In this paper, we study a class of first-order mean-field games (MFGs) that model price formation. Using Poincaré lemma, we eliminate one of the equations of the MFGs system and obtain a variational problem for a single function. We prove the uniqueness of the solutions to the variational problem and address the existence of solutions by applying relaxation arguments. Moreover, we establish a correspondence between solutions of the MFGs system and the variational problem. Based on this correspondence, we introduce an alternative numerical approach for the solution of the original MFGs problem. We end the paper with numerical results for a linear-quadratic model.
{"title":"A variational approach for price formation models in one dimension","authors":"Yuri Ashrafyan, Tigran Bakaryan, Diogo Gomes, Julian Gutierrez","doi":"10.4310/cms.2024.v22.n1.a10","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n1.a10","url":null,"abstract":"In this paper, we study a class of first-order mean-field games (MFGs) that model price formation. Using Poincaré lemma, we eliminate one of the equations of the MFGs system and obtain a variational problem for a single function. We prove the uniqueness of the solutions to the variational problem and address the existence of solutions by applying relaxation arguments. Moreover, we establish a correspondence between solutions of the MFGs system and the variational problem. Based on this correspondence, we introduce an alternative numerical approach for the solution of the original MFGs problem. We end the paper with numerical results for a linear-quadratic model.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"82 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138556117","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-07DOI: 10.4310/cms.2024.v22.n1.a5
Hao Wang, Guangqing Wang
This paper is concerned with the Cauchy problem on the Vlasov–Poisson–Boltzmann system in the torus domain. The Boltzmann collision kernel is assumed to be angular non-cutoff with $0 leq gamma lt 1$ and $1/2 leq s lt 1$, where $gamma, s$ are two parameters describing the kinetic and angular singularities, respectively. We obtain the global-in-time unique mild solutions, and prove that the solutions converge to the global Maxwellian with the large-time decay rate of $mathcal{O}(e^{-lambda t})$ in the $L^1_k L^2_v$-norm for some $lambda gt 0$. Furthermore, we justify the property of propagation of regularity of solutions in the spatial variable.
{"title":"Global mild solutions of the non-cutoff Vlasov–Poisson–Boltzmann system","authors":"Hao Wang, Guangqing Wang","doi":"10.4310/cms.2024.v22.n1.a5","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n1.a5","url":null,"abstract":"This paper is concerned with the Cauchy problem on the Vlasov–Poisson–Boltzmann system in the torus domain. The Boltzmann collision kernel is assumed to be angular non-cutoff with $0 leq gamma lt 1$ and $1/2 leq s lt 1$, where $gamma, s$ are two parameters describing the kinetic and angular singularities, respectively. We obtain the global-in-time unique mild solutions, and prove that the solutions converge to the global Maxwellian with the large-time decay rate of $mathcal{O}(e^{-lambda t})$ in the $L^1_k L^2_v$-norm for some $lambda gt 0$. Furthermore, we justify the property of propagation of regularity of solutions in the spatial variable.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"4 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138556118","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-07DOI: 10.4310/cms.2024.v22.n1.a1
Yue Feng, Ying Ma
We establish the improved uniform error bound on the time-splitting Fourier pseudospectral (TSFP) method for the long-time dynamics of the generalized fractional nonlinear Schrödinger equation (FNLSE) with $O(varepsilon^2)$-nonlinearity, where $varepsilon in (0,1]$ is a dimensionless parameter. Numerically, we discretize the FNLSE by the second-order Strang splitting method in time and Fourier pseudospectral method in space. Combining with energy method, we utilize the regularity compensation oscillation (RCO) technique to rigorously prove the improved uniform error bound at $O(h^{m_0} + varepsilon^2 tau^2)$ with the mesh size $h$ and time step $tau$ up to the long-time at $O(1 / varepsilon^2)$, which gains an additional $varepsilon^2$ in time compared with classical error estimates. The key idea behind the RCO technique is to analyze low frequency modes by phase cancellation and control high frequency modes by the regularity of the exact solution. With the help of the RCO technique, we relax some constraints in the previous proof for the improved uniform error bound and extend the result to more general cases. Finally, numerical examples are provided to confirm our improved uniform error bound and demonstrate its suitability in different cases.
{"title":"Improved uniform error bound on the time-splitting method for the long-time dynamics of the fractional nonlinear Schrödinger equation","authors":"Yue Feng, Ying Ma","doi":"10.4310/cms.2024.v22.n1.a1","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n1.a1","url":null,"abstract":"We establish the improved uniform error bound on the time-splitting Fourier pseudospectral (TSFP) method for the long-time dynamics of the generalized fractional nonlinear Schrödinger equation (FNLSE) with $O(varepsilon^2)$-nonlinearity, where $varepsilon in (0,1]$ is a dimensionless parameter. Numerically, we discretize the FNLSE by the second-order Strang splitting method in time and Fourier pseudospectral method in space. Combining with energy method, we utilize the regularity compensation oscillation (RCO) technique to rigorously prove the improved uniform error bound at $O(h^{m_0} + varepsilon^2 tau^2)$ with the mesh size $h$ and time step $tau$ up to the long-time at $O(1 / varepsilon^2)$, which gains an additional $varepsilon^2$ in time compared with classical error estimates. The key idea behind the RCO technique is to analyze low frequency modes by phase cancellation and control high frequency modes by the regularity of the exact solution. With the help of the RCO technique, we relax some constraints in the previous proof for the improved uniform error bound and extend the result to more general cases. Finally, numerical examples are provided to confirm our improved uniform error bound and demonstrate its suitability in different cases.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"60 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138556847","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-07DOI: 10.4310/cms.2024.v22.n1.a3
Lingling Wang, Chong Lai
A multi-agent wealth exchange model, which considers a varying trading propensity and a control of wealth inequality, is adopted to investigate the wealth distribution under infectious disease. Using the feedback control method, two saturated nonlinear incidence rates are obtained to explore the impact of the government contact control measures on epidemic dynamics and wealth distribution. We find that the contact control measures may reduce the peak of the infected fraction and make more people remain uninfected, but prolong the duration of the epidemic and increase wealth inequality. In a closed (an open) economy, the large-time behavior of wealth distribution presents a Pareto tail, and examples of trading propensity depending on wealth suggest that an increase in the savings of the wealthy may increase wealth inequality. In addition, our simulation results illustrate that the government’s tax and redistribution measures can alleviate the wealth inequality caused by the contact control and improve the wealth of agents at low and middle levels.
{"title":"Epidemic dynamics and wealth inequality under two feedback control strategies","authors":"Lingling Wang, Chong Lai","doi":"10.4310/cms.2024.v22.n1.a3","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n1.a3","url":null,"abstract":"A multi-agent wealth exchange model, which considers a varying trading propensity and a control of wealth inequality, is adopted to investigate the wealth distribution under infectious disease. Using the feedback control method, two saturated nonlinear incidence rates are obtained to explore the impact of the government contact control measures on epidemic dynamics and wealth distribution. We find that the contact control measures may reduce the peak of the infected fraction and make more people remain uninfected, but prolong the duration of the epidemic and increase wealth inequality. In a closed (an open) economy, the large-time behavior of wealth distribution presents a Pareto tail, and examples of trading propensity depending on wealth suggest that an increase in the savings of the wealthy may increase wealth inequality. In addition, our simulation results illustrate that the government’s tax and redistribution measures can alleviate the wealth inequality caused by the contact control and improve the wealth of agents at low and middle levels.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"40 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138556886","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-07DOI: 10.4310/cms.2024.v22.n1.a9
Olga S. Rozanova, Marko K. Turzynsky
We study the affine solutions of the equations of plane oscillations of cold plasma, which, under the assumption of electrostaticity, correspond to the Euler–Poisson equations in the repulsive case. It is proved that the zero equilibrium state of the cold plasma equations, both with and without the assumption of electrostaticity, is unstable in the class of all affine solutions. It is also shown that an arbitrary perturbation of an axially symmetric electrostatic solution leads to a finite time blow-up.
{"title":"On the properties of affine solutions of cold plasma equations","authors":"Olga S. Rozanova, Marko K. Turzynsky","doi":"10.4310/cms.2024.v22.n1.a9","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n1.a9","url":null,"abstract":"We study the affine solutions of the equations of plane oscillations of cold plasma, which, under the assumption of electrostaticity, correspond to the Euler–Poisson equations in the repulsive case. It is proved that the zero equilibrium state of the cold plasma equations, both with and without the assumption of electrostaticity, is unstable in the class of all affine solutions. It is also shown that an arbitrary perturbation of an axially symmetric electrostatic solution leads to a finite time blow-up.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"111 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138557142","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-07DOI: 10.4310/cms.2024.v22.n1.a11
Xia Ye, Mingxuan Zhu
This paper is concerned with the initial value problem of the three-dimensional nonhomogeneous incompressible liquid crystal system with vacuum and density-dependent viscosity. We prove the existence of global strong solution on $mathbb{R}^3 times (0,infty)$ under the initial norm ${lVert u_0 rVert}_{dot{H}^alpha} + {lVert nabla d_0 rVert}_{dot{H}^alpha} (1/2 lt alpha leq 1)$ being suitably small. In addition, the algebraic decay rate estimates of the global strong solution are obtained.
{"title":"Existence and decay of global strong solutions to the nonhomogeneous incompressible liquid crystal system with vacuum and density-dependent viscosity","authors":"Xia Ye, Mingxuan Zhu","doi":"10.4310/cms.2024.v22.n1.a11","DOIUrl":"https://doi.org/10.4310/cms.2024.v22.n1.a11","url":null,"abstract":"This paper is concerned with the initial value problem of the three-dimensional nonhomogeneous incompressible liquid crystal system with vacuum and density-dependent viscosity. We prove the existence of global strong solution on $mathbb{R}^3 times (0,infty)$ under the initial norm ${lVert u_0 rVert}_{dot{H}^alpha} + {lVert nabla d_0 rVert}_{dot{H}^alpha} (1/2 lt alpha leq 1)$ being suitably small. In addition, the algebraic decay rate estimates of the global strong solution are obtained.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"248 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138556121","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-15DOI: 10.4310/cms.2023.v21.n8.a1
Antoine Benoit, Jean-François Coulombel
We study the stability of the two-dimensional Lax–Wendroff scheme with a stabilizer that approximates solutions to the transport equation. The problem is first analyzed in the whole space in order to show that the so-called energy method yields an optimal stability criterion for this finite difference scheme. We then deal with the case of a half-space when the transport operator is outgoing. At the numerical level, we enforce the Neumann extrapolation boundary condition and show that the corresponding scheme is stable. Eventually we analyze the case of a quarter-space when the transport operator is outgoing with respect to both sides. We then enforce the Neumann extrapolation boundary condition on each side of the boundary and propose an extrapolation boundary condition at the numerical corner in order to maintain stability for the whole numerical scheme.
{"title":"The Neumann boundary condition for the two-dimensional Lax–Wendroff scheme","authors":"Antoine Benoit, Jean-François Coulombel","doi":"10.4310/cms.2023.v21.n8.a1","DOIUrl":"https://doi.org/10.4310/cms.2023.v21.n8.a1","url":null,"abstract":"We study the stability of the two-dimensional Lax–Wendroff scheme with a stabilizer that approximates solutions to the transport equation. The problem is first analyzed in the whole space in order to show that the so-called energy method yields an optimal stability criterion for this finite difference scheme. We then deal with the case of a half-space when the transport operator is outgoing. At the numerical level, we enforce the Neumann extrapolation boundary condition and show that the corresponding scheme is stable. Eventually we analyze the case of a quarter-space when the transport operator is outgoing with respect to both sides. We then enforce the Neumann extrapolation boundary condition on each side of the boundary and propose an extrapolation boundary condition at the numerical corner in order to maintain stability for the whole numerical scheme.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138519800","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-15DOI: 10.4310/cms.2023.v21.n8.a6
Jean Bussac
This paper concerns the study of a full non-equilibrium model for a compressible mixture of any number of phases. Miscible conditions are considered in one phase, which lead to non-symmetric constraints on the statistical fractions. These models are subject to the choice of interfacial and source terms. We show that under a standard assumption on the interfacial velocity, the interfacial pressures are uniquely defined. The model is hyperbolic and symmetrizable under nonresonance conditions. Classes of entropy-consistent source terms are then proposed.
{"title":"A non-equilibrium multi-component model with miscible conditions","authors":"Jean Bussac","doi":"10.4310/cms.2023.v21.n8.a6","DOIUrl":"https://doi.org/10.4310/cms.2023.v21.n8.a6","url":null,"abstract":"This paper concerns the study of a full non-equilibrium model for a compressible mixture of any number of phases. Miscible conditions are considered in one phase, which lead to non-symmetric constraints on the statistical fractions. These models are subject to the choice of interfacial and source terms. We show that under a standard assumption on the interfacial velocity, the interfacial pressures are uniquely defined. The model is hyperbolic and symmetrizable under nonresonance conditions. Classes of entropy-consistent source terms are then proposed.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"76 3","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138519806","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-09DOI: 10.4310/cms.2023.v21.n7.a13
Renata Bunoiu, Karim Ramdani, Claudia Timofte
We study the homogenization of a scalar problem posed in a composite medium made up of two materials, a positive and a negative one. An important feature is the presence of a flux jump across their oscillating interface. The main difficulties of this study are due to the sign-changing coefficients and the appearance of an unsigned surface integral term in the variational formulation. A proof by contradiction (nonstandard in this context) and T‑coercivity technics are used in order to cope with these difficulties.
{"title":"Homogenization of a transmission problem with sign-changing coefficients and interfacial flux jump","authors":"Renata Bunoiu, Karim Ramdani, Claudia Timofte","doi":"10.4310/cms.2023.v21.n7.a13","DOIUrl":"https://doi.org/10.4310/cms.2023.v21.n7.a13","url":null,"abstract":"We study the homogenization of a scalar problem posed in a composite medium made up of two materials, a positive and a negative one. An important feature is the presence of a flux jump across their oscillating interface. The main difficulties of this study are due to the sign-changing coefficients and the appearance of an unsigned surface integral term in the variational formulation. A proof by contradiction (nonstandard in this context) and T‑coercivity technics are used in order to cope with these difficulties.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"1200 ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138515179","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-09-22DOI: 10.4310/cms.2023.v21.n6.a6
Jing An, Lexing Ying
Many machine learning and data science tasks require solving non-convex optimization problems. When the loss function is a sum of multiple terms, a popular method is the stochastic gradient descent. Viewed as a process for sampling the loss function landscape, the stochastic gradient descent is known to prefer flat minima. Though this is desired for certain optimization problems such as in deep learning, it causes issues when the goal is to find the global minimum, especially if the global minimum resides in a sharp valley. Illustrated with a simple motivating example, we show that the fundamental reason is that the difference in the Lipschitz constants of multiple terms in the loss function causes stochastic gradient descent to experience different gradient variances at different minima. In order to mitigate this effect and perform faithful optimization, we propose a combined resampling-reweighting scheme to balance the variance at local minima and extend to general loss functions. We explain from the numerical stability perspective how the proposed scheme is more likely to select the true global minimum, and from the local convergence analysis perspective how it converges to a minimum faster when compared with the vanilla stochastic gradient descent. Experiments from robust statistics and computational chemistry are provided to demonstrate the theoretical findings.
{"title":"Combining resampling and reweighting for faithful stochastic optimization","authors":"Jing An, Lexing Ying","doi":"10.4310/cms.2023.v21.n6.a6","DOIUrl":"https://doi.org/10.4310/cms.2023.v21.n6.a6","url":null,"abstract":"Many machine learning and data science tasks require solving non-convex optimization problems. When the loss function is a sum of multiple terms, a popular method is the stochastic gradient descent. Viewed as a process for sampling the loss function landscape, the stochastic gradient descent is known to prefer flat minima. Though this is desired for certain optimization problems such as in deep learning, it causes issues when the goal is to find the global minimum, especially if the global minimum resides in a sharp valley. Illustrated with a simple motivating example, we show that the fundamental reason is that the difference in the Lipschitz constants of multiple terms in the loss function causes stochastic gradient descent to experience different gradient variances at different minima. In order to mitigate this effect and perform faithful optimization, we propose a combined resampling-reweighting scheme to balance the variance at local minima and extend to general loss functions. We explain from the numerical stability perspective how the proposed scheme is more likely to select the true global minimum, and from the local convergence analysis perspective how it converges to a minimum faster when compared with the vanilla stochastic gradient descent. Experiments from robust statistics and computational chemistry are provided to demonstrate the theoretical findings.","PeriodicalId":50659,"journal":{"name":"Communications in Mathematical Sciences","volume":"1219 ","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138515178","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：