{"ID":2875217,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.01855","arxiv_id":"2509.01855","title":"A Million-Point Fast Trajectory Optimization Solver","abstract":"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.","short_abstract":"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 c...","url_abs":"https://arxiv.org/abs/2509.01855","url_pdf":"https://arxiv.org/pdf/2509.01855v1","authors":"[\"A. Javeed\",\"D. P. Kouri\",\"D. Ridzal\",\"J. D. Steinman\",\"I. M. Ross\"]","published":"2025-09-02T00:47:59Z","proceeding":"math.NA","tasks":"[\"math.NA\",\"cs.MS\",\"eess.SY\",\"math.OC\"]","methods":"[]","has_code":false}