{"ID":2858432,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.08782","arxiv_id":"2510.08782","title":"A generalized alternating NGMRES method for PDE-constrained optimization problems governed by transport equations","abstract":"In this work, we propose a generalized alternating nonlinear generalized minimal residual method (GA-NGMRES) to accelerate first-order optimization schemes for PDE-constrained optimization problems governed by transport equations. We apply GA-NGMRES to a preconditioned first-order optimization scheme by interpreting the update rule as a fixed-point (FP) iteration. Our approach introduces a novel periodic mixing strategy that integrates NGMRES updates with FP steps. This new scheme improves efficiency in terms of both iteration count and runtime compared to the state-of-the-art. We include a comparison to first-order preconditioned gradient descent and preconditioned, inexact Gauss--Newton--Krylov methods. Since the proposed optimization scheme only relies on first-order derivative information, its implementation is straightforward. We evaluate performance as a function of hyperparameters, the mesh size, and the regularization parameter. We consider advection, incompressible flows, and mass-preserving transport (i.e., optimal transport-type problems) as PDE models. Stipulating adequate smoothness requirements based on variational regularization of the control variable ensures that the computed transport maps are diffeomorphic. Numerical experiments on real-world and synthetic problems highlight the robustness and effectiveness of the proposed method. Our approach yields runtimes that are up to 5x faster than state-of-the-art Newton--Krylov methods, without sacrificing accuracy. Additionally, our GA-NGMRES algorithm outperforms the well-known Anderson acceleration for the models and numerical approach considered in this work.","short_abstract":"In this work, we propose a generalized alternating nonlinear generalized minimal residual method (GA-NGMRES) to accelerate first-order optimization schemes for PDE-constrained optimization problems governed by transport equations. We apply GA-NGMRES to a preconditioned first-order optimization scheme by interpreting th...","url_abs":"https://arxiv.org/abs/2510.08782","url_pdf":"https://arxiv.org/pdf/2510.08782v1","authors":"[\"Yunhui He\",\"Andreas Mang\"]","published":"2025-10-09T19:59:39Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[]","has_code":false}