Well-Posed and Ill-Posed Abstract Cauchy Problems: The Concept of Regularization

In this paper, we consider a nonhomogeneous differential operator equation of first order u ′ t + A u t = f t . The coefficient operator A is linear unbounded and self-adjoint in a Hilbert space. We assume that the operator does not have a fixed sign. We associate to this equation the initial or final conditions u 0 = Φ or u T = Φ . We note that the Cauchy problem is severely ill-posed in the sense that the solution if it exists does not depend continuously on the given data. Using a quasi-boundary value method, we obtain an approximate nonlocal problem depending on a small parameter. We show that regularized problem is well-posed and has a strongly solution. Finally, some convergence results are provided.

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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.

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

Journal of Inverse and Ill-Posed Problems ◽

10.1515/jiip-2020-0003 ◽

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.

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

Nanomaterials ◽

10.3390/nano11030573 ◽

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.

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