process, with the purpose of recovering the gas temperature and the concentration of burned gas simultaneously using injected lasers along two directions with multiple frequencies. After establishing the nonlinear integral equations describing the energy absorption process, this ill-posed model is transformed into a nonlinear optimization problem with some penalty terms. Then we present an alternative iteration scheme (AIS) to solve this problem. The convergence of the iterative sequence for AIS algorithm together with the estimate on the value of cost functional is established, ensuring that AIS can indeed generate a satisfactory approximate solution to the original optimization problem. Numerical implementations using simulant data are presented to show the validity of the proposed scheme.

A new operational matrix method based on the Bernstein polynomials for solving the backward inverse heat conduction problems

Purpose – The purpose of this paper is to introduce a novel approach based on the high-order matrix derivative of the Bernstein basis and collocation method and its employment to solve an interesting and ill-posed model in the heat conduction problems, homogeneous backward heat conduction problem (BHCP). Design/methodology/approach – By using the properties of the Bernstein polynomials the problems are reduced to an ill-conditioned linear system of equations. To overcome the unstability of the standard methods for solving the system of equations an efficient technique based on the Tikhonov regularization technique with GCV function method is used for solving the ill-condition system. Findings – The presented numerical results through table and figures demonstrate the validity and applicability and accuracy of the technique. Originality/value – A novel method based on the high-order matrix derivative of the Bernstein basis and collocation method is developed and well-used to obtain the numerical solutions of an interesting and ill-posed model in heat conduction problems, homogeneous BHCP with high accuracy.

Three-dimensional inversion of CSEM data: Water leak detection using a small-loop EM method

Abstract Electromagnetic (EM) data sets were acquired using ground-based small-loop electromagnetic (EM) sensors over an artificially constructed sandy ground in Tsukuba, Japan, for three conditions of water saturation. The research aim was to evaluate the imaging capability of the geophysical method for detecting a water leak. Because the sensitivity analysis showed a considerable three-dimensional (3D) distribution of sensitivity around the source–receiver location, 3D inversion was applied to the field data sets. We used an accurate 3D controlled-source electromagnetic (CSEM) inversion algorithm, which adopted an edge-based finite element method as a forward scheme. To obtain the best inversion strategy for our field data and for the selection of the optimal regularization method, 3D inversion was applied to synthetic data sets prior to the inversion of the field data; the synthetic data sets were based on synthetic resistivity models used to simulate the three water-saturation phases of the real test site. Based on the results of sensitivity analysis, different regularization methods were systematically tested and the inversion results were qualitatively analyzed. For our monitoring scenario, the best performance was achieved by combining spatially varying regularization parameters, structural constraints, and the reference model method. The reference model parameters were derived from the less ill-posed model, instead of the main target model, which allowed for more accurate imaging of the main region of interest. This optimal inversion strategy was applied to the test-site field data. The water leak was successfully visualized by small-loop EM data, with systematic application of a priori information as constraints.

The Instability of Optimal Control Problems to Time Delay

This paper considers a general optimal control problem with a small time delay $\Delta$ in both the state and control. The objective is to find a correct way to neglect the delay or, in other words, to construct a well-posed instantaneous approximation of the delayed problem. The natural way to do so is by merely ignoring the delay in the model. It is shown that this way is almost always incorrect: it gives rise to an error that does not vanish as $\Delta \to +0$, and so results in an ill-posed model. A proper approximation of the delayed problem is offered.

Adaptive Strategy for the Damping Parameters in an Iteratively Regularized Gauss–Newton Method

In this paper, a convergence analysis of an adaptive choice of the sequence of damping parameters in the iteratively regularized Gauss–Newton method for solving nonlinear ill-posed operator equations is presented. The selection criterion is motivated from the damping parameter choice criteria, which are used for the efficient solution of nonlinear least-square problems. The performance of this selection criterion is tested for the solution of nonlinear ill-posed model problems.

Models for thin viscous sheets

Leading-order equations governing the dynamics of a two-dimensional thin viscous sheet are derived. The inclusion of inertia effects is found to result in an ill-posed model when the sheet is compressed, and the resulting paradox is resolved by rescaling the equations over new length-and timescales which depend on the Reynolds number of the flow and the aspect ratio of the sheet. Physically this implies a dominant lengthscale for transverse displacements during viscous buckling. The theory is generalized to give new models for fully three-dimensional sheets.

Study of Numerical Solution Generated by an Ill-Posed Two-Fluid Model.

Two-phase flow dynamics plays an important role in understanding the kinetics of nuclear reactors and boilers. Difficulty has been reported in obtaining the numerical solution of two-phase flow by a two-fluid model which was described as ill-posed. In the present paper, all analysis is performed for an ill-posed nonlinear partial differential equation which can be a reasonable simplified alternative to the two-fluid model. We attempt to gain a practical understanding of the ill posedness by numerical implementation of the present equation with the finite difference method.