A characterization of ill-posed data instances for convex programming

2002 ◽  
Vol 91 (2) ◽  
pp. 375-390 ◽  
Author(s):  
Manuel A. Nunez
Keyword(s):  
Convex Programming ◽  
Ill Posed

scholarly journals Machine learning powered ellipsometry

2021 ◽  
Vol 10 (1) ◽  
Author(s):  
Jinchao Liu ◽  
Di Zhang ◽  
Dianqiang Yu ◽  
Mengxin Ren ◽  
Jingjun Xu
Keyword(s):  
Machine Learning ◽  
Optical Constants ◽  
Optical Characterization ◽  
Superior Performance ◽  
Trial And Error ◽  
Powerful Method ◽  
Human In The Loop ◽  
Ill Posed ◽  
Fully Automatic

AbstractEllipsometry is a powerful method for determining both the optical constants and thickness of thin films. For decades, solutions to ill-posed inverse ellipsometric problems require substantial human–expert intervention and have become essentially human-in-the-loop trial-and-error processes that are not only tedious and time-consuming but also limit the applicability of ellipsometry. Here, we demonstrate a machine learning based approach for solving ellipsometric problems in an unambiguous and fully automatic manner while showing superior performance. The proposed approach is experimentally validated by using a broad range of films covering categories of metals, semiconductors, and dielectrics. This method is compatible with existing ellipsometers and paves the way for realizing the automatic, rapid, high-throughput optical characterization of films.

Proximal interior point approach in convex programming (ill-posed problems)*†

Optimization ◽  
1999 ◽  
Vol 45 (1-4) ◽  
pp. 117-147 ◽  
Author(s):  
A. kaplan ◽  
R. Tichatschke
Keyword(s):  
Convex Programming ◽  
Interior Point ◽  
Ill Posed

Characterization of the solution for the problems of strong and weak convex programming

2021 ◽  
Vol 212 (6) ◽  
Author(s):  
Sergey Ivanovitch Dudov ◽  
Mikhail Anatol'evich Osiptsev
Keyword(s):  
Convex Programming

scholarly journals Projection Methods for Ill-Posed Problems Revisited

2016 ◽  
Vol 16 (2) ◽  
pp. 257-276 ◽  
Author(s):  
Stefan Kindermann
Keyword(s):  
Exact Solution ◽  
Global Convergence ◽  
Least Squares ◽  
Local Convergence ◽  
Sufficient Conditions ◽  
Projection Methods ◽  
Bounded Sequence ◽  
Oblique Projection ◽  
Ill Posed

AbstractWe consider the discretization of least-squares problems for linear ill-posed operator equations in Hilbert spaces. The main subject of this article concerns conditions for convergence of the associated discretized minimum-norm least-squares solution to the exact solution using exact attainable data. The two cases of global convergence (convergence for all exact solutions) or local convergence (convergence for a specific exact solution) are investigated. We review the existing results and prove new equivalent conditions when the discretized solution always converges to the exact solution. An important tool is to recognize the discrete solution operator as an oblique projection. Hence, global convergence can be characterized by certain subspaces having uniformly bounded angles. We furthermore derive practically useful conditions when this holds and put them into the context of known results. For local convergence, we generalize results on the characterization of weak or strong convergence and state some new sufficient conditions. We furthermore provide an example of a bounded sequence of discretized solutions which does not converge at all, not even weakly.

scholarly journals Prevalence of white matter pathways coming into a single diffusion MRI voxel orientation: the bottleneck issue in tractography

2021 ◽  
Author(s):  
Kurt Schilling ◽  
Chantal M.W. Tax ◽  
Francois M.W. Rheault ◽  
Bennett A Landman ◽  
Adam W Anderson ◽  
...  
Keyword(s):  
White Matter ◽  
Human Brain ◽  
Diffusion Mri ◽  
Brain Connectivity ◽  
False Negative ◽  
Fiber Tractography ◽  
Single Voxel ◽  
Ill Posed ◽  
Pass Through

Characterizing and understanding the limitations of diffusion MRI fiber tractography is a prerequisite for methodological advances and innovations which will allow these techniques to accurately map the connections of the human brain. The so-called "crossing fiber problem" has received tremendous attention and has continuously triggered the community to develop novel approaches for disentangling distinctly oriented fiber populations. Perhaps an even greater challenge occurs when multiple white matter bundles converge within a single voxel, or throughout a single brain region, and share the same parallel orientation, before diverging and continuing towards their final cortical or sub-cortical terminations. These so-called "bottleneck" regions contribute to the ill-posed nature of the tractography process, and lead to both false positive and false negative estimated connections. Yet, as opposed to the extent of crossing fibers, a thorough characterization of bottleneck regions has not been performed. The aim of this study is to quantify the prevalence of bottleneck regions. To do this, we use diffusion tractography to segment known white matter bundles of the brain, and assign each bundle to voxels they pass through and to specific orientations within those voxels (i.e. fixels). We demonstrate that bottlenecks occur in greater than 50-70% of fixels in the white matter of the human brain. We find that all projection, association, and commissural fibers contribute to, and are affected by, this phenomenon, and show that even regions traditionally considered "single fiber voxels" often contain multiple fiber populations. Together, this study shows that a majority of white matter presents bottlenecks for tractography which may lead to incorrect or erroneous estimates of brain connectivity or quantitative tractography (i.e., tractometry), and underscores the need for a paradigm shift in the process of tractography and bundle segmentation for studying the fiber pathways of the human brain.

scholarly journals Inverse problems: A Bayesian perspective

Acta Numerica ◽  
2010 ◽  
Vol 19 ◽  
pp. 451-559 ◽  
Author(s):  
A. M. Stuart
Keyword(s):  
Inverse Problems ◽  
Differential Equations ◽  
Bayesian Approach ◽  
Practical Importance ◽  
Full Characterization ◽  
Ill Posed ◽  
The Subject ◽  
Computational Resources ◽  
The Bayesian Approach

The subject of inverse problems in differential equations is of enormous practical importance, and has also generated substantial mathematical and computational innovation. Typically some form of regularization is required to ameliorate ill-posed behaviour. In this article we review the Bayesian approach to regularization, developing a function space viewpoint on the subject. This approach allows for a full characterization of all possible solutions, and their relative probabilities, whilst simultaneously forcing significant modelling issues to be addressed in a clear and precise fashion. Although expensive to implement, this approach is starting to lie within the range of the available computational resources in many application areas. It also allows for the quantification of uncertainty and risk, something which is increasingly demanded by these applications. Furthermore, the approach is conceptually important for the understanding of simpler, computationally expedient approaches to inverse problems.

scholarly journals Duality theorems for convex programming without constraint qualification

1984 ◽  
Vol 36 (2) ◽  
pp. 253-266 ◽  
Author(s):  
P. Kanniappan
Keyword(s):  
Convex Function ◽  
Convex Programming ◽  
Constraint Qualification ◽  
Constraint Qualifications ◽  
Convex Programming Problem ◽  
Duality Theorems ◽  
Finite Dimensional ◽  
Lower Semi ◽  
Continuous Convex Function

AbstractInvoking a recent characterization of Optimality for a convex programming problem with finite dimensional range without any constraint qualification given by Borwein and Wolkowicz, we establish duality theorems. These duality theorems subsume numerous earlier duality results with constraint qualifications. We apply our duality theorems in the case of the objective function being the sum of a positively homogeneous, lower-semi-continuous, convex function and a subdifferentiable convex function. We also study specific problems of the above type in this setting.

Inverse Problems in Heat Transfer: New Trends on Solution Methodologies and Applications

2010 ◽  
Author(s):  
Helcio R. B. Orlande
Keyword(s):  
Inverse Problems ◽  
Research Work ◽  
Practical Interest ◽  
Physical Problem ◽  
Analysis And Design ◽  
Oil Pipelines ◽  
Regularization Techniques ◽  
Ill Posed ◽  
Well Posed

Systematic methods for the solution of inverse problems have developed significantly during the last twenty years and have become a powerful tool for analysis and design in engineering. Inverse analysis is nowadays a common practice in which the groups involved with experiments and numerical simulation synergistically collaborate throughout the research work, in order to obtain the maximum of information regarding the physical problem under study. Inverse problems are mathematically classified as ill-posed, that is, their solutions do not satisfy either one of the requirements of existence, uniqueness or stability. The solution approaches generally consist of the reformulation of the inverse problem in terms of an approximate well-posed problem. In this paper we briefly review various approaches for the solution of inverse problems, including those based on classical regularization techniques and those based on the Bayesian statistics. Applications of inverse problems are then presented for cases of practical interest, such as the characterization of non-homogeneous materials and the prediction of the temperature field in oil pipelines.

On A characterization of optimality in convex programming

1976 ◽  
Vol 11 (1) ◽  
pp. 81-88 ◽  
Author(s):  
A. Ben-Israel ◽  
A. Ben-Tal
Keyword(s):  
Convex Programming ◽  
Characterization Of Optimality
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