Stable manifolds to bounded solutions in possibly ill-posed PDEs

Despite the skill and determination of electro-optical system designers, the images acquired using their best designs often suffer from blur and noise. The aim of an “image enhancer” such as myself is to improve these poor images, usually by digital means, such that they better resemble the true, “optical object,” input to the system. This problem is notoriously “ill-posed,” i.e. any direct approach at inversion of the image data suffers strongly from the presence of even a small amount of noise in the data. In fact, the fluctuations engendered in neighboring output values tend to be strongly negative-correlated, so that the output spatially oscillates up and down, with large amplitude, about the true object. What can be done about this situation? As we shall see, various concepts taken from statistical communication theory have proven to be of real use in attacking this problem. We offer below a brief summary of these concepts.

Download Full-text

Methods of solving ill-posed inverse problems

Journal of Engineering Physics ◽

10.1007/bf01254725 ◽

1983 ◽

Vol 45 (5) ◽

pp. 1237-1245 ◽

Cited By ~ 12

Author(s):

O. M. Alifanov

Keyword(s):

Inverse Problems ◽

Ill Posed

Download Full-text

A Nonlinearly Ill-Posed Problem of Reconstructing the Temperature from Interior Data

Numerical Functional Analysis and Optimization ◽

10.1080/01630560802000983 ◽

2008 ◽

Vol 29 (3-4) ◽

pp. 445-469

Author(s):

Pham Hoang Quan ◽

Dang Duc Trong ◽

Alain Pham Ngoc Dinh

Keyword(s):

Interior Data ◽

Ill Posed

Download Full-text

Convergence Rates of First- and Higher-Order Dynamics for Solving Linear Ill-Posed Problems

Foundations of Computational Mathematics ◽

10.1007/s10208-021-09536-6 ◽

2021 ◽

Author(s):

Radu Boţ ◽

Guozhi Dong ◽

Peter Elbau ◽

Otmar Scherzer

Keyword(s):

Linear Operator ◽

Operator Equation ◽

Convex Analysis ◽

Convergence Rates ◽

Linear Operator Equation ◽

Optimal Convergence Rates ◽

Ill Posed ◽

Thresholding Algorithm ◽

The Given ◽

Theoretical Results

AbstractRecently, there has been a great interest in analysing dynamical flows, where the stationary limit is the minimiser of a convex energy. Particular flows of great interest have been continuous limits of Nesterov’s algorithm and the fast iterative shrinkage-thresholding algorithm, respectively. In this paper, we approach the solutions of linear ill-posed problems by dynamical flows. Because the squared norm of the residual of a linear operator equation is a convex functional, the theoretical results from convex analysis for energy minimising flows are applicable. However, in the restricted situation of this paper they can often be significantly improved. Moreover, since we show that the proposed flows for minimising the norm of the residual of a linear operator equation are optimal regularisation methods and that they provide optimal convergence rates for the regularised solutions, the given rates can be considered the benchmarks for further studies in convex analysis.

Download Full-text

Leveraging Label Information in a Knowledge-Driven Approach for Rolling-Element Bearings Remaining Useful Life Prediction

Energies ◽

10.3390/en14082163 ◽

2021 ◽

Vol 14 (8) ◽

pp. 2163

Author(s):

Tarek Berghout ◽

Mohamed Benbouzid ◽

Leïla-Hayet Mouss

Keyword(s):

Transfer Learning ◽

Short Term Memory ◽

Remaining Useful Life ◽

Accelerated Life Tests ◽

Learning Path ◽

Accelerated Life ◽

Unseen Data ◽

Label Information ◽

Life Tests ◽

Ill Posed

Since bearing deterioration patterns are difficult to collect from real, long lifetime scenarios, data-driven research has been directed towards recovering them by imposing accelerated life tests. Consequently, insufficiently recovered features due to rapid damage propagation seem more likely to lead to poorly generalized learning machines. Knowledge-driven learning comes as a solution by providing prior assumptions from transfer learning. Likewise, the absence of true labels was able to create inconsistency related problems between samples, and teacher-given label behaviors led to more ill-posed predictors. Therefore, in an attempt to overcome the incomplete, unlabeled data drawbacks, a new autoencoder has been designed as an additional source that could correlate inputs and labels by exploiting label information in a completely unsupervised learning scheme. Additionally, its stacked denoising version seems to more robustly be able to recover them for new unseen data. Due to the non-stationary and sequentially driven nature of samples, recovered representations have been fed into a transfer learning, convolutional, long–short-term memory neural network for further meaningful learning representations. The assessment procedures were benchmarked against recent methods under different training datasets. The obtained results led to more efficiency confirming the strength of the new learning path.

Download Full-text

A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation

Open Mathematics ◽

10.1515/math-2020-0111 ◽

2020 ◽

Vol 18 (1) ◽

pp. 1685-1697

Author(s):

Zhenyu Zhao ◽

Lei You ◽

Zehong Meng

Keyword(s):

Cauchy Problem ◽

Laplace Equation ◽

Tikhonov Regularization ◽

Regularization Method ◽

Regularization Parameter ◽

Solution Process ◽

Tikhonov Regularization Method ◽

Hermite Expansion ◽

The Cauchy Problem ◽

Ill Posed

Abstract In this paper, a Cauchy problem for the Laplace equation is considered. We develop a modified Tikhonov regularization method based on Hermite expansion to deal with the ill posed-ness of the problem. The regularization parameter is determined by a discrepancy principle. For various smoothness conditions, the solution process of the method is uniform and the convergence rate can be obtained self-adaptively. Numerical tests are also carried out to verify the effectiveness of the method.

Download Full-text

Global stability result for parabolic Cauchy problems

Journal of Inverse and Ill-Posed Problems ◽

10.1515/jiip-2020-0026 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Mourad Choulli ◽

Masahiro Yamamoto

Keyword(s):

New York ◽

Partial Differential Equations ◽

Global Stability ◽

Classical Problem ◽

Stability Result ◽

Cauchy Problems ◽

Smooth Solutions ◽

Linear Partial Differential Equations ◽

Ill Posed ◽

The Stability

AbstractUniqueness of parabolic Cauchy problems is nowadays a classical problem and since Hadamard [Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Dover, New York, 1953], these kind of problems are known to be ill-posed and even severely ill-posed. Until now, there are only few partial results concerning the quantification of the stability of parabolic Cauchy problems. We bring in the present work an answer to this issue for smooth solutions under the minimal condition that the domain is Lipschitz.

Download Full-text

A Factorization of Least-Squares Projection Schemes for Ill-Posed Problems

Computational Methods in Applied Mathematics ◽

10.1515/cmam-2019-0173 ◽

2020 ◽

Vol 20 (4) ◽

pp. 783-798

Author(s):

Shukai Du ◽

Nailin Du

Keyword(s):

Least Squares ◽

Rate Of Convergence ◽

Convergence Conditions ◽

Principal Angles ◽

Factorization Formula ◽

Ill Posed

AbstractWe give a factorization formula to least-squares projection schemes, from which new convergence conditions together with formulas estimating the rate of convergence can be derived. We prove that the convergence of the method (including the rate of convergence) can be completely determined by the principal angles between {T^{\dagger}T(X_{n})} and {T^{*}T(X_{n})}, and the principal angles between {X_{n}\cap(\mathcal{N}(T)\cap X_{n})^{\perp}} and {(\mathcal{N}(T)+X_{n})\cap\mathcal{N}(T)^{\perp}}. At the end, we consider several specific cases and examples to further illustrate our theorems.

Download Full-text

Complex wavefront sensing based on coherent diffraction imaging using vortex modulation

Scientific Reports ◽

10.1038/s41598-021-88523-x ◽

2021 ◽

Vol 11 (1) ◽

Author(s):

Rujia Li ◽

Liangcai Cao

Keyword(s):

Phase Retrieval ◽

Dynamic Range ◽

A Priori ◽

Measured Intensity ◽

Physical Constraints ◽

Coherent Diffraction Imaging ◽

Effective Dynamic ◽

Ill Posed ◽

Retrieval Problem ◽

Diffraction Imaging

AbstractPhase retrieval seeks to reconstruct the phase from the measured intensity, which is an ill-posed problem. A phase retrieval problem can be solved with physical constraints by modulating the investigated complex wavefront. Orbital angular momentum has been recently employed as a type of reliable modulation. The topological charge l is robust during propagation when there is atmospheric turbulence. In this work, topological modulation is used to solve the phase retrieval problem. Topological modulation offers an effective dynamic range of intensity constraints for reconstruction. The maximum intensity value of the spectrum is reduced by a factor of 173 under topological modulation when l is 50. The phase is iteratively reconstructed without a priori knowledge. The stagnation problem during the iteration can be avoided using multiple topological modulations.

Download Full-text

Inverse Infiltration Modeling of Dike Covers Made of Dredged Material Using PEST and AMALGAM

Geosciences ◽

10.3390/geosciences11020041 ◽

2021 ◽

Vol 11 (2) ◽

pp. 41

Author(s):

Tim Jurisch ◽

Stefan Cantré ◽

Fokke Saathoff

Keyword(s):

Large Scale ◽

Measurement Data ◽

Fine Grained ◽

Dredged Materials ◽

Measurement Results ◽

Large Scale Field ◽

Eigenvalue Ratio ◽

Ill Posed ◽

Materials Used ◽

Installation Technology

A variety of studies recently proved the applicability of different dried, fine-grained dredged materials as replacement material for erosion-resistant sea dike covers. In Rostock, Germany, a large-scale field experiment was conducted, in which different dredged materials were tested with regard to installation technology, stability, turf development, infiltration, and erosion resistance. The infiltration experiments to study the development of a seepage line in the dike body showed unexpected measurement results. Due to the high complexity of the problem, standard geo-hydraulic models proved to be unable to analyze these results. Therefore, different methods of inverse infiltration modeling were applied, such as the parameter estimation tool (PEST) and the AMALGAM algorithm. In the paper, the two approaches are compared and discussed. A sensitivity analysis proved the presumption of a non-linear model behavior for the infiltration problem and the Eigenvalue ratio indicates that the dike infiltration is an ill-posed problem. Although this complicates the inverse modeling (e.g., termination in local minima), parameter sets close to an optimum were found with both the PEST and the AMALGAM algorithms. Together with the field measurement data, this information supports the rating of the effective material properties of the applied dredged materials used as dike cover material.

Download Full-text