Bregman-projected mirror methods for regularized stationary mean-field games

Mathematics > Numerical Analysis [Submitted on 17 Jun 2026] Title:Bregman-projected mirror methods for regularized stationary mean-field games View PDF HTML (experimental)Abstract:We develop and analyze a Bregman-projected mirror iteration for low-order regularizations of stationary mean-field game (MFG) systems in their natural Banach space setting. For separable Hamiltonians of the form \(H(x,p,m)=H_0(x,p)-g(m)\), with quadratic or super-quadratic Hamiltonian growth and linear or super-linear density couplings, we formulate a low-order \(\bar\gamma\)-Laplacian regularization of the stationary MFG system as a variational inequality on \(L^{\bar\beta}(\mathbb T^d)\times W^{1,\bar\gamma}(\mathbb T^d)\). To approximate solutions of this regularized variational inequality, we introduce a Bregman geometry matched to the mixed Lebesgue--Sobolev exponents of the problem and analyze a constrained two-step mirror method with frozen operator evaluation. For the exact constrained iteration and each fixed regularization parameter \(\epsi>0\), we derive a one-step Bregman inequality and use it to prove that the constrained iteration converges strongly to the unique solution of the regularized variational inequality under natural summability conditions on the step sizes. Numerical experiments on one- and two-dimensional models, validated against exact test solutions, illustrate residual decay under mesh refinement and suggest improved practical performance of the two-step implementation in the tested discretizations. Submission history From: Hussain Al Abdulaziz [view email][v1] Wed, 17 Jun 2026 21:33:47 UTC (4,396 KB) Current browse context: math.NA References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.