An Optimization Method in Inverse Elastic Scattering for One-Dimensional Grating Profiles

2012 ◽  
Vol 12 (5) ◽  
pp. 1434-1460 ◽  
Author(s):  
Johannes Elschner ◽  
Guanghui Hu
Keyword(s):  
Piecewise Linear ◽  
Diffraction Problem ◽  
Optimization Method ◽  
Ill Posed ◽  
Least Squares Optimization ◽  
Step Algorithm ◽  
Navier Equation ◽  
Inverse Diffraction ◽  
Electromagnetic Diffraction ◽  
Well Posed

AbstractConsider the inverse diffraction problem to determine a two-dimensional periodic structure from scattered elastic waves measured above the structure. We formulate the inverse problem as a least squares optimization problem, following the two-step algorithm by G. Bruckner and J. Elschner [Inverse Probl., 19 (2003), 315-329] for electromagnetic diffraction gratings. Such a method is based on the Kirsch-Kress optimization scheme and consists of two parts: a linear severely ill-posed problem and a nonlinear well-posed one. We apply this method to both smooth (C2) and piecewise linear gratings for the Dirichlet boundary value problem of the Navier equation. Numerical reconstructions from exact and noisy data illustrate the feasibility of the method.

An Inverse Diffraction Problem: Shape Reconstruction

2015 ◽  
Vol 5 (4) ◽  
pp. 342-360 ◽  
Author(s):  
Yanfeng Kong ◽  
Zhenping Li ◽  
Xiangtuan Xiong
Keyword(s):  
Error Estimates ◽  
Diffraction Problem ◽  
Shape Reconstruction ◽  
Numerical Examples ◽  
Tikhonov Method ◽  
Ill Posed ◽  
Tikhonov Regularisation ◽  
Slow Evolution ◽  
Inverse Diffraction

AbstractAn inverse diffraction problem is considered. Both classical Tikhonov regularisation and a slow-evolution-from-the-continuation-boundary (SECB) method are used to solve the ill-posed problem. Regularisation error estimates for the two methods are compared, and the SECB method is seen to be an improvement on the classical Tikhonov method. Two numerical examples demonstrate their feasibility and efficiency.

Convergence and stability analysis of the half thresholding based few-view CT reconstruction

2020 ◽  
Vol 28 (6) ◽  
pp. 829-847
Author(s):  
Hua Huang ◽  
Chengwu Lu ◽  
Lingli Zhang ◽  
Weiwei Wang
Keyword(s):  
Noise Suppression ◽  
Reconstructed Image ◽  
Reference Image ◽  
Projection Data ◽  
Ct Reconstruction ◽  
Reconstruction Algorithms ◽  
Ill Posed ◽  
Thresholding Algorithm ◽  
The Stability ◽  
Well Posed

AbstractThe projection data obtained using the computed tomography (CT) technique are often incomplete and inconsistent owing to the radiation exposure and practical environment of the CT process, which may lead to a few-view reconstruction problem. Reconstructing an object from few projection views is often an ill-posed inverse problem. To solve such problems, regularization is an effective technique, in which the ill-posed problem is approximated considering a family of neighboring well-posed problems. In this study, we considered the {\ell_{1/2}} regularization to solve such ill-posed problems. Subsequently, the half thresholding algorithm was employed to solve the {\ell_{1/2}} regularization-based problem. The convergence analysis of the proposed method was performed, and the error bound between the reference image and reconstructed image was clarified. Finally, the stability of the proposed method was analyzed. The result of numerical experiments demonstrated that the proposed method can outperform the classical reconstruction algorithms in terms of noise suppression and preserving the details of the reconstructed image.

scholarly journals Elastostatics of Bernoulli–Euler Beams Resting on Displacement-Driven Nonlocal Foundation

Nanomaterials ◽  
2021 ◽  
Vol 11 (3) ◽  
pp. 573
Author(s):  
Marzia Sara Vaccaro ◽  
Francesco Paolo Pinnola ◽  
Francesco Marotti de Sciarra ◽  
Raffaele Barretta
Keyword(s):  
Boundary Conditions ◽  
Structural Engineering ◽  
Beam Deflection ◽  
Soil Reaction ◽  
Differential Problem ◽  
Reaction Field ◽  
Ill Posed ◽  
Elastic Responses ◽  
Benchmark Solutions ◽  
Well Posed

The simplest elasticity model of the foundation underlying a slender beam under flexure was conceived by Winkler, requiring local proportionality between soil reactions and beam deflection. Such an approach leads to well-posed elastostatic and elastodynamic problems, but as highlighted by Wieghardt, it provides elastic responses that are not technically significant for a wide variety of engineering applications. Thus, Winkler’s model was replaced by Wieghardt himself by assuming that the beam deflection is the convolution integral between soil reaction field and an averaging kernel. Due to conflict between constitutive and kinematic compatibility requirements, the corresponding elastic problem of an inflected beam resting on a Wieghardt foundation is ill-posed. Modifications of the original Wieghardt model were proposed by introducing fictitious boundary concentrated forces of constitutive type, which are physically questionable, being significantly influenced on prescribed kinematic boundary conditions. Inherent difficulties and issues are overcome in the present research using a displacement-driven nonlocal integral strategy obtained by swapping the input and output fields involved in Wieghardt’s original formulation. That is, nonlocal soil reaction fields are the output of integral convolutions of beam deflection fields with an averaging kernel. Equipping the displacement-driven nonlocal integral law with the bi-exponential averaging kernel, an equivalent nonlocal differential problem, supplemented with non-standard constitutive boundary conditions involving nonlocal soil reactions, is established. As a key implication, the integrodifferential equations governing the elastostatic problem of an inflected elastic slender beam resting on a displacement-driven nonlocal integral foundation are replaced with much simpler differential equations supplemented with kinematic, static, and new constitutive boundary conditions. The proposed nonlocal approach is illustrated by examining and analytically solving exemplar problems of structural engineering. Benchmark solutions for numerical analyses are also detected.

scholarly journals Fast Quadratic Shape From Shading With Unknown Light Direction

2021 ◽  
Author(s):  
Kuros Yalpani
Keyword(s):  
Linear Optimization ◽  
Quadratic Function ◽  
Optimization Method ◽  
Shape From Shading ◽  
Surface Patch ◽  
Algebraic Complexity ◽  
Surface Patches ◽  
Quadratic Surface ◽  
Least Squares Optimization ◽  
Light Source Direction

An algorithm is proposed that extracts 3D shape from shading information in a digital image. The algorithm assumes that there is only a single source of light producing the image, that the surface of the shape giving rise to the image is Lambertian (matte) and that its shape can be locally approximated by a quadratic function. Previous work shows that under these assumptions, robust shape from shading is possible, though slow for large images because a non-linear optimization method is applied in order to estimate local quadratic surface patches from image intensities. The work presented here shows that local quadratic surface patch estimates can be computed, without prior knowledge of the light source direction, via a linear least squares optimization, thus greatly improving the algebraic complexity and run-time of this existing algorithms.

scholarly journals Inverse source problems for time-fractional mixed parabolic-hyperbolic-type equations

2015 ◽  
Vol 23 (4) ◽  
Author(s):  
Pengbin Feng ◽  
Erkinjon T. Karimov
Keyword(s):  
Wave Equation ◽  
Mixed Type ◽  
Orthonormal System ◽  
Type Equation ◽  
Hyperbolic Type ◽  
Initial Time ◽  
Inverse Source Problem ◽  
Ill Posed ◽  
Inverse Source Problems ◽  
Well Posed

AbstractIn the present paper we consider an inverse source problem for a time-fractional mixed parabolic-hyperbolic equation with Caputo derivatives. In the case when the hyperbolic part of the considered mixed-type equation is the wave equation, the uniqueness of the source and the solution are strongly influenced by the initial time and the problem is generally ill-posed. However, when the hyperbolic part is time-fractional, the problem is well-posed if the end time is large. Our method relies on the orthonormal system of eigenfunctions of the operator with respect to the space variables. Finally, we prove uniqueness and stability of certain weak solutions for the problems under consideration.

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