Ill-posed Cases Research Articles

Uniqueness of solutions in multivariate Chebyshev approximation problems

We study the solution set to multivariate Chebyshev approximation problem, focussing on the ill-posed case when the uniqueness of solutions can not be established via strict polynomial separation. We obtain an upper bound on the dimension of the solution set and show that nonuniqueness is generic for ill-posed problems on discrete domains. Moreover, given a prescribed set of points of minimal and maximal deviation we construct a function for which the dimension of the set of best approximating polynomials is maximal for any choice of domain. We also present several examples that illustrate the aforementioned phenomena, demonstrate practical application of our results and propose a number of open questions.

A mollification regularization method with Dirichlet kernel to solve potential-free field inverse Schrödinger Cauchy problem

We consider solving the Cauchy problem of the Schrödinger equation with potential-free field by a mollification regularization method in this work. By convolving the measured data with the Dirichlet kernel, the ill-posed case is turned into a well-posed one. Convergent estimates are gained via the priori and the posteriori parameter selection rules. Finally, three simulation experiment results are shown to prove the feasibility and stability of our presented procedure.

A Non-Iteration Solution for Solving the Backward-in-Time Two-Dimensional Burgers’ equation with a Large Reynolds Number

This article proposes a noniteration solution based on the Lie-group shooting method (LGSM) to solve the backward-in-time two-dimensional Burgers’ equation with a large Reynolds number. The backward problem is famous for seriously ill-posed cases because the solution is generally unstable and highly dependent on the input data. Small perturbations in the input data, such as random errors inherent to the measurements in the analysis, can cause large oscillations in the solution. To handle a large Reynolds number under long time spans, it is very difficult to integrate towards the time direction. To avoid time integration and numerical iteration, the noniteration vector solution based on a two-point equation of the LGSM, including the initial and final conditions and boundary conditions (BCs) at the initial and terminal times, can be constructed. When the vector solution can be obtained from the ratios of the wave fronts on the BCs at the initial and terminal times, this solver can avoid the numerical iteration and numerical divergence of the conventional LGSM. Two benchmark examples in one and two variables are examined to illustrate the performance of the proposed method. The numerical results of this research are very consistent with the exact solutions when considering disturbances from noisy data. Even when the Reynolds number reaches 10E12, from the noisy final and boundary data, the noniteration solution can efficiently address the nonlinear Burgers’ problem with or without disturbances. This method does not use any transformation techniques, iterative processes, or regularization processes to avoid numerical instability. Hence, a noniterative solution is more stable and accurate for the unsteady nonlinear Burgers’ equation than currently used methods.

MAP Estimation With Bernoulli Randomness, and Its Application to Text Analysis and Recommender Systems

MAP estimation plays an important role in many probabilistic models. However, in many cases, the MAP problem is non-convex and intractable. In this work, we propose a novel algorithm, called BOPE, which uses Bernoulli randomness for Online Maximum a Posteriori Estimation. We show that BOPE has a fast convergence rate. In particular, BOPE implicitly employs a prior which plays as regularization. Such a prior is different from the one of the MAP problem and will be vanishing as BOPE does more iterations. This property of BOPE is significant and enables to reduce severe overfitting for probabilistic models in ill-posed cases, including short text, sparse data, and noisy data. We validate the practical efficiency of BOPE in two contexts: text analysis and recommender systems. Both contexts show the superior of BOPE over the baselines.

Reconstruction Algorithms for Photoacoustic Tomography in Heterogeneous Damping Media

In this article, we study several reconstruction methods for the inverse source problem of photoacoustic tomography with spatially variable sound speed and damping. The backbone of these methods is the adjoint operators, which we thoroughly analyze in both the $$L^2$$ - and $$H^1$$ -settings. They are casted in the form of a nonstandard wave equation. We derive the well posedness of the aforementioned wave equation in a natural functional space and also prove the finite speed of propagation. Under the uniqueness and visibility condition, our formulations of the standard iterative reconstruction methods, such as Landweber’s and conjugate gradients (CG), achieve a linear rate of convergence in either $$L^2$$ - or $$H^1$$ -norm. When the visibility condition is not satisfied, the problem is severely ill posed and one must apply a regularization technique to stabilize the solutions. To that end, we study two classes of regularization methods: (i) iterative and (ii) variational regularization. In the case of full data, our simulations show that the CG method works best; it is very fast and robust. In the ill-posed case, the CG method behaves unstably. Total variation regularization method (TV), in this case, significantly improves the reconstruction quality.

Insights on Asymptotic Behavior of Characteristic Roots of Quasi-Polynomials with Delay-Dependent Coefficients in Some Ill-Posed Cases

This work focus on the asymptotic behavior of critical roots of linear time-delay systems with delay-dependent coefficients. Our main goal is to derive a simple methodology to characterize some ill-posed cases from the open literature. In order to accomplish such a task, our main tool will be to express the solutions as a convergent power series. More specifically, by means of an appropriate change of variable, the expansion of unbounded solutions is obtained, which allows to derive an appropriate auxiliary solution that serves to analyze the behavior of such solutions. One of the main benefits in adopting such an approach, is that stability analysis in the ill-posed cases can also be covered.

Chopping Time of the FPU $${\alpha }$$ α -Model

We study, both numerically and analytically, the time needed to observe the breaking of an FPU $\alpha$-chain in two or more pieces, starting from an unbroken configuration at a given temperature. It is found that such a "chopping" time is given by a formula that, at low temperatures, is of the Arrhenius-Kramers form, so that the chain does not break up on an observable time-scale. The result explains why the study of the FPU problem is meaningful also in the ill-posed case of the $\alpha$-model

Coronal loop density profile estimated by forward modelling of EUV intensity

Aims. The transverse density structuring of coronal loops was recently calculated for the first time using the general damping profile for kink oscillations. This seismological method assumes a density profile with a linear transition region. We consider to what extent this density profile accounts for the observed intensity profile of the loop, and how the transverse intensity profile may be used to complement the seismological technique.Methods. We use isothermal and optically transparent approximations for which the intensity of extreme ultraviolet (EUV) emission is proportional to the square of the plasma density integrated along the line of sight. We consider four different models for the transverse density profile; the generalised Epstein profile, the step function, the linear transition region profile, and a Gaussian profile. The effect of the point spread function is included and Bayesian analysis is used for comparison of the models.Results. The two profiles with finite transitions are found to be preferable to the step function profile, which supports the interpretation of kink mode damping as being due to mode coupling. The estimate of the transition layer width using forward modelling is consistent with the seismological estimate.Conclusions. For wide loops, that is those observed with sufficiently high spatial resolution, this method can provide an independent estimate of density profile parameters for comparison with seismological estimates. In the ill-posed case of only one of the Gaussian or exponential damping regimes being observed, it may provide additional information to allow a seismological inversion to be performed. Alternatively, it may be used to obtain structuring information for loops that do not oscillate.

Patch Ordering as a Regularization for Inverse Problems in Image Processing

Recent work in image processing suggests that operating on (overlapping) patches in an image may lead to state-of-the-art results. This has been demonstrated for a variety of problems including denoising, inpainting, deblurring, and super-resolution. The work reported in [I. Ram, I. Cohen, and M. Elad, IEEE Trans. Image Process., 23 (2014), pp. 2779--2792] and [I. Ram, M. Elad, and I. Cohen, IEEE Trans. Image Process., 22 (2013), pp. 2764--2774] takes an extra step forward by showing that ordering these patches to form an approximate shortest path can be leveraged for better processing. The core idea is to apply a simple filter on the resulting 1D smoothed signal obtained after the patch-permutation. This idea has been also explored in combination with a wavelet pyramid, leading eventually to a sophisticated and highly effective regularizer for inverse problems in imaging. In this work we further study the patch-permutation concept and harness it to propose a new simple yet effective regularization for image restoration problems. Our approach builds on the classic maximum a posteriori probability (MAP), with a penalty function consisting of a regular log-likelihood term and a novel permutation-based regularization term. Using a plain 1D Laplacian, the proposed regularization forces robust smoothness ($L1$) on the permuted pixels. Since the permutation originates from patch ordering, we propose to accumulate the smoothness terms over all of the patches' pixels. Furthermore, we take into account the found distances between adjacent patches in the ordering by weighting the Laplacian outcome. We demonstrate the proposed scheme on the following diverse set of problems: (i) severe Poisson image denoising, (ii) Gaussian image denoising, (iii) image deblurring, and (iv) single image super-resolution. In all of these cases, we use recent methods that handle these problems as initialization to our scheme. This is followed by an L-BFGS optimization of the above-described penalty function, leading to state-of-the-art results, especially for highly ill-posed cases.

Maize Leaf Area Index Retrieval from Synthetic Quad Pol SAR Time Series Using the Water Cloud Model

In order to monitor crop growth along the season with synthetic aperture radar (SAR) images, radiative transfer models were developed to retrieve key biophysical parameters, such as the Leaf Area Index (LAI). The semi-empirical water cloud model (WCM) can be used to estimate LAI values from SAR data and surface soil moisture information. Nevertheless, instability problems can occur during the model calibration, which subsequently reduce its transferability in both time and space. To avoid these ill-posed cases, three calibration methodologies are benchmarked in the present study. The accuracy of the retrieved LAI values for each methodology was analyzed, as well as the sensitivity of the signal to LAI for different soil moisture values. The sensitivity of the cross-polarization was highlighted especially for high LAI. The VV polarization was found sensitive for LAI values inferior to 2 m 2 /m 2 . Given the differential sensitivity of the C-band backscatter to maize canopies in each polarization, a Bayesian fusion of the LAI estimates in linear polarizations was developed. This fusion gives lower weights to estimates with a high uncertainty. This method systematically reduces the error and its associated variance. When considering all polarizations, the RMSE on LAI estimation decreased by 0.32 m 2 /m 2 , i.e., one fourth of the error value, as compared to the best estimation from a single polarization, and the associated uncertainty was reduced by a factor of two. Focusing on the two most sensitive polarizations to maize canopies (VV-HV), the error diminished by a third. This fusion framework shows thus a great potential to improve the accuracy and reliability of LAI retrieval of C-band quad-polarized data, as well as dual-polarized data, such as Sentinel-1.

A BIEM using the Trefftz test functions for solving the inverse Cauchy and source recovery problems

In this paper we develop a global domain/boundary integral equation method for the Laplace and Poisson equations, which is based on the Green׳s second identity. A derived global relation links the source term to the Dirichlet and Neumann boundary conditions into a single integral equation in terms of the Trefftz test functions. By suitably choosing the Trefftz test functions, which are not the usual Green functions as that used in the conventional boundary integral method, the present boundary integral equation method (BIEM) can find the unknown boundary conditions for the inverse Cauchy problems very well. Even under a large noise to 10% and the data over-specified in a 25% portion of the whole boundary, the recovered result is still accurate. The inverse source problems of the Poisson equation are resolved numerically by using the BIEM which is stable and effective for strongly ill-posed case with a large noise being imposed on the supplementary data.

Making use of a partial order in solving inverse problems: II.

Mathematical formulations of applied inverse problems often involve operator equations in normed functional spaces. In many cases, these spaces can, in addition, be endowed with a partial order relation, which turns them into Banach lattices. The fact that two tools, such as a partial order relation and a monotone (with respect to this partial order) norm, are available to the researchers, gives them a clear advantage of having more freedom in the problem formulations. For instance, errors in the approximate data are sometimes easier to describe in terms of pointwise bounds. Inverse problems in partially ordered normed spaces (Banach lattices) have been studied before in the case when a compact set containing the unknown exact solution is available a priori. It turned out that under this assumption it is possible, even in the ill-posed case, to compute ‘pointwise’ bounds for the unknown exact solution (or rather, bounds by means of the appropriate partial order), thus providing an error estimate by means of the partial order. Also, a useful property of this approach was that one was able to quantify the uncertainty in the operator by means of linear inequalities that were included in the corresponding optimization problems as (linear) constraints and made the computations easier. However, the compactness assumption might sometimes be too demanding in practice. This paper aims at revealing the possibilities and advantages of using partial order in solving inverse problems in the case when no compact set of prior restrictions is available, concentrating on linear inverse (possibly ill-posed) problems.

Regularising Ill-posed Discrete Optimisation: Quests with P Systems

We propose a novel approach to justify and guide regularisation of an ill-posed one-dimensional global optimisation with multiple solutions using a massively parallel P system model of the solution space. Classical optimisation assumes a well-posed problem with a stable unique solution. Most of important practical problems are ill posed due to an unstable or non-unique global optimum and are regularised to get a unique best-suited solution. Whilst regularisation theory exists largely for unstable unique solutions, its recommendations are often routinely applied to inverse optical problems with essentially non-unique solutions, e.g. computer stereo vision or image segmentation, typically formulated in terms of global energy minimisation. In these cases the recommended regularisation becomes purely heuristic and does not guarantee a unique solution. As a result, classical optimisation algorithms: dynamic programming DP and belief propagation BP --meet with difficulties. Our recent concurrent propagation CP, leaning upon the P systems paradigm, extends DP and BP to always detect whether the problem is ill posed or not and store in the ill-posed case an entire space of solutions that yield the same global optimum. This suggests a radically new path to proper regularisation: select the best-suited unique solution by exploring statistical and structural features of this space. We propose a P systems based implementation of CP and set out as a case study an application of CP to the image matching problem in stereo vision.

Continuous-time method and its discretization to inverse problem of intensity-modulated radiation therapy treatment planning

We propose a novel approach for solving box-constrained inverse problems in intensity-modulated radiation therapy (IMRT) treatment planning based on the idea of continuous dynamical methods and split-feasibility algorithms. Our method can compute a feasible solution without the second derivative of an objective function, which is required for gradient-based optimization algorithms. We prove theoretically that a double Kullback–Leibler divergence can be used as the Lyapunov function for the IMRT planning system.Moreover, we propose a non-negatively constrained iterative method formulated by discretizing a differential equation in the continuous method. We give proof for the convergence of a desired solution in the discretized system, theoretically. The proposed method not only reduces computational costs but also does not produce a solution with an unphysical negative radiation beam weight in solving IMRT planning inverse problems.The convergence properties of solutions for an ill-posed case are confirmed by numerical experiments using phantom data simulating a clinical setup.

Bayesian inverse problems with non-conjugate priors

We investigate the frequentist posterior contraction rate of nonparametric Bayesian procedures in linear inverse problems in both the mildly and severely ill-posed cases. A theorem is proved in a general Hilbert space setting under approximation-theoretic assumptions on the prior. The result is applied to non-conjugate priors, notably sieve and wavelet series priors, as well as in the conjugate setting. In the mildly ill-posed setting minimax optimal rates are obtained, with sieve priors being rate adaptive over Sobolev classes. In the severely ill-posed setting, oversmoothing the prior yields minimax rates. Previously established results in the conjugate setting are obtained using this method. Examples of applications include deconvolution, recovering the initial condition in the heat equation and the Radon transform.

Theory Study by CGM for Moving Force Identification

The conjugate gradient method (CGM) is proposed in this paper to improve the ill-posed problem of moving force identification. Based on theoretical analysis, the CGM formula is deduced and applied in moving force identification. Related research shows that the CGM have higher identification accuracy and robust noise immunity as well as producing an acceptable solution to ill-posed cases to some extent comparing with the time domain method (TDM), when they are used to identify the moving force. The theoretical study results by CGM are practical significant to selection properly method for moving force identification.

Theory Study for Moving Force Identification

Three identification methods, the time domain method (TDM)、the conjugate gradient method (CGM)and the pre-treatment conjugate gradient method (PCGM) are employed for moving force identification. Related research shows that the PCGM have higher identification accuracy and robust noise immunity as well as producing an acceptable solution to ill-posed cases to some extent when they are used to identify the moving force. However, the pre-treatment matrix is very important to the PCGM because it affects the identification accuracy and robust noise immunity as well as ill-posed cases to some extent. The theory study results are practical significant to selection properly pre-treatment matrix.

Uniform confidence bands for functions estimated nonparametrically with instrumental variables

This paper is concerned with developing uniform confidence bands for functions estimated nonparametrically with instrumental variables. We show that a sieve nonparametric instrumental variables estimator is pointwise asymptotically normally distributed. The asymptotic normality result holds in both mildly and severely ill-posed cases. We present methods to obtain a uniform confidence band and show that the bootstrap can be used to obtain the required critical values. Monte Carlo experiments illustrate the finite-sample performance of the uniform confidence band.

A one-dimensional model for the interaction between cell-to-cell adhesion and chemotactic signalling

We develop and analyse a discrete, one-dimensional model of cell motility which incorporates the effects of volume filling, cell-to-cell adhesion and chemotaxis. The formal continuum limit of the model is a non-linear generalisation of the parabolic-elliptic Keller–Segel equations, with a diffusivity which can become negative if the adhesion coefficient is large. The consequent ill-posedness results in the appearance of spatial oscillations and the development of plateaus in numerical solutions of the underlying discrete model. A global-existence result is obtained for the continuum equations in the case of favourable parameter values and data, and a steady-state analysis, which, amongst other things, accounts for high-adhesion plateaus, is carried out. For ill-posed cases, a singular Stefan-problem formulation of the continuum limit is written down and solved numerically, and the numerical solutions are compared with those of the original discrete model.

Solution properties and approximate Riemann solvers for two-layer shallow flow models

The 4 × 4 system of governing equations for two-layer shallow flow models is known to exhibit particular behaviours such as loss of hyperbolicity under certain flow configurations. An eigenvalue analysis of the conservation part of the equations shows that the loss of hyperbolicity is due only to the reaction exerted by each fluid onto the other at the interface between the fluids. Three Riemann solvers derived from the HLL formalism are presented. In the first solver, the pressure-induced terms are accounted for by the source term; in the second solver, they are incorporated into the fluxes; the third solver uses the same formulation as the first, except that the mass and momentum balance for the bottom layer are replaced with the balance equations for the system formed by the two layers as a whole. Numerical results using the three solvers are presented for (1) static conditions such as two fluids of identical densities at rest above each other, (2) dam-break flows involving the collapse of a body of light fluid over a uniform layer of a denser fluid, and (3) Liska and Wendroff’s ill-posed test cases [24] involving two-layer flows over a topographic bump. The three solvers produce quasi-undistinguishable results for the dam-break flows, and produce sharp solutions over the full range of density ratio, from 0 to 1. However, only the third solver allows a strict preservation of static configurations. Moreover, a method is proposed to assess the convergence of the numerical solutions in the configurations for which no analytical solution can be obtained.