Pub Date : 2023-04-21DOI: 10.1142/s0219530523500197
P. Colli, G. Gilardi, G. Marinoschi, E. Rocca
This paper is concerned with the well-posedness and optimal control problem of a reaction-diffusion system for an epidemic Susceptible-Infected-Recovered-Susceptible (SIRS) mathematical model in which the dynamics develops in a spatially heterogeneous environment. Using as control variables the transmission rates $u_{i}$ and $u_{e}$ of contagion resulting from the contact with both asymptomatic and symptomatic persons, respectively, we optimize the number of exposed and infected individuals at a final time $T$ of the controlled evolution of the system. More precisely, we search for the optimal $u_{i}$ and $u_{e}$ such that the number of infected plus exposed does not exceed at the final time a threshold value $Lambda$, fixed a priori. We prove here the existence of optimal controls in a proper functional framework and we derive the first-order necessary optimality conditions in terms of the adjoint variables.
{"title":"Optimal control of a reaction-diffusion model related to the spread of COVID-19","authors":"P. Colli, G. Gilardi, G. Marinoschi, E. Rocca","doi":"10.1142/s0219530523500197","DOIUrl":"https://doi.org/10.1142/s0219530523500197","url":null,"abstract":"This paper is concerned with the well-posedness and optimal control problem of a reaction-diffusion system for an epidemic Susceptible-Infected-Recovered-Susceptible (SIRS) mathematical model in which the dynamics develops in a spatially heterogeneous environment. Using as control variables the transmission rates $u_{i}$ and $u_{e}$ of contagion resulting from the contact with both asymptomatic and symptomatic persons, respectively, we optimize the number of exposed and infected individuals at a final time $T$ of the controlled evolution of the system. More precisely, we search for the optimal $u_{i}$ and $u_{e}$ such that the number of infected plus exposed does not exceed at the final time a threshold value $Lambda$, fixed a priori. We prove here the existence of optimal controls in a proper functional framework and we derive the first-order necessary optimality conditions in terms of the adjoint variables.","PeriodicalId":55519,"journal":{"name":"Analysis and Applications","volume":" ","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46527749","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-13DOI: 10.1142/s0219530523500033
Shengda Zeng, Yunru Bai, Nikolaos S. Papageorgiou, Vicentiu D. Rădulescu
{"title":"Double phase implicit obstacle problems with convection term and multivalued operator","authors":"Shengda Zeng, Yunru Bai, Nikolaos S. Papageorgiou, Vicentiu D. Rădulescu","doi":"10.1142/s0219530523500033","DOIUrl":"https://doi.org/10.1142/s0219530523500033","url":null,"abstract":"","PeriodicalId":55519,"journal":{"name":"Analysis and Applications","volume":" ","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44755171","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-13DOI: 10.1142/s0219530523500021
Michael K. Ng, A. Yip
{"title":"Stability and Generalization of Graph Convolutional Networks in Eigen-domains","authors":"Michael K. Ng, A. Yip","doi":"10.1142/s0219530523500021","DOIUrl":"https://doi.org/10.1142/s0219530523500021","url":null,"abstract":"","PeriodicalId":55519,"journal":{"name":"Analysis and Applications","volume":" ","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45064452","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-26DOI: 10.1142/s0219530522400085
Tong Mao, Zhongjie Shi, Ding-Xuan Zhou
Deep convolutional neural networks (DCNNs) have achieved great empirical success in many fields such as natural language processing, computer vision, and pattern recognition. But there still lacks theoretical understanding of the flexibility and adaptivity of DCNNs in various learning tasks, and the power of DCNNs at feature extraction. We propose a generic DCNN structure consisting of two groups of convolutional layers associated with two downsampling operators, and a fully connected layer, which is determined only by three structural parameters. Our generic DCNNs are capable of extracting various features including not only polynomial features but also general smooth features. We also show that the curse of dimensionality can be circumvented by our DCNNs for target functions of the compositional form with (symmetric) polynomial features, spatially sparse smooth features, and interaction features. These demonstrate the expressive power of our DCNN structure, while the model selection can be relaxed comparing with other deep neural networks since there are only three hyperparameters controlling the architecture to tune.
{"title":"Approximating functions with multi-features by deep convolutional neural networks","authors":"Tong Mao, Zhongjie Shi, Ding-Xuan Zhou","doi":"10.1142/s0219530522400085","DOIUrl":"https://doi.org/10.1142/s0219530522400085","url":null,"abstract":"Deep convolutional neural networks (DCNNs) have achieved great empirical success in many fields such as natural language processing, computer vision, and pattern recognition. But there still lacks theoretical understanding of the flexibility and adaptivity of DCNNs in various learning tasks, and the power of DCNNs at feature extraction. We propose a generic DCNN structure consisting of two groups of convolutional layers associated with two downsampling operators, and a fully connected layer, which is determined only by three structural parameters. Our generic DCNNs are capable of extracting various features including not only polynomial features but also general smooth features. We also show that the curse of dimensionality can be circumvented by our DCNNs for target functions of the compositional form with (symmetric) polynomial features, spatially sparse smooth features, and interaction features. These demonstrate the expressive power of our DCNN structure, while the model selection can be relaxed comparing with other deep neural networks since there are only three hyperparameters controlling the architecture to tune.","PeriodicalId":55519,"journal":{"name":"Analysis and Applications","volume":" ","pages":""},"PeriodicalIF":2.2,"publicationDate":"2022-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45605324","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-25DOI: 10.1142/s021953052250018x
Zhan Yu
{"title":"Block coordinate type methods for optimization and learning","authors":"Zhan Yu","doi":"10.1142/s021953052250018x","DOIUrl":"https://doi.org/10.1142/s021953052250018x","url":null,"abstract":"","PeriodicalId":55519,"journal":{"name":"Analysis and Applications","volume":" ","pages":""},"PeriodicalIF":2.2,"publicationDate":"2022-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45806669","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-18DOI: 10.1142/s0219530522400139
S. Villa, Simon Matet, B. Vu, L. Rosasco
{"title":"Implicit regularization with strongly convex bias: stability and acceleration","authors":"S. Villa, Simon Matet, B. Vu, L. Rosasco","doi":"10.1142/s0219530522400139","DOIUrl":"https://doi.org/10.1142/s0219530522400139","url":null,"abstract":"","PeriodicalId":55519,"journal":{"name":"Analysis and Applications","volume":" ","pages":""},"PeriodicalIF":2.2,"publicationDate":"2022-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44292836","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-09-08DOI: 10.1142/s0219530522400061
G. Strang, D. Drucker
{"title":"Three Matrix Factorizations from the Steps of Elimination","authors":"G. Strang, D. Drucker","doi":"10.1142/s0219530522400061","DOIUrl":"https://doi.org/10.1142/s0219530522400061","url":null,"abstract":"","PeriodicalId":55519,"journal":{"name":"Analysis and Applications","volume":" ","pages":""},"PeriodicalIF":2.2,"publicationDate":"2022-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42421554","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-08-19DOI: 10.1142/s021953052240005x
H. Feichtinger
{"title":"Translation and modulation invariant banach spaces of tempered distributions satisfy the metric approximation property","authors":"H. Feichtinger","doi":"10.1142/s021953052240005x","DOIUrl":"https://doi.org/10.1142/s021953052240005x","url":null,"abstract":"","PeriodicalId":55519,"journal":{"name":"Analysis and Applications","volume":" ","pages":""},"PeriodicalIF":2.2,"publicationDate":"2022-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44574322","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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