BDF schemes for accelerated gradient flows in projection-free approximation of nonconvex constrained variational minimization
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913889746157568 |
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| author | Dong, Guozhi Gong, Zikang Xie, Ziqing Yang, Shuo |
| author_facet | Dong, Guozhi Gong, Zikang Xie, Ziqing Yang, Shuo |
| contents | We propose novel algorithms combining accelerated gradient flows with linearized projection-free treatments of non-convex constraints and BDF pseudo-temporal discretization for quadratic energy minimization. A general framework is developed to analyze constraint violations in such projection-free techniques for quadratic constraints. This analysis proves to be universal to all projection-free iterative methods, and constraint error bounds depend solely on iterate regularity. For BDF-k(k=1,2,3,4), we derive both unconditional and conditional high-order constraint violation estimates for accelerated gradient flows using our framework. We further discover a new family of BDF-k accelerated gradient methods achieving modified energy stability for arbitrary positive integer k. Numerical experiments validate our theoretical results and demonstrate superior efficiency and accuracy compared to existing gradient flow approaches. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_14670 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | BDF schemes for accelerated gradient flows in projection-free approximation of nonconvex constrained variational minimization Dong, Guozhi Gong, Zikang Xie, Ziqing Yang, Shuo Numerical Analysis Optimization and Control 65B99, 65K10, 65M15 We propose novel algorithms combining accelerated gradient flows with linearized projection-free treatments of non-convex constraints and BDF pseudo-temporal discretization for quadratic energy minimization. A general framework is developed to analyze constraint violations in such projection-free techniques for quadratic constraints. This analysis proves to be universal to all projection-free iterative methods, and constraint error bounds depend solely on iterate regularity. For BDF-k(k=1,2,3,4), we derive both unconditional and conditional high-order constraint violation estimates for accelerated gradient flows using our framework. We further discover a new family of BDF-k accelerated gradient methods achieving modified energy stability for arbitrary positive integer k. Numerical experiments validate our theoretical results and demonstrate superior efficiency and accuracy compared to existing gradient flow approaches. |
| title | BDF schemes for accelerated gradient flows in projection-free approximation of nonconvex constrained variational minimization |
| topic | Numerical Analysis Optimization and Control 65B99, 65K10, 65M15 |
| url | https://arxiv.org/abs/2409.14670 |