SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 376-399, February 2024. Abstract. We develop structure preserving schemes for a class of nonlinear mobility continuity equation. When the mobility is a concave function, this equation admits a form of gradient flow with respect to a Wasserstein-like transport metric. Our numerical schemes build upon such formulation and utilize modern large-scale optimization algorithms. There are two distinctive features of our approach compared to previous ones. On the one hand, the essential properties of the solution, including positivity, global bounds, mass conservation, and energy dissipation, are all guaranteed by construction. On the other hand, our approach enjoys sufficient flexibility when applied to a large variety of problems including different free energy functionals, general wetting boundary conditions, and degenerate mobilities. The performance of our methods is demonstrated through a suite of examples.
{"title":"Structure Preserving Primal Dual Methods for Gradient Flows with Nonlinear Mobility Transport Distances","authors":"José A. Carrillo, Li Wang, Chaozhen Wei","doi":"10.1137/23m1562068","DOIUrl":"https://doi.org/10.1137/23m1562068","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 376-399, February 2024. <br/> Abstract. We develop structure preserving schemes for a class of nonlinear mobility continuity equation. When the mobility is a concave function, this equation admits a form of gradient flow with respect to a Wasserstein-like transport metric. Our numerical schemes build upon such formulation and utilize modern large-scale optimization algorithms. There are two distinctive features of our approach compared to previous ones. On the one hand, the essential properties of the solution, including positivity, global bounds, mass conservation, and energy dissipation, are all guaranteed by construction. On the other hand, our approach enjoys sufficient flexibility when applied to a large variety of problems including different free energy functionals, general wetting boundary conditions, and degenerate mobilities. The performance of our methods is demonstrated through a suite of examples.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"236 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139695658","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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 353-375, February 2024. Abstract. Quasiperiodic systems are important space-filling ordered structures, without decay and translational invariance. How to solve quasiperiodic systems accurately and efficiently is a great challenge. A useful approach, the projection method (PM) [J. Comput. Phys., 256 (2014), pp. 428–440], has been proposed to compute quasiperiodic systems. Various studies have demonstrated that the PM is an accurate and efficient method to solve quasiperiodic systems. However, there is a lack of theoretical analysis of the PM. In this paper, we present a rigorous convergence analysis of the PM by establishing a mathematical framework of quasiperiodic functions and their high-dimensional periodic functions. We also give a theoretical analysis of the quasiperiodic spectral method (QSM) based on this framework. Results demonstrate that the PM and QSM both have exponential decay, and the QSM (PM) is a generalization of the periodic Fourier spectral (pseudospectral) method. Then, we analyze the computational complexity of the PM and QSM in calculating quasiperiodic systems. The PM can use a fast Fourier transform, while the QSM cannot. Moreover, we investigate the accuracy and efficiency of the PM, QSM, and periodic approximation method in solving the linear time-dependent quasiperiodic Schrödinger equation.
{"title":"Numerical Methods and Analysis of Computing Quasiperiodic Systems","authors":"Kai Jiang, Shifeng Li, Pingwen Zhang","doi":"10.1137/22m1524783","DOIUrl":"https://doi.org/10.1137/22m1524783","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 353-375, February 2024. <br/> Abstract. Quasiperiodic systems are important space-filling ordered structures, without decay and translational invariance. How to solve quasiperiodic systems accurately and efficiently is a great challenge. A useful approach, the projection method (PM) [J. Comput. Phys., 256 (2014), pp. 428–440], has been proposed to compute quasiperiodic systems. Various studies have demonstrated that the PM is an accurate and efficient method to solve quasiperiodic systems. However, there is a lack of theoretical analysis of the PM. In this paper, we present a rigorous convergence analysis of the PM by establishing a mathematical framework of quasiperiodic functions and their high-dimensional periodic functions. We also give a theoretical analysis of the quasiperiodic spectral method (QSM) based on this framework. Results demonstrate that the PM and QSM both have exponential decay, and the QSM (PM) is a generalization of the periodic Fourier spectral (pseudospectral) method. Then, we analyze the computational complexity of the PM and QSM in calculating quasiperiodic systems. The PM can use a fast Fourier transform, while the QSM cannot. Moreover, we investigate the accuracy and efficiency of the PM, QSM, and periodic approximation method in solving the linear time-dependent quasiperiodic Schrödinger equation.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"8 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139660060","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}
Valeria Banica, Georg Maierhofer, Katharina Schratz
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 322-352, February 2024. Abstract. We introduce a numerical approach to computing the Schrödinger map (SM) based on the Hasimoto transform which relates the SM flow to a cubic nonlinear Schrödinger (NLS) equation. In exploiting this nonlinear transform we are able to introduce the first fully explicit unconditionally stable symmetric integrators for the SM equation. Our approach consists of two parts: an integration of the NLS equation followed by the numerical evaluation of the Hasimoto transform. Motivated by the desire to study rough solutions to the SM equation, we also introduce a new symmetric low-regularity integrator for the NLS equation. This is combined with our novel fast low-regularity Hasimoto (FLowRH) transform, based on a tailored analysis of the resonance structures in the Magnus expansion and a fast realization based on block-Toeplitz partitions, to yield an efficient low-regularity integrator for the SM equation. This scheme in particular allows us to obtain approximations to the SM in a more general regime (i.e., under lower-regularity assumptions) than previously proposed methods. The favorable properties of our methods are exhibited both in theoretical convergence analysis and in numerical experiments.
SIAM 数值分析期刊》第 62 卷第 1 期第 322-352 页，2024 年 2 月。 摘要。我们介绍了一种基于 Hasimoto 变换的薛定谔图（SM）数值计算方法，该变换将薛定谔图流与立方非线性薛定谔（NLS）方程联系起来。利用这种非线性变换，我们能够为 SM 方程引入第一个完全明确的无条件稳定对称积分器。我们的方法由两部分组成：对 NLS 方程进行积分，然后对 Hasimoto 变换进行数值评估。出于研究 SM 方程粗糙解的愿望，我们还为 NLS 方程引入了一种新的对称低规则积分器。它与我们新颖的快速低规则性哈希莫托（FlowRH）变换相结合，基于对马格努斯展开中共振结构的定制分析和基于块-托普利兹分区的快速实现，产生了一种高效的 SM 方程低规则性积分器。与之前提出的方法相比，这一方案尤其能让我们在更一般的情况下（即在低规则性假设下）获得 SM 的近似值。我们的方法在理论收敛分析和数值实验中都表现出了良好的特性。
{"title":"Numerical Integration of Schrödinger Maps via the Hasimoto Transform","authors":"Valeria Banica, Georg Maierhofer, Katharina Schratz","doi":"10.1137/22m1531555","DOIUrl":"https://doi.org/10.1137/22m1531555","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 322-352, February 2024. <br/> Abstract. We introduce a numerical approach to computing the Schrödinger map (SM) based on the Hasimoto transform which relates the SM flow to a cubic nonlinear Schrödinger (NLS) equation. In exploiting this nonlinear transform we are able to introduce the first fully explicit unconditionally stable symmetric integrators for the SM equation. Our approach consists of two parts: an integration of the NLS equation followed by the numerical evaluation of the Hasimoto transform. Motivated by the desire to study rough solutions to the SM equation, we also introduce a new symmetric low-regularity integrator for the NLS equation. This is combined with our novel fast low-regularity Hasimoto (FLowRH) transform, based on a tailored analysis of the resonance structures in the Magnus expansion and a fast realization based on block-Toeplitz partitions, to yield an efficient low-regularity integrator for the SM equation. This scheme in particular allows us to obtain approximations to the SM in a more general regime (i.e., under lower-regularity assumptions) than previously proposed methods. The favorable properties of our methods are exhibited both in theoretical convergence analysis and in numerical experiments.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"121 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139655653","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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 295-321, February 2024. Abstract. We introduce an efficient numerical method for second-order linear ODEs whose solution may vary between highly oscillatory and slowly changing over the solution interval. In oscillatory regions the solution is generated via a nonoscillatory phase function that obeys the nonlinear Riccati equation. We propose a defect correction iteration that gives an asymptotic series for such a phase function; this is numerically approximated on a Chebyshev grid with a small number of nodes. For analytic coefficients we prove that each iteration, up to a certain maximum number, reduces the residual by a factor of order of the local frequency. The algorithm adapts both the stepsize and the choice of method, switching to a conventional spectral collocation method away from oscillatory regions. In numerical experiments we find that our proposal outperforms other state-of-the-art oscillatory solvers, most significantly at low to intermediate frequencies and at low tolerances, where it may use up to [math] times fewer function evaluations. Even in high-frequency regimes, our implementation is on average 10 times faster than other specialized solvers.
{"title":"An Adaptive Spectral Method for Oscillatory Second-Order Linear ODEs with Frequency-Independent Cost","authors":"Fruzsina J. Agocs, Alex H. Barnett","doi":"10.1137/23m1546609","DOIUrl":"https://doi.org/10.1137/23m1546609","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 295-321, February 2024. <br/> Abstract. We introduce an efficient numerical method for second-order linear ODEs whose solution may vary between highly oscillatory and slowly changing over the solution interval. In oscillatory regions the solution is generated via a nonoscillatory phase function that obeys the nonlinear Riccati equation. We propose a defect correction iteration that gives an asymptotic series for such a phase function; this is numerically approximated on a Chebyshev grid with a small number of nodes. For analytic coefficients we prove that each iteration, up to a certain maximum number, reduces the residual by a factor of order of the local frequency. The algorithm adapts both the stepsize and the choice of method, switching to a conventional spectral collocation method away from oscillatory regions. In numerical experiments we find that our proposal outperforms other state-of-the-art oscillatory solvers, most significantly at low to intermediate frequencies and at low tolerances, where it may use up to [math] times fewer function evaluations. Even in high-frequency regimes, our implementation is on average 10 times faster than other specialized solvers.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"343 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139573530","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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 248-272, February 2024. Abstract. Surface Stokes and Navier–Stokes equations are used to model fluid flow on surfaces. They have attracted significant recent attention in the numerical analysis literature because approximation of their solutions poses significant challenges not encountered in the Euclidean context. One challenge comes from the need to simultaneously enforce tangentiality and [math] conformity (continuity) of discrete vector fields used to approximate solutions in the velocity-pressure formulation. Existing methods in the literature all enforce one of these two constraints weakly either by penalization or by use of Lagrange multipliers. Missing so far is a robust and systematic construction of surface Stokes finite element spaces which employ nodal degrees of freedom, including MINI, Taylor–Hood, Scott–Vogelius, and other composite elements which can lead to divergence-conforming or pressure-robust discretizations. In this paper we construct surface MINI spaces whose velocity fields are tangential. They are not [math]-conforming, but do lie in [math] and do not require penalization to achieve optimal convergence rates. We prove stability and optimal-order energy-norm convergence of the method and demonstrate optimal-order convergence of the velocity field in [math] via numerical experiments. The core advance in the paper is the construction of nodal degrees of freedom for the velocity field. This technique also may be used to construct surface counterparts to many other standard Euclidean Stokes spaces, and we accordingly present numerical experiments indicating optimal-order convergence of nonconforming tangential surface Taylor–Hood [math] elements.
{"title":"A Tangential and Penalty-Free Finite Element Method for the Surface Stokes Problem","authors":"Alan Demlow, Michael Neilan","doi":"10.1137/23m1583995","DOIUrl":"https://doi.org/10.1137/23m1583995","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 248-272, February 2024. <br/> Abstract. Surface Stokes and Navier–Stokes equations are used to model fluid flow on surfaces. They have attracted significant recent attention in the numerical analysis literature because approximation of their solutions poses significant challenges not encountered in the Euclidean context. One challenge comes from the need to simultaneously enforce tangentiality and [math] conformity (continuity) of discrete vector fields used to approximate solutions in the velocity-pressure formulation. Existing methods in the literature all enforce one of these two constraints weakly either by penalization or by use of Lagrange multipliers. Missing so far is a robust and systematic construction of surface Stokes finite element spaces which employ nodal degrees of freedom, including MINI, Taylor–Hood, Scott–Vogelius, and other composite elements which can lead to divergence-conforming or pressure-robust discretizations. In this paper we construct surface MINI spaces whose velocity fields are tangential. They are not [math]-conforming, but do lie in [math] and do not require penalization to achieve optimal convergence rates. We prove stability and optimal-order energy-norm convergence of the method and demonstrate optimal-order convergence of the velocity field in [math] via numerical experiments. The core advance in the paper is the construction of nodal degrees of freedom for the velocity field. This technique also may be used to construct surface counterparts to many other standard Euclidean Stokes spaces, and we accordingly present numerical experiments indicating optimal-order convergence of nonconforming tangential surface Taylor–Hood [math] elements.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"155 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139551046","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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 273-294, February 2024. Abstract. This paper presents a novel Fourier spectral method that utilizes optimization techniques to ensure the positivity and conservation of moments in the space of trigonometric polynomials. We rigorously analyze the accuracy of the new method and prove that it maintains spectral accuracy. To solve the optimization problem, we propose an efficient Newton solver that has a quadratic convergence rate. Numerical examples are provided to demonstrate the high accuracy of the proposed method. Our method is also integrated into the spectral solver of the Boltzmann equation, showing the benefit of our approach in applications.
{"title":"A Positive and Moment-Preserving Fourier Spectral Method","authors":"Zhenning Cai, Bo Lin, Meixia Lin","doi":"10.1137/23m1563918","DOIUrl":"https://doi.org/10.1137/23m1563918","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 273-294, February 2024. <br/> Abstract. This paper presents a novel Fourier spectral method that utilizes optimization techniques to ensure the positivity and conservation of moments in the space of trigonometric polynomials. We rigorously analyze the accuracy of the new method and prove that it maintains spectral accuracy. To solve the optimization problem, we propose an efficient Newton solver that has a quadratic convergence rate. Numerical examples are provided to demonstrate the high accuracy of the proposed method. Our method is also integrated into the spectral solver of the Boltzmann equation, showing the benefit of our approach in applications.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"5 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139551093","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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 229-247, February 2024. Abstract. We propose two novel unbiased estimators of the integral [math] for a function [math], which depend on a smoothness parameter [math]. The first estimator integrates exactly the polynomials of degrees [math] and achieves the optimal error [math] (where [math] is the number of evaluations of [math]) when [math] is [math] times continuously differentiable. The second estimator is also optimal in terms of convergence rate and has the advantage of being computationally cheaper, but it is restricted to functions that vanish on the boundary of [math]. The construction of the two estimators relies on a combination of cubic stratification and control variates based on numerical derivatives. We provide numerical evidence that they show good performance even for moderate values of [math].
{"title":"Higher-Order Monte Carlo through Cubic Stratification","authors":"Nicolas Chopin, Mathieu Gerber","doi":"10.1137/22m1532287","DOIUrl":"https://doi.org/10.1137/22m1532287","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 229-247, February 2024. <br/> Abstract. We propose two novel unbiased estimators of the integral [math] for a function [math], which depend on a smoothness parameter [math]. The first estimator integrates exactly the polynomials of degrees [math] and achieves the optimal error [math] (where [math] is the number of evaluations of [math]) when [math] is [math] times continuously differentiable. The second estimator is also optimal in terms of convergence rate and has the advantage of being computationally cheaper, but it is restricted to functions that vanish on the boundary of [math]. The construction of the two estimators relies on a combination of cubic stratification and control variates based on numerical derivatives. We provide numerical evidence that they show good performance even for moderate values of [math].","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"61 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139544242","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}
Sergio Gomez, Lorenzo Mascotto, Andrea Moiola, Ilaria Perugia
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 199-228, February 2024. Abstract. We propose and analyze a space-time virtual element method for the discretization of the heat equation in a space-time cylinder, based on a standard Petrov–Galerkin formulation. Local discrete functions are solutions to a heat equation problem with polynomial data. Global virtual element spaces are nonconforming in space, so that the analysis and the design of the method are independent of the spatial dimension. The information between time slabs is transmitted by means of upwind terms involving polynomial projections of the discrete functions. We prove well posedness and optimal error estimates for the scheme, and validate them with several numerical tests.
{"title":"Space-Time Virtual Elements for the Heat Equation","authors":"Sergio Gomez, Lorenzo Mascotto, Andrea Moiola, Ilaria Perugia","doi":"10.1137/22m154140x","DOIUrl":"https://doi.org/10.1137/22m154140x","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 199-228, February 2024. <br/> Abstract. We propose and analyze a space-time virtual element method for the discretization of the heat equation in a space-time cylinder, based on a standard Petrov–Galerkin formulation. Local discrete functions are solutions to a heat equation problem with polynomial data. Global virtual element spaces are nonconforming in space, so that the analysis and the design of the method are independent of the spatial dimension. The information between time slabs is transmitted by means of upwind terms involving polynomial projections of the discrete functions. We prove well posedness and optimal error estimates for the scheme, and validate them with several numerical tests.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"7 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139489571","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}
Elisabetta Carlini, Francisco J. Silva, Ahmad Zorkot
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 167-198, February 2024. Abstract. In this work, we consider a first order mean field game system with nonlocal couplings. A Lagrange–Galerkin scheme for the continuity equation, coupled with a semi-Lagrangian scheme for the Hamilton–Jacobi–Bellman equation, is proposed to discretize the mean field games system. The convergence of solutions to the scheme towards a solution to the mean field game system is established in arbitrary space dimensions. The scheme is implemented to approximate two mean field games systems in dimensions one and two.
{"title":"A Lagrange–Galerkin Scheme for First Order Mean Field Game Systems","authors":"Elisabetta Carlini, Francisco J. Silva, Ahmad Zorkot","doi":"10.1137/23m1561762","DOIUrl":"https://doi.org/10.1137/23m1561762","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 167-198, February 2024. <br/> Abstract. In this work, we consider a first order mean field game system with nonlocal couplings. A Lagrange–Galerkin scheme for the continuity equation, coupled with a semi-Lagrangian scheme for the Hamilton–Jacobi–Bellman equation, is proposed to discretize the mean field games system. The convergence of solutions to the scheme towards a solution to the mean field game system is established in arbitrary space dimensions. The scheme is implemented to approximate two mean field games systems in dimensions one and two.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"7 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139480610","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}
SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 138-166, February 2024. Abstract. The formulation of mean field games (MFG) typically requires continuous differentiability of the Hamiltonian in order to determine the advective term in the Kolmogorov–Fokker–Planck equation for the density of players. However, in many cases of practical interest, the underlying optimal control problem may exhibit bang-bang controls, which typically lead to nondifferentiable Hamiltonians. We develop the analysis and numerical analysis of stationary MFG for the general case of convex, Lipschitz, but possibly nondifferentiable Hamiltonians. In particular, we propose a generalization of the MFG system as a partial differential inclusion (PDI) based on interpreting the derivative of the Hamiltonian in terms of subdifferentials of convex functions. We establish the existence of a weak solution to the MFG PDI system, and we further prove uniqueness under a similar monotonicity condition to the one considered by Lasry and Lions. We then propose a monotone finite element discretization of the problem, and we prove strong [math]-norm convergence of the approximations of the value function and strong [math]-norm convergence of the approximations of the density function. We illustrate the performance of the numerical method in numerical experiments featuring nonsmooth solutions.
{"title":"Analysis and Numerical Approximation of Stationary Second-Order Mean Field Game Partial Differential Inclusions","authors":"Yohance A. P. Osborne, Iain Smears","doi":"10.1137/22m1519274","DOIUrl":"https://doi.org/10.1137/22m1519274","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 1, Page 138-166, February 2024. <br/> Abstract. The formulation of mean field games (MFG) typically requires continuous differentiability of the Hamiltonian in order to determine the advective term in the Kolmogorov–Fokker–Planck equation for the density of players. However, in many cases of practical interest, the underlying optimal control problem may exhibit bang-bang controls, which typically lead to nondifferentiable Hamiltonians. We develop the analysis and numerical analysis of stationary MFG for the general case of convex, Lipschitz, but possibly nondifferentiable Hamiltonians. In particular, we propose a generalization of the MFG system as a partial differential inclusion (PDI) based on interpreting the derivative of the Hamiltonian in terms of subdifferentials of convex functions. We establish the existence of a weak solution to the MFG PDI system, and we further prove uniqueness under a similar monotonicity condition to the one considered by Lasry and Lions. We then propose a monotone finite element discretization of the problem, and we prove strong [math]-norm convergence of the approximations of the value function and strong [math]-norm convergence of the approximations of the density function. We illustrate the performance of the numerical method in numerical experiments featuring nonsmooth solutions.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"40 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139435461","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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