On a level-set method for ill-posed problems with piecewise non-constant coefficients

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 of the 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. Keywords: Level Set Methods, Regularization, Ill-Posed Problems, Piecewise Non-Constant CoefficientsComment: Accepte

A Fractional SIRC Model For The Spread Of Diseases In Two Interacting Populations

In this contribution we address the following question: what is the behavior of a disease spreading between two distinct populations that interact, under the premise that both populations have only partial immunity to circulating stains of the disease? Our approach consists of proposing and analyzing a multi-fractional Susceptible (S), Infected  (I), Recovered (R) and Cross-immune (C)  compartmental model, assuming that the dynamics between the compartments of the same population is governed by a fractional derivative, while the interaction between distinct populations is characterized by the proportion of interaction between susceptible and infected individuals of both populations. We prove the well-posedness of the proposed dynamics, which is complemented with simulated scenarios showing the effects of fractional order derivatives (memory) on the dynamics

Identification of Nano-Beams Rigidity Coefficient: A Numerical Analysis Using the Landweber Method

Due to their supporting function, beams are one of the main elements in structural projects. With the intense technological development in the field of nanotechnology, beams at micro- and nanoscales have become objects of intense study and research interest, see for example [8]. In this approach, we analyze numerically the inverse problem of identifying the stiffness coefficient in micro-nano-beams as a function that implicitly depends on the fractal media map for the continuum from strain measurements. Such a problem is unstable with respect to noise in strain measurements, which is inherent in practical problems. We introduce the equations that compose Landweber's iterative regularization method as a strategy to obtain a stable and convergent approximate solution with respect to the noise level in the measurements. We show some scenarios with simulated data for identifying the stiffness coefficient for different noise levels in measurements and for different coefficient of transformation of fractal medium. The results found numerically show that Landweber's method is a regularization strategy for the problem of identifying the stiffness coefficient in micro/nano-beams

On the Choice of the Tikhonov Regularization Parameter and the Discretization Level: A Discrepancy-Based Strategy

We address the classical issue of appropriate choice of the regularization and discretization level for the Tikhonov regularization of an inverse problem with imperfectly measured data. We focus on the fact that the proper choice of the discretization level in the domain together with the regularization parameter is a key feature in adequate regularization. We propose a discrepancy-based choice for these quantities by applying a relaxed version of Morozov's discrepancy principle. Indeed, we prove the existence of the discretization level and the regularization parameter satisfying such discrepancy. We also prove associated regularizing properties concerning the Tikhonov minimizers.Comment: 28 pages, 4 figure

A Fractional SIRC Model For The Spread Of Diseases In Two Interacting Populations

In this contribution we address the following question: what is the behavior of a disease spreading between two distinct populations that interact, under the premise that both populations have only partial immunity to circulating stains of the disease? Our approach consists of proposing and analyzing a multi-fractional Susceptible (S), Infected  (I), Recovered (R) and Cross-immune (C)  compartmental model, assuming that the dynamics between the compartments of the same population is governed by a fractional derivative, while the interaction between distinct populations is characterized by the proportion of interaction between susceptible and infected individuals of both populations. We prove the well-posedness of the proposed dynamics, which is complemented with simulated scenarios showing the effects of fractional order derivatives (memory) on the dynamics

Identification of Nano-Beams Rigidity Coefficient: A Numerical Analysis Using the Landweber Method

Due to their supporting function, beams are one of the main elements in structural projects. With the intense technological development in the field of nanotechnology, beams at micro- and nanoscales have become objects of intense study and research interest, see for example [8]. In this approach, we analyze numerically the inverse problem of identifying the stiffness coefficient in micro-nano-beams as a function that implicitly depends on the fractal media map for the continuum from strain measurements. Such a problem is unstable with respect to noise in strain measurements, which is inherent in practical problems. We introduce the equations that compose Landweber's iterative regularization method as a strategy to obtain a stable and convergent approximate solution with respect to the noise level in the measurements. We show some scenarios with simulated data for identifying the stiffness coefficient for different noise levels in measurements and for different coefficient of transformation of fractal medium. The results found numerically show that Landweber's method is a regularization strategy for the problem of identifying the stiffness coefficient in micro/nano-beams

Homogeneização Assintótica e Cálculo Fracionário na modelagem de meios micro-heterogêneos: uma introdução com o caso de uma barra funcionalmente graduada, microperiódica e linear

The study of materials with complex structure, like the functionally graded, is a field of increasing interest, what happens mostly because the importance of these materials in the industry. In this work, the Asymptotic Homogenization Method and Fractional Calculus are both applied in a problem which models the behaviour of a micro-heterogeneous material, like the functionally graded. The goal of this work is the study of the association possibilities between these two tools, since which one are providing important results in the mathematical modelling of complex structures. The results show that each methodology reproduce a different aspect of the phenomenon: the Homogenization stays in the microstructure details and the fractional derivative takes care of a macroscopic behaviour, which nature is possibly dissipative. Here are important information, but a deeper and more diverse approach is necessary to provide strong e more general statements about this theme.O estudo de materiais com estrutura complexa, como os funcionalmente graduados, tem cada vez mais chamado a atenção, seja pela dificuldade em obter os resultados ou pela importância de tais materiais em diversos ramos da indústria. Neste trabalho, o Método de Homogeneização Assintótica e ferramentas do Cálculo Fracionário são aplicados para modelar o comportamento um material micro-heterogêneo, como os funcionalmente graduados. O interesse principal desse trabalho é encontrar uma forma de associar ambas metodologias, que têm fornecido bons resultados quando aplicadas em problemas envolvendo estruturas complexas, mas de forma separada. Os resultados obtidos mostram que cada metodologia reproduz diferentes aspectos do fenômeno: a Homogenização está nos detalhes da microestrutura, enquanto que a derivada fracionária se ocupa de um comportamento macroscópico, cuja natureza pode ser dissipativa. Aqui estão resultados importantes, porém uma abordagem mais profunda e diversificada é necessária a fim de fornecer conclusões mais fortes e generalizadas acerca do tema