Pub Date : 2022-06-01DOI: 10.4208/csiam-am.so-2021-0023
Yihui Han null, Haitao Leng
{"title":"Adaptive $H$(div)-Conforming Embedded-Hybridized Discontinuous Galerkin Finite Element Methods for the Stokes Problems","authors":"Yihui Han null, Haitao Leng","doi":"10.4208/csiam-am.so-2021-0023","DOIUrl":"https://doi.org/10.4208/csiam-am.so-2021-0023","url":null,"abstract":"","PeriodicalId":29749,"journal":{"name":"CSIAM Transactions on Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46361792","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 : 2022-06-01DOI: 10.4208/csiam-am.so-2021-0008
Jie Shen null, Nan Zheng
We develop in this paper efficient numerical schemes for solving the micropolar Navier-Stokes equations by combining the SAV approach and pressure-correction method. Our firstand second-order semi-discrete schemes enjoy remarkable properties such as (i) unconditional energy stable with a modified energy, and (ii) only a sequence of decoupled linear equations with constant coefficients need to be solved at each time step. We also construct fully discrete versions of these schemes with a special spectral discretization which preserve the essential properties of the semi-discrete schemes. Numerical experiments are presented to validate the proposed schemes. AMS subject classifications: 65M12, 65M70, 35Q30, 76A05
{"title":"Efficient and Unconditional Energy Stable Schemes for the Micropolar Navier-Stokes Equations","authors":"Jie Shen null, Nan Zheng","doi":"10.4208/csiam-am.so-2021-0008","DOIUrl":"https://doi.org/10.4208/csiam-am.so-2021-0008","url":null,"abstract":"We develop in this paper efficient numerical schemes for solving the micropolar Navier-Stokes equations by combining the SAV approach and pressure-correction method. Our firstand second-order semi-discrete schemes enjoy remarkable properties such as (i) unconditional energy stable with a modified energy, and (ii) only a sequence of decoupled linear equations with constant coefficients need to be solved at each time step. We also construct fully discrete versions of these schemes with a special spectral discretization which preserve the essential properties of the semi-discrete schemes. Numerical experiments are presented to validate the proposed schemes. AMS subject classifications: 65M12, 65M70, 35Q30, 76A05","PeriodicalId":29749,"journal":{"name":"CSIAM Transactions on Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45428075","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 : 2022-02-28DOI: 10.4208/csiam-am.so-2022-0033
F. Jiang, Song Jiang, Youyi Zhao
It is still open whether the phenomenon of inhibition of Rayleigh--Taylor (RT) instability by a horizontal magnetic field can be mathematically verified for a non-resistive emph{viscous} magnetohydrodynamic (MHD) fluid in a two-dimensional (2D) horizontal slab domain, since it was roughly proved in the linearized case by Wang in cite{WYC}. In this paper, we prove such inhibition phenomenon by the (nonlinear) inhomogeneous, incompressible, emph{viscous case} with emph{Navier (slip) boundary condition}. More precisely, we show that there is a critical number of field strength $m_{mm{C}}$, such that if the strength $|m|$ of a horizontal magnetic field is bigger than $m_{mm{C}}$, then the small perturbation solution around the magnetic RT equilibrium state is {algebraically} stable in time. In addition, we also provide a nonlinear instability result for the case $|m|in[0, m_{mm{C}})$. The instability result presents that a horizontal magnetic field can not inhibit the RT instability, if it's strength is too small.
{"title":"On Inhibition of the Rayleigh-Taylor Instability by a Horizontal Magnetic Field in 2D Non-Resistive MHD Fluids: The Viscous Case","authors":"F. Jiang, Song Jiang, Youyi Zhao","doi":"10.4208/csiam-am.so-2022-0033","DOIUrl":"https://doi.org/10.4208/csiam-am.so-2022-0033","url":null,"abstract":"It is still open whether the phenomenon of inhibition of Rayleigh--Taylor (RT) instability by a horizontal magnetic field can be mathematically verified for a non-resistive emph{viscous} magnetohydrodynamic (MHD) fluid in a two-dimensional (2D) horizontal slab domain, since it was roughly proved in the linearized case by Wang in cite{WYC}. In this paper, we prove such inhibition phenomenon by the (nonlinear) inhomogeneous, incompressible, emph{viscous case} with emph{Navier (slip) boundary condition}. More precisely, we show that there is a critical number of field strength $m_{mm{C}}$, such that if the strength $|m|$ of a horizontal magnetic field is bigger than $m_{mm{C}}$, then the small perturbation solution around the magnetic RT equilibrium state is {algebraically} stable in time. In addition, we also provide a nonlinear instability result for the case $|m|in[0, m_{mm{C}})$. The instability result presents that a horizontal magnetic field can not inhibit the RT instability, if it's strength is too small.","PeriodicalId":29749,"journal":{"name":"CSIAM Transactions on Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2022-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44366264","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 : 2021-11-10DOI: 10.4208/csiam-am.so-2021-0040
Yuhong Dai, Liwei Zhang
Minimax optimization problems are an important class of optimization problems arising from both modern machine learning and from traditional research areas. We focus on the stability of constrained minimax optimization problems based on the notion of local minimax point by Dai and Zhang (2020). Firstly, we extend the classical Jacobian uniqueness conditions of nonlinear programming to the constrained minimax problem and prove that this set of properties is stable with respect to small $C^2$ perturbation. Secondly, we provide a set of conditions, called Property A, which does not require the strict complementarity condition for the upper level constraints. Finally, we prove that Property A is a sufficient condition for the strong regularity of the Kurash-Kuhn-Tucker (KKT) system at the KKT point, and it is also a sufficient condition for the local Lipschitzian homeomorphism of the Kojima mapping near the KKT point.
{"title":"Stability for Constrained Minimax Optimization","authors":"Yuhong Dai, Liwei Zhang","doi":"10.4208/csiam-am.so-2021-0040","DOIUrl":"https://doi.org/10.4208/csiam-am.so-2021-0040","url":null,"abstract":"Minimax optimization problems are an important class of optimization problems arising from both modern machine learning and from traditional research areas. We focus on the stability of constrained minimax optimization problems based on the notion of local minimax point by Dai and Zhang (2020). Firstly, we extend the classical Jacobian uniqueness conditions of nonlinear programming to the constrained minimax problem and prove that this set of properties is stable with respect to small $C^2$ perturbation. Secondly, we provide a set of conditions, called Property A, which does not require the strict complementarity condition for the upper level constraints. Finally, we prove that Property A is a sufficient condition for the strong regularity of the Kurash-Kuhn-Tucker (KKT) system at the KKT point, and it is also a sufficient condition for the local Lipschitzian homeomorphism of the Kojima mapping near the KKT point.","PeriodicalId":29749,"journal":{"name":"CSIAM Transactions on Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2021-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45845729","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 : 2021-07-17DOI: 10.4208/csiam-am.so-2022-0028
Jin Cheng, Shuai Lu, Masahiro Yamamoto
We consider a diffusion and a wave equations: $$ partial_t^ku(x,t) = Delta u(x,t) + mu(t)f(x), quad xin Omega, , t>0, quad k=1,2 $$ with the zero initial and boundary conditions, where $Omega subset mathbb{R}^d$ is a bounded domain. We establish uniqueness and/or stability results for inverse problems of 1. determining $mu(t)$, $0
{"title":"Determination of Source Terms in Diffusion and Wave Equations by Observations After Incidents: Uniqueness and Stability","authors":"Jin Cheng, Shuai Lu, Masahiro Yamamoto","doi":"10.4208/csiam-am.so-2022-0028","DOIUrl":"https://doi.org/10.4208/csiam-am.so-2022-0028","url":null,"abstract":"We consider a diffusion and a wave equations: $$ partial_t^ku(x,t) = Delta u(x,t) + mu(t)f(x), quad xin Omega, , t>0, quad k=1,2 $$ with the zero initial and boundary conditions, where $Omega subset mathbb{R}^d$ is a bounded domain. We establish uniqueness and/or stability results for inverse problems of 1. determining $mu(t)$, $0<t<T$ with given $f(x)$; 2. determining $f(x)$, $xin Omega$ with given $mu(t)$ end{itemize} by data of $u$: $u(x_0,cdot)$ with fixed point $x_0in Omega$ or Neumann data on subboundary over time interval. In our inverse problems, data are taken over time interval $T_1<t<T_1$, by assuming that $T<T_1<T_2$ and $mu(t)=0$ for $tge T$, which means that the source stops to be active after the time $T$ and the observations are started only after $T$. This assumption is practical by such a posteriori data after incidents, although inverse problems had been well studied in the case of $T=0$. We establish the non-uniqueness, the uniqueness and conditional stability for a diffusion and a wave equations. The proofs are based on eigenfunction expansions of the solutions $u(x,t)$, and we rely on various knowledge of the generalized Weierstrass theorem on polynomial approximation, almost periodic functions, Carleman estimate, non-harmonic Fourier series.","PeriodicalId":29749,"journal":{"name":"CSIAM Transactions on Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2021-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49437655","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 : 2021-06-13DOI: 10.1101/2021.06.09.21258668
J. Jia, S. Liu, Y. Liu, R. Shan, K. Zennir, R. Zhang
In this paper, we formulate a special epidemic dynamic model to describe the transmission of COVID-19 in Algeria. We derive the threshold parameter control reproduction number (R0c ), and present the effective control reproduction number (Rc(t)) as a real-time index for evaluating the epidemic under different control strategies. Due to the limitation of the reported data, we redefine the number of accumulative confirmed cases with diagnostic shadow and then use the processed data to do the optimal numerical simulations. According to the control measures, we divide the whole research period into six stages. And then the corresponding medical resource estimations and the average effective control reproduction numbers for each stage are given. Meanwhile, we use the parameter values which are obtained from the optimal numerical simulations to forecast the whole epidemic tendency under different control strategies.
{"title":"Modeling and reviewing analysis of the COVID-19 epidemic in Algeria with diagnostic shadow","authors":"J. Jia, S. Liu, Y. Liu, R. Shan, K. Zennir, R. Zhang","doi":"10.1101/2021.06.09.21258668","DOIUrl":"https://doi.org/10.1101/2021.06.09.21258668","url":null,"abstract":"In this paper, we formulate a special epidemic dynamic model to describe the transmission of COVID-19 in Algeria. We derive the threshold parameter control reproduction number (R0c ), and present the effective control reproduction number (Rc(t)) as a real-time index for evaluating the epidemic under different control strategies. Due to the limitation of the reported data, we redefine the number of accumulative confirmed cases with diagnostic shadow and then use the processed data to do the optimal numerical simulations. According to the control measures, we divide the whole research period into six stages. And then the corresponding medical resource estimations and the average effective control reproduction numbers for each stage are given. Meanwhile, we use the parameter values which are obtained from the optimal numerical simulations to forecast the whole epidemic tendency under different control strategies.","PeriodicalId":29749,"journal":{"name":"CSIAM Transactions on Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2021-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43336705","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 : 2021-06-12DOI: 10.4208/csiam-am.so-2021-0030
Dong Li, Chaoyu Quan, T. Tang, Wen Yang
We consider numerical solutions for the Allen-Cahn equation with standard double well potential and periodic boundary conditions. Surprisingly it is found that using standard numerical discretizations with high precision computational solutions may converge to completely incorrect steady states. This happens for very smooth initial data and state-of-the-art algorithms. We analyze this phenomenon and showcase the resolution of this problem by a new symmetry-preserving filter technique. We develop a new theoretical framework and rigorously prove the convergence to steady states for the filtered solutions.
{"title":"On Symmetry Breaking of Allen-Cahn","authors":"Dong Li, Chaoyu Quan, T. Tang, Wen Yang","doi":"10.4208/csiam-am.so-2021-0030","DOIUrl":"https://doi.org/10.4208/csiam-am.so-2021-0030","url":null,"abstract":"We consider numerical solutions for the Allen-Cahn equation with standard double well potential and periodic boundary conditions. Surprisingly it is found that using standard numerical discretizations with high precision computational solutions may converge to completely incorrect steady states. This happens for very smooth initial data and state-of-the-art algorithms. We analyze this phenomenon and showcase the resolution of this problem by a new symmetry-preserving filter technique. We develop a new theoretical framework and rigorously prove the convergence to steady states for the filtered solutions.","PeriodicalId":29749,"journal":{"name":"CSIAM Transactions on Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2021-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49025514","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 : 2021-06-01DOI: 10.4208/CSIAM-AM.2020-0212
Jun Hou
{"title":"Identification of Corrupted Data via $k$-Means Clustering for Function Approximation","authors":"Jun Hou","doi":"10.4208/CSIAM-AM.2020-0212","DOIUrl":"https://doi.org/10.4208/CSIAM-AM.2020-0212","url":null,"abstract":"","PeriodicalId":29749,"journal":{"name":"CSIAM Transactions on Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44601437","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 : 2021-06-01DOI: 10.4208/CSIAM-AM.2020-0042
Yanan Zhang
{"title":"Two Modified Schemes for the Primal Dual Fixed Point Method","authors":"Yanan Zhang","doi":"10.4208/CSIAM-AM.2020-0042","DOIUrl":"https://doi.org/10.4208/CSIAM-AM.2020-0042","url":null,"abstract":"","PeriodicalId":29749,"journal":{"name":"CSIAM Transactions on Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.3,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45814410","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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