2021/02/26 by Jiajia Yu, Yu, Jiajia, Rongjie Lai +5 · 2 citations
Computer Science · Mathematics · Physics and Astronomy · #49M25 #49M41 #65K10 #Advanced Optimization Algorithms Research #FOS: Mathematics #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Optimization and Control (math.OC) #Polynomial and algebraic computation #cs.NA #math.NA #math.OC #msc:49M25 #msc:49M41 #msc:65K10
paper · pdf · doi:10.48550/arxiv.2102.13260
38 pages, 9 figures
arxiv created 2021/02/26 · openalex publication_date 2021/02/26 · arxiv updated 2021/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we propose an efficient and flexible algorithm to solve dynamic mean-field planning problems based on an accelerated proximal gradient method. Besides an easy-to-implement gradient descent step in this algorithm, a crucial projection step becomes solving an elliptic equation whose solution can be obtained by conventional methods efficiently. By induction on iterations used in the algorithm, we theoretically show that the proposed discrete solution converges to the underlying continuous solution as the grid size increases. Furthermore, we generalize our algorithm to mean-field game problems and accelerate it using multilevel and multigrid strategies. We conduct comprehensive numerical experiments to confirm the convergence analysis of the proposed algorithm, to show its efficiency and mass preservation property by comparing it with state-of-the-art methods, and to illustrates its flexibility for handling various mean-field variational problems.