A Million-Point Fast Trajectory Optimization Solver
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arXiv
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| Format: | Preprint |
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2025
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| author | Javeed, A. Kouri, D. P. Ridzal, D. Steinman, J. D. Ross, I. M. |
| author_facet | Javeed, A. Kouri, D. P. Ridzal, D. Steinman, J. D. Ross, I. M. |
| contents | One might argue that solving a trajectory optimization problem over a million grid points is preposterous. How about solving such a problem at an incredibly fast computational time? On a small form-factor processor? Algorithmic details that make possible this trifecta of breakthroughs are presented in this paper. The computational mathematics that deliver these advancements are: (i) a Birkhoff-theoretic discretization of optimal control problems, (ii) matrix-free linear algebra leveraging Krylov-subspace methods, and (iii) a near-perfect Birkhoff preconditioner that helps achieve $\mathcal{O}(1)$ iteration speed with respect to the grid size,~$N$. A key enabler of this high performance is the computation of Birkhoff matrix-vector products at $\mathcal{O}(N\log(N))$ time using fast Fourier transform techniques that eliminate traditional computational bottlenecks. A numerical demonstration of this unprecedented scale and speed is illustrated for a practical astrodynamics problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_01855 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Million-Point Fast Trajectory Optimization Solver Javeed, A. Kouri, D. P. Ridzal, D. Steinman, J. D. Ross, I. M. Numerical Analysis Mathematical Software Systems and Control Optimization and Control One might argue that solving a trajectory optimization problem over a million grid points is preposterous. How about solving such a problem at an incredibly fast computational time? On a small form-factor processor? Algorithmic details that make possible this trifecta of breakthroughs are presented in this paper. The computational mathematics that deliver these advancements are: (i) a Birkhoff-theoretic discretization of optimal control problems, (ii) matrix-free linear algebra leveraging Krylov-subspace methods, and (iii) a near-perfect Birkhoff preconditioner that helps achieve $\mathcal{O}(1)$ iteration speed with respect to the grid size,~$N$. A key enabler of this high performance is the computation of Birkhoff matrix-vector products at $\mathcal{O}(N\log(N))$ time using fast Fourier transform techniques that eliminate traditional computational bottlenecks. A numerical demonstration of this unprecedented scale and speed is illustrated for a practical astrodynamics problem. |
| title | A Million-Point Fast Trajectory Optimization Solver |
| topic | Numerical Analysis Mathematical Software Systems and Control Optimization and Control |
| url | https://arxiv.org/abs/2509.01855 |