Pub Date : 2024-06-01DOI: 10.4208/aam.oa-2024-0009
Boling Guo and Ying Zhang
{"title":"Existence of Global Attractor for Weakly Damped FDS Nonlinear Wave Equations","authors":"Boling Guo and Ying Zhang","doi":"10.4208/aam.oa-2024-0009","DOIUrl":"https://doi.org/10.4208/aam.oa-2024-0009","url":null,"abstract":"","PeriodicalId":517399,"journal":{"name":"Annals of Applied Mathematics","volume":"43 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141277361","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-01DOI: 10.4208/aam.oa-2024-0003
Weishi Yin,Zhaobin Xu,Pinchao Meng, Hongyu Liu
In this paper, we are concerned with the inverse transmission eigenvalue problem to recover the shape as well as the constant refractive index of a penetrable medium scatterer. The linear sampling method is employed to determine the transmission eigenvalues within a certain wavenumber interval based on far-field measurements. Based on a prior information given by the linear sampling method, the neural network approach is proposed for the reconstruction of the unknown scatterer. We divide the wavenumber intervals into several subintervals, ensuring that each transmission eigenvalue is located in its corresponding subinterval. In each such subinterval, the wavenumber that yields the maximum value of the indicator functional will be included in the input set during the generation of the training data. This technique for data generation effectively ensures the consistent dimensions of model input. The refractive index and shape are taken as the output of the network. Due to the fact that transmission eigenvalues considered in our method are relatively small, certain super-resolution effects can also be generated. Numerical experiments are presented to verify the effectiveness and promising features of the proposed method in two and three dimensions.
{"title":"On a Hybrid Method for Inverse Transmission Eigenvalue Problems","authors":"Weishi Yin,Zhaobin Xu,Pinchao Meng, Hongyu Liu","doi":"10.4208/aam.oa-2024-0003","DOIUrl":"https://doi.org/10.4208/aam.oa-2024-0003","url":null,"abstract":"In this paper, we are concerned with the inverse transmission eigenvalue problem to recover the shape as well as the constant refractive index of\u0000a penetrable medium scatterer. The linear sampling method is employed to\u0000determine the transmission eigenvalues within a certain wavenumber interval\u0000based on far-field measurements. Based on a prior information given by the\u0000linear sampling method, the neural network approach is proposed for the reconstruction of the unknown scatterer. We divide the wavenumber intervals\u0000into several subintervals, ensuring that each transmission eigenvalue is located\u0000in its corresponding subinterval. In each such subinterval, the wavenumber that\u0000yields the maximum value of the indicator functional will be included in the\u0000input set during the generation of the training data. This technique for data\u0000generation effectively ensures the consistent dimensions of model input. The\u0000refractive index and shape are taken as the output of the network. Due to the\u0000fact that transmission eigenvalues considered in our method are relatively small,\u0000certain super-resolution effects can also be generated. Numerical experiments\u0000are presented to verify the effectiveness and promising features of the proposed\u0000method in two and three dimensions.","PeriodicalId":517399,"journal":{"name":"Annals of Applied Mathematics","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140932874","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-01DOI: 10.4208/aam.oa-2024-0013
Mejdi Azaiez, Kévin Le Balc’h
We investigate the numerical approximation for stabilizing the semidiscrete linearized Boussinesq system around an unstable stationary state. Stabilization is attained through internal feedback controls applied to the velocity and temperature equations, localized within an arbitrary open subset. This study follows the framework presented in [14], considering the continuous linearized Boussinesq system. The primary objective is to explore the penalization-based approximation of the free divergence condition in the semidiscrete case and provide a numerical validation of these results in a two-dimensional setting.
{"title":"Numerical Study of Semidiscrete Penalty Approach for Stabilizing Boussinesq System with Localized Feedback Control","authors":"Mejdi Azaiez, Kévin Le Balc’h","doi":"10.4208/aam.oa-2024-0013","DOIUrl":"https://doi.org/10.4208/aam.oa-2024-0013","url":null,"abstract":"We investigate the numerical approximation for stabilizing the\u0000semidiscrete linearized Boussinesq system around an unstable stationary state.\u0000Stabilization is attained through internal feedback controls applied to the velocity and temperature equations, localized within an arbitrary open subset. This\u0000study follows the framework presented in [14], considering the continuous linearized Boussinesq system. The primary objective is to explore the penalization-based approximation of the free divergence condition in the semidiscrete case\u0000and provide a numerical validation of these results in a two-dimensional setting.","PeriodicalId":517399,"journal":{"name":"Annals of Applied Mathematics","volume":"131 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140932887","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-01DOI: 10.4208/aam.oa-2024-0007
Logan J. Cross, Xiangxiong Zhang
The monotonicity of discrete Laplacian, i.e., inverse positivity of stiffness matrix, implies discrete maximum principle, which is in general not true for high order accurate schemes on unstructured meshes. On the other hand, it is possible to construct high order accurate monotone schemes on structured meshes. All previously known high order accurate inverse positive schemes are or can be regarded as fourth order accurate finite difference schemes, which is either an M-matrix or a product of two M-matrices. For the $Q^3$ spectral element method for the two-dimensional Laplacian, we prove its stiffness matrix is a product of four M-matrices thus it is unconditionally monotone. Such a scheme can be regarded as a fifth order accurate finite difference scheme. Numerical tests suggest that the unconditional monotonicity of $Q^k$ spectral element methods will be lost for $k≥9$ in two dimensions, and for $k≥4$ in three dimensions. In other words, for obtaining a high order monotone scheme, only $Q^2$ and $Q^3$ spectral element methods can be unconditionally monotone in three dimensions.
离散拉普拉卡矩阵的单调性，即刚度矩阵的逆正性，意味着离散最大原则，这对于非结构网格上的高阶精确方案来说一般是不正确的。另一方面，在结构网格上构建高阶精确单调方案是可能的。所有之前已知的高阶精确逆正方案都是或可以看作是四阶精确有限差分方案，它要么是一个 M 矩阵，要么是两个 M 矩阵的乘积。对于二维拉普拉斯矩的 $Q^3$ 光谱元素法，我们证明其刚度矩阵是四个 M 矩阵的乘积，因此它是无条件单调的。这种方案可视为五阶精确有限差分方案。数值测试表明，在二维中，当 $k≥9$ 时，$Q^k$ 谱元法的无条件单调性将消失；在三维中，当 $k≥4$ 时，$Q^k$ 谱元法的无条件单调性将消失。换句话说，要获得高阶单调方案，只有 $Q^2$ 和 $Q^3$ 光谱元方法在三维空间可以无条件单调。
{"title":"On the Monotonicity of $Q^3$ Spectral Element Method for Laplacian","authors":"Logan J. Cross, Xiangxiong Zhang","doi":"10.4208/aam.oa-2024-0007","DOIUrl":"https://doi.org/10.4208/aam.oa-2024-0007","url":null,"abstract":"The monotonicity of discrete Laplacian, i.e., inverse positivity of\u0000stiffness matrix, implies discrete maximum principle, which is in general not true\u0000for high order accurate schemes on unstructured meshes. On the other hand,\u0000it is possible to construct high order accurate monotone schemes on structured\u0000meshes. All previously known high order accurate inverse positive schemes are\u0000or can be regarded as fourth order accurate finite difference schemes, which is\u0000either an M-matrix or a product of two M-matrices. For the $Q^3$ spectral element\u0000method for the two-dimensional Laplacian, we prove its stiffness matrix is a\u0000product of four M-matrices thus it is unconditionally monotone. Such a scheme\u0000can be regarded as a fifth order accurate finite difference scheme. Numerical tests\u0000suggest that the unconditional monotonicity of $Q^k$ spectral element methods will\u0000be lost for $k≥9$ in two dimensions, and for $k≥4$ in three dimensions. In other\u0000words, for obtaining a high order monotone scheme, only $Q^2$ and $Q^3$ spectral\u0000element methods can be unconditionally monotone in three dimensions.","PeriodicalId":517399,"journal":{"name":"Annals of Applied Mathematics","volume":"84 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140932911","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-02-01DOI: 10.4208/aam.oa-2023-0035
Dawei Wu, Zhennan Zhou
The stochastic growth-fragmentation model describes the temporal evolution of a structured cell population through a discrete-time and continuous-state Markov chain. The simulations of this stochastic process and its invariant measure are of interest. In this paper, we propose a numerical scheme for both the simulation of the process and the computation of the invariant measure, and show that under appropriate assumptions, the numerical chain converges to the continuous growth-fragmentation chain with an explicit error bound. With a triangle inequality argument, we are also able to quantitatively estimate the distance between the invariant measures of these two Markov chains.
{"title":"A Convergent Numerical Algorithm for the Stochastic Growth-Fragmentation Problem","authors":"Dawei Wu, Zhennan Zhou","doi":"10.4208/aam.oa-2023-0035","DOIUrl":"https://doi.org/10.4208/aam.oa-2023-0035","url":null,"abstract":"The stochastic growth-fragmentation model describes the temporal evolution of a structured cell population through a discrete-time and\u0000continuous-state Markov chain. The simulations of this stochastic process and\u0000its invariant measure are of interest. In this paper, we propose a numerical\u0000scheme for both the simulation of the process and the computation of the invariant measure, and show that under appropriate assumptions, the numerical\u0000chain converges to the continuous growth-fragmentation chain with an explicit\u0000error bound. With a triangle inequality argument, we are also able to quantitatively estimate the distance between the invariant measures of these two\u0000Markov chains.","PeriodicalId":517399,"journal":{"name":"Annals of Applied Mathematics","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140011661","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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