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Numerical Analysis Seminar
Seminar information archive ～09/15｜Next seminar｜Future seminars 09/16～
| Date, time & place | Tuesday 16:30 - 18:00 002Room #002 (Graduate School of Math. Sci. Bldg.) |
|---|---|
| Organizer(s) | Norikazu Saito, Takahito Kashiwabara |
Future seminars
2026/10/06
16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Yohance Osborne (Durham University)
Numerical approximation of mean field games systems (English)
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
Yohance Osborne (Durham University)
Numerical approximation of mean field games systems (English)
[ Abstract ]
Mean field games (MFG) are models for Nash equilibria of competitive differential games of optimal control involving a continuum of players. MFG equilibria are often described by systems of partial differential equations, where a Hamilton–Jacobi–Bellman (HJB) equation for the representative player’s value function is coupled nonlinearly with a Kolmogorov–Fokker–Planck (KFP) equation that determines the player distribution. MFG systems are particularly challenging to approximate computationally because they exhibit the following difficulties all at once: nonlinear couplings between the HJB and KFP equations, a temporal forward–backward structure, non-negativity of absolutely continuous player distributions, as well as potential nondifferentiability of the Hamiltonian. This talk addresses the numerical approximation of MFG systems for two classes of numerical schemes: (1) monotone stabilized finite element methods for second-order MFG systems with nondifferentiable Hamiltonians, and (2) particle methods for first-order order MFG systems under displacement monotonicity. I will present theorems on the strong convergence of these numerical schemes and their rates of convergence, along with discussion of some numerical experiments.
[ Reference URL ]Mean field games (MFG) are models for Nash equilibria of competitive differential games of optimal control involving a continuum of players. MFG equilibria are often described by systems of partial differential equations, where a Hamilton–Jacobi–Bellman (HJB) equation for the representative player’s value function is coupled nonlinearly with a Kolmogorov–Fokker–Planck (KFP) equation that determines the player distribution. MFG systems are particularly challenging to approximate computationally because they exhibit the following difficulties all at once: nonlinear couplings between the HJB and KFP equations, a temporal forward–backward structure, non-negativity of absolutely continuous player distributions, as well as potential nondifferentiability of the Hamiltonian. This talk addresses the numerical approximation of MFG systems for two classes of numerical schemes: (1) monotone stabilized finite element methods for second-order MFG systems with nondifferentiable Hamiltonians, and (2) particle methods for first-order order MFG systems under displacement monotonicity. I will present theorems on the strong convergence of these numerical schemes and their rates of convergence, along with discussion of some numerical experiments.
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
2026/10/08
16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Note that the seminar will be held on Thursday.
Marcus Grote (University of Basel)
Adaptive FEM with Explicit Time Integration For the Wave Equation
(English)
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
Note that the seminar will be held on Thursday.
Marcus Grote (University of Basel)
Adaptive FEM with Explicit Time Integration For the Wave Equation
(English)
[ Abstract ]
Starting from a recent a posteriori error estimator for the finite element solution of the wave equation with explicit time integration [Grote, Lakkis, Santos, 2024], we devise a space-time adaptive strategy which includes both time evolving meshes and local time-stepping to overcome any overly stringent CFL stability restriction on the time-step due to local mesh refinement. Moreover, at each time-step the adaptive algorithm monitors the accuracy thanks to the error indicators and recomputes the current step on a refined mesh until the desired tolerance is met; meanwhile, the mesh is coarsened in regions of smaller errors. Leapfrog based local time-stepping is applied in all regions of local mesh refinement to incorporate adaptivity into fully explicit time integration with mesh change while retaining efficiency. Numerical results illustrate the optimal rate of convergence of the a posteriori error estimators on time evolving meshes.
[ Reference URL ]Starting from a recent a posteriori error estimator for the finite element solution of the wave equation with explicit time integration [Grote, Lakkis, Santos, 2024], we devise a space-time adaptive strategy which includes both time evolving meshes and local time-stepping to overcome any overly stringent CFL stability restriction on the time-step due to local mesh refinement. Moreover, at each time-step the adaptive algorithm monitors the accuracy thanks to the error indicators and recomputes the current step on a refined mesh until the desired tolerance is met; meanwhile, the mesh is coarsened in regions of smaller errors. Leapfrog based local time-stepping is applied in all regions of local mesh refinement to incorporate adaptivity into fully explicit time integration with mesh change while retaining efficiency. Numerical results illustrate the optimal rate of convergence of the a posteriori error estimators on time evolving meshes.
https://sites.google.com/g.ecc.u-tokyo.ac.jp/utnas-bulletin-board/
