scholarly journals On a Level-Set Method for Ill-Posed Problems with Piecewise Nonconstant Coefficients

2013 ◽  
Vol 2013 ◽  
pp. 1-15 ◽  
Author(s):  
A. De Cezaro
Keyword(s):  
Inverse Problems ◽  
Level Set Method ◽  
Level Set ◽  
Regularization Method ◽  
Level Sets ◽  
Approximate Solutions ◽  
Type Method ◽  
Ill Posed ◽  
Level Set Approach ◽  
Level Set Framework

We investigate a level-set-type method for solving ill-posed problems, with the assumption that the solutions are piecewise, but not necessarily constant functions with unknown level sets and unknown level values. In order to get stable approximate solutions of the inverse problem, we propose a Tikhonov-type regularization approach coupled with a level-set framework. We prove the existence of generalized minimizers for the Tikhonov functional. Moreover, we prove convergence and stability for regularized solutions with respect to the noise level, characterizing the level-set approach as a regularization method for inverse problems. We also show the applicability of the proposed level-set method in some interesting inverse problems arising in elliptic PDE models.

The Level Set Method for Etching and Deposition

1997 ◽  
Vol 07 (08) ◽  
pp. 1153-1186 ◽  
Author(s):  
D. Adalsteinsson ◽  
L. C. Evans ◽  
H. Ishii
Keyword(s):  
Level Set Method ◽  
Level Set ◽  
Numerical Implementation ◽  
Level Set Approach ◽  
Level Set Framework

We provide a rigorous interpretation of the level set approach to certain nonlocal geometric motions modelling etching effects in manufacture. The shadowing of certain parts of a surface by other parts gives rise to a nonlocal Hamilton–Jacobi type PDE, with a multivalued Hamiltonian. We also show that deposition effects do not fall within the conventional level set framework, and accordingly must be reinterpreted for numerical implementation.

scholarly journals Range-relaxed criteria for choosing the Lagrange multipliers in nonstationary iterated Tikhonov method

2018 ◽  
Vol 40 (1) ◽  
pp. 606-627 ◽  
Author(s):  
R Boiger ◽  
A Leitão ◽  
B F Svaiter
Keyword(s):  
Lagrange Multipliers ◽  
Approximate Solutions ◽  
Identification Problem ◽  
Linear Operators ◽  
Type Method ◽  
Geometrical Properties ◽  
Regularization Parameters ◽  
New Strategy ◽  
Parameter Identification Problem ◽  
Ill Posed

Abstract In this article we propose a novel nonstationary iterated Tikhonov (NIT)-type method for obtaining stable approximate solutions to ill-posed operator equations modeled by linear operators acting between Hilbert spaces. Geometrical properties of the problem are used to derive a new strategy for choosing the sequence of regularization parameters (Lagrange multipliers) for the NIT iteration. Convergence analysis for this new method is provided. Numerical experiments are presented for two distinct applications: (I) a two-dimensional elliptic parameter identification problem (inverse potential problem); and (II) an image-deblurring problem. The results obtained validate the efficiency of our method compared with standard implementations of the NIT method (where a geometrical choice is typically used for the sequence of Lagrange multipliers).

scholarly journals A weighted region-based level set method for image segmentation with intensity inhomogeneity

PLoS ONE ◽  
2021 ◽  
Vol 16 (8) ◽  
pp. e0255948
Author(s):  
Haiping Yu ◽  
Ping Sun ◽  
Fazhi He ◽  
Zhihua Hu
Keyword(s):  
Image Segmentation ◽  
Level Set Method ◽  
Level Set ◽  
State Of The Art ◽  
Intensity Inhomogeneity ◽  
Curve Evolution ◽  
Pressure Force ◽  
The Stability ◽  
Level Set Framework ◽  
Robust To Noise

Image segmentation is a fundamental task in image processing and is still a challenging problem when processing images with high noise, low resolution and intensity inhomogeneity. In this paper, a weighted region-based level set method, which is based on the techniques of local statistical theory, level set theory and curve evolution, is proposed. Specifically, a new weighted pressure force function (WPF) is first presented to flexibly drive the closed contour to shrink or expand outside and inside of the object. Second, a faster and smoother regularization term is added to ensure the stability of the curve evolution and that there is no need for initialization in curve evolution. Third, the WPF is integrated into the region-based level set framework to accelerate the speed of the curve evolution and improve the accuracy of image segmentation. Experimental results on medical and natural images demonstrate that the proposed segmentation model is more efficient and robust to noise than other state-of-the-art models.

Lagrange principle and its regularization as a theoretical basis of stable solving optimal control and inverse problems

2021 ◽  
pp. 151-171
Author(s):  
Mikhail I. Sumin
Keyword(s):  
Optimal Control ◽  
Inverse Problems ◽  
Theoretical Basis ◽  
Optimization Problems ◽  
Existence Theorems ◽  
General Structure ◽  
Approximate Solutions ◽  
Lagrange Principle ◽  
Ill Posed ◽  
Final Observation

The paper is devoted to the regularization of the classical optimality conditions (COC) — the Lagrange principle and the Pontryagin maximum principle in a convex optimal control problem for a parabolic equation with an operator (pointwise state) equality-constraint at the final time. The problem contains distributed, initial and boundary controls, and the set of its admissible controls is not assumed to be bounded. In the case of a specific form of the quadratic quality functional, it is natural to interpret the problem as the inverse problem of the final observation to find the perturbing effect that caused this observation. The main purpose of regularized COCs is stable generation of minimizing approximate solutions (MAS) in the sense of J. Warga. Regularized COCs are: 1) formulated as existence theorems of the MASs in the original problem with a simultaneous constructive representation of specific MASs; 2) expressed in terms of regular classical Lagrange and Hamilton–Pontryagin functions; 3) are sequential generalizations of the COCs and retain the general structure of the latter; 4) “overcome” the ill-posedness of the COCs, are regularizing algorithms for solving optimization problems, and form the theoretical basis for the stable solving modern meaningful ill-posed optimization and inverse problems.

Concurrent Optimization of Structure Topology and Infill Properties With a Cardinal-Function-Based Parametric Level Set Method

2018 ◽  
Author(s):  
Long Jiang ◽  
Shikui Chen ◽  
Peng Wei
Keyword(s):  
Level Set Method ◽  
Level Set ◽  
Material Property ◽  
Kernel Function ◽  
Basis Function ◽  
Nonlinear Problems ◽  
Structural Topology ◽  
Distance Information ◽  
Level Set Function ◽  
Level Set Approach

In this paper, a parametric level-set-based topology optimization framework is proposed, to concurrently optimize structural topology at the macroscale as well as the infill material properties at the mesoscale. With the parametric level set framework, both the boundary evolution and the material property optimization during the optimization process are driven by the Method of Moving Asymptotes (MMA) optimizer, which is more efficient than the PDE-driven level set approach when handling nonlinear problems with multiple constraints. Rather than using a radial basis function (RBF) for the level set parameterization, a new type of cardinal basis function (CBF) was constructed as the kernel function for the proposed parametric level set approach. With this CBF kernel function, the bounds of the design variables can be defined explicitly, which is a great advantage compared with the RBF-based level set method. A variational approach was conducted to regulate the level set function to be a distance-regularized shape for a better material property interpolation accuracy and higher design robustness. With the embedded distance information from the level set model, boundary layer and the infill can be naturally discriminated.

Regularization of Ill-Posed Problems Via the Level Set Approach

1998 ◽  
Vol 58 (6) ◽  
pp. 1689-1706 ◽  
Author(s):  
Eduard Harabetian ◽  
Stanley Osher
Keyword(s):  
Level Set ◽  
Ill Posed ◽  
Level Set Approach

A level set method for inverse problems

2001 ◽  
Vol 17 (5) ◽  
pp. 1327-1355 ◽  
Author(s):  
Martin Burger
Keyword(s):  
Inverse Problems ◽  
Level Set Method ◽  
Level Set

A Level Set Method for Image Restoration

2011 ◽  
Vol 50-51 ◽  
pp. 880-884
Author(s):  
Wai Bin Huang ◽  
Wei Ran Lin
Keyword(s):  
Image Quality ◽  
Image Restoration ◽  
Level Set Method ◽  
Level Set ◽  
Level Sets ◽  
Detection Technique ◽  
Level Lines

This paper presents a level set method for image restoration. In the light of the loss of image quality caused by occlusion or part stain, it adopts the method of picking-up image level sets and filling in level sets of occlusion to reconstruct the image. In the process of linking level lines, besides the traditional geodesic curves, it makes use of the Meaningful beeline detection technique. The experiment results show this method is of great importance in the preservation of images.

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