Sobolev regularization of a nonlinear ill-posed parabolic problem as a model for aggregating populations

Padron, Victor

doi:10.1080/03605309808821353

Cited by 72 publications

(11 citation statements)

References 9 publications

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“…The partial differential equation in problem (1-1) can be regarded as the regularization of the forwardbackward parabolic equation u t = ϕ(u), which leads to ill-posed problems. The latter equation and its regularizations arise in several applications, such as edge detection in image processing [Perona and Malik 1990], aggregation models in population dynamics [Padrón 1998], and stratified turbulent shear flow [Barenblatt et al 1993a].…”

Section: Introductionmentioning

confidence: 99%

Pseudoparabolic regularization of forward-backward parabolic equations: A logarithmic nonlinearity

2013

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with measure-valued initial data, assuming that the regularizing term ψ has logarithmic growth (the case of power-type ψ was dealt with in an earlier work). We prove that this case is intermediate between the case of power-type ψ and that of bounded ψ, to be addressed in a forthcoming paper. Specifically, the support of the singular part of the solution with respect to the Lebesgue measure remains constant in time (as in the case of power-type ψ), although the singular part itself need not be constant (as in the case of bounded ψ, where the support of the singular part can also increase). However, it turns out that the concentrated part of the solution with respect to the Newtonian capacity remains constant.

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Section: Introductionmentioning

confidence: 99%

Pseudoparabolic regularization of forward-backward parabolic equations: A logarithmic nonlinearity

2013

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“…where B = (I − ∆) −1 is a nonlocal operator [33]. According to the above two non-local effects, equations like (1) have had a high profile in the study of many phenomena such as biological species dynamics, nonlinear elasticity, non-stationary fluid, image recovery,... (see [28,1,5,6,21] and references therein). The third non-local term comes from the source |u| q−1 u − 1 |Ω| Ω |u| q−1 udx, which leads to the conservation property Ω u = 0, and points out that the solutions may change sign.…”

mentioning

confidence: 99%

Initial boundary value problem of a class of mixed pseudo-parabolic Kirchhoff equations

Cao

Zhao

2021

era

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<p style='text-indent:20px;'>In this paper, we consider the initial boundary value problem for a mixed pseudo-parabolic Kirchhoff equation. Due to the comparison principle being invalid, we use the potential well method to give a threshold result of global existence and non-existence for the sign-changing weak solutions with initial energy <inline-formula><tex-math id="M1">\begin{document}$ J(u_0)\leq d $\end{document}</tex-math></inline-formula>. When the initial energy <inline-formula><tex-math id="M2">\begin{document}$ J(u_0)>d $\end{document}</tex-math></inline-formula>, we find another criterion for the vanishing solution and blow-up solution. Our interest also lies in the discussion of the exponential decay rate of the global solution and life span of the blow-up solution.</p>

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“…The smoothing term in Equation 1 is required to regularize the solution in the presence of sparse data. Previously, a Sobolev smoothing term [12, 23] has been used for this purpose [6]; however, this has the disadvantage of being sensitive to large rotations and motions, as are often seen around the RV, leaving artifacts in the model surfaces. In this work we used a D-Affine smoothing term [5], which penalized change in deformation across the model.…”

Section: Methodsmentioning

confidence: 99%

4D modelling for rapid assessment of biventricular function in congenital heart disease

Gilbert

Pontré

Occleshaw

et al. 2017

Int J Cardiovasc Imaging

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Although more patients with congenital heart disease (CHD) are now living longer due to better surgical interventions, they require regular imaging to monitor cardiac performance. There is a need for robust clinical tools which can accurately assess cardiac function of both the left and right ventricles in these patients. We have developed methods to rapidly quantify 4D (3D + time) biventricular function from standard cardiac MRI examinations. A finite element model was interactively customized to patient images using guide-point modelling. Computational efficiency and ability to model large deformations was improved by predicting cardiac motion for the left ventricle and epicardium with a polar model. In addition, large deformations through the cycle were more accurately modeled using a Cartesian deformation penalty term. The model was fitted to user-defined guide points and image feature tracking displacements throughout the cardiac cycle. We tested the methods in 60 cases comprising a variety of congenital heart diseases and showed good correlation with the gold standard manual analysis, with acceptable inter-observer error. The algorithm was considerably faster than standard analysis and shows promise as a clinical tool for patients with CHD.

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Sobolev regularization of a nonlinear ill-posed parabolic problem as a model for aggregating populations

Cited by 72 publications

References 9 publications

Pseudoparabolic regularization of forward-backward parabolic equations: A logarithmic nonlinearity

Pseudoparabolic regularization of forward-backward parabolic equations: A logarithmic nonlinearity

Initial boundary value problem of a class of mixed pseudo-parabolic Kirchhoff equations

4D modelling for rapid assessment of biventricular function in congenital heart disease

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