A generalized alternating NGMRES method for PDE-constrained optimization problems governed by transport equations
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| Format: | Preprint |
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2025
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| author | He, Yunhui Mang, Andreas |
| author_facet | He, Yunhui Mang, Andreas |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_08782 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A generalized alternating NGMRES method for PDE-constrained optimization problems governed by transport equations He, Yunhui Mang, Andreas Optimization and Control 49J20, 49K20, 49N45, 65B30, 65K10 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. |
| title | A generalized alternating NGMRES method for PDE-constrained optimization problems governed by transport equations |
| topic | Optimization and Control 49J20, 49K20, 49N45, 65B30, 65K10 |
| url | https://arxiv.org/abs/2510.08782 |