牛顿法与皮卡德求解器在平均场博弈PDE系统中的计算权衡
Picard-Based Acceleration of Newton Continuation for Mean Field Game PDE Systems
- New York University Shanghai（纽约大学上海）
- NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai（纽约大学上海华东师范大学数学科学联合研究院）
- Department of Mathematics, New York University Shanghai（纽约大学上海数学系）
- Department of Data Science, New York University Shanghai（纽约大学上海数据科学系）
- Department of Mathematics, New York University（纽约大学数学系）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
研究平均场博弈PDE系统中牛顿法与皮卡德法的计算权衡，发现皮卡德法收敛快但低粘度易失败，牛顿法更鲁棒，并提出混合延拓策略。
AI中文摘要：
我们研究了同一半隐式有限差分离散化下两种求解器在平均场博弈（MFGs）中产生的正反向偏微分方程（PDE）系统中的计算权衡。皮卡德方法使用外部不动点迭代，交替进行前向Fokker-Planck求解和后向Hamilton-Jacobi-Bellman求解。牛顿法则直接将牛顿法应用于耦合的非线性时空系统。在一维和二维MFG基准测试中，我们发现皮卡德方法在收敛时具有更低的墙钟成本，但在足够低的粘度下可能失败，并且在时间冲击下可能需要强阻尼。通过参数延拓，牛顿法在这些情况下更鲁棒，但代价是更大的耦合线性系统。我们将这些权衡与可分离、局部不可分离和非局部哈密顿量产生的残差、雅可比块和稀疏结构联系起来。我们还在二维双阱MFG基准测试上测试了一种混合延拓策略，在切换到牛顿延拓之前，使用皮卡德迭代进行粘度下降的廉价部分。
英文摘要：
We develop a method that uses Picard iterations to accelerate Newton continuation for the semi-implicit finite-difference discretization of forward-backward partial differential equation (PDE) systems arising in mean field games (MFGs). We first investigate the computational properties of the Picard and Newton methods, which are widely used separately in the MFG literature but whose comparative cost and robustness across different regimes remain insufficiently explored and documented. The Picard method uses an outer fixed-point iteration that alternates a forward Fokker-Planck solve and a backward Hamilton-Jacobi-Bellman solve. The Newton method instead applies Newton's method directly to the coupled nonlinear space-time system. Across one- and two-dimensional MFG benchmarks, Picard offers substantial computational savings in favorable regimes, but may fail at sufficiently low viscosity or require strong damping under temporal shocks, making Newton continuation preferable. With parameter continuation in the viscosity parameter, the Newton method is more robust in these regimes, at the cost of larger coupled linear systems. We relate these trade-offs to the residuals, Jacobian blocks, and sparsity structures produced by separable, local nonseparable, and nonlocal Hamiltonians. We then demonstrate how to combine inexpensive Picard iterations with Newton continuation in a hybrid method to reduce the total computational cost for a two- dimensional double-well problem.
