高维哈密顿-雅可比方程的可扩展固定点框架
Scalable Fixed-Point Framework for High-Dimensional Hamilton-Jacobi Equations
AI总结：
本文提出了一种无网格、无梯度的固定点方法，用于高效求解高维哈密顿-雅可比方程的粘性解，展示了其在高维问题中的高精度和高效性。
摘要：
我们提出了一种新颖的、无网格的、无梯度的固定点方法，用于计算高维哈密顿-雅可比（HJ）方程的粘性解。通过利用霍普夫-拉克斯公式，我们的方法通过皮卡迭代法求解相关的变分问题，从而高效地评估解及其对应的控制，而无需依赖网格、特征线或微分。我们通过最多100维的数值实验展示了该方法的实用性和可扩展性，包括控制问题和非光滑解。我们的结果表明，所提出的方法具有高精度、高效性，并且计算时间在很大程度上与维度无关，突显了其在高维问题中的适用性。
英文摘要：
We propose a novel, mesh-free, and gradient-free fixed-point approach for computing viscosity solutions of high-dimensional Hamilton-Jacobi (HJ) equations. By leveraging the Hopf-Lax formula, our approach iteratively solves the associated variational problem via a Picard iteration, enabling efficient evaluation of both the solution and its corresponding control without relying on grids, characteristics, or differentiation. We demonstrate the practical efficacy and scalability of the approach through numerical experiments in up to 100 dimensions, including control problems and non-smooth solutions. Our results show that the proposed scheme achieves high accuracy, is highly efficient, and exhibits computational times that are largely independent of dimensionality, highlighting its suitability for high-dimensional problems.
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