HJ-Gauss: A Monte-Carlo HJ Reachability Scheme

[2605.18566] HJ-Gauss: A Monte-Carlo HJ Reachability Scheme Skip to main content Search Submit Donate Log in Electrical Engineering and Systems Science > Systems and Control arXiv:2605.18566 (eess) [Submitted on 18 May 2026 ( v1 ), last revised 27 Aug 2026 (this version, v7)] Title: HJ-Gauss: A Monte-Carlo HJ Reachability Scheme Authors: Lekan Molu , Venkatraman Renganathan , Namhoon Cho View PDF HTML (experimental) Abstract: Backward reachable sets or tubes (BRS/Ts), evaluated with grid-based level-set methods over viscous Hamilton-Jacobi (HJ) equations, furnish principled reachability certificates for learning-enabled control. However, these methods incur an $O(M^n)$ memory cost, where $M$ is the number of grid points for every $n$-state dimension, which has precluded their applications in high-dimensional physical phenomena. Towards scalable reachability analysis, we propose a frozen-coefficient Picard iterative Gaussian sampling scheme: with a Cole-Hopf-type transformation, the HJ PDE effectively reduces to a sequence of linear heat equations, whose values are then iteratively recovered via Gaussian heat-kernel expectations. In this sentiment, Monte Carlo roll-outs over Gaussian densities ultimately recover the (approximate) HJ value and its spatial gradient. Ours is a storage- and discretization-free algorithm whose memory footprint scales (for \textit{i.i.d} samples, $N$) as $N\cdot n$; we show that sampling is a polynomial power law in $N$ and provide a conditional linear convergence analysis to the \textit{consistent} viscosity solution. Furthermore, we provide a quasilinearization defect per iteration between frozen coefficient samples to make our HJ sampling scheme tight. Our theoretical machinery is rigorously benchmarked on rocket launch and Dubins pursuit-evasion (P-E) games over Holm-Bonferroni-informed experiments: for a $45D$-multi-rocket launch P-E game, we find an \textit{almost zero} Picard residual floor. Certifying the safety of \textbf{$100,000$} European starlings (\textit{sturnus vulgaris}) in murmurations over their collective value functions reveals vacuole nucleation, cordon formation, and flock splitting on the resulting BRT zero-levelset phase topology as predator attacks evolve , demonstrating the scalability of our approach to large-scale multi-agent systems. Subjects: Systems and Control (eess.SY) Cite as: arXiv:2605.18566 [eess.SY] (or arXiv:2605.18566v7 [eess.SY] for this version) https://doi.org/10.48550/arXiv.2605.18566 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Lekan Molu [ view email ] [v1] Mon, 18 May 2026 15:43:59 UTC (1,922 KB) [v2] Mon, 29 Jun 2026 02:18:28 UTC (4,234 KB) [v3] Tue, 28 Jul 2026 02:53:17 UTC (3,043 KB) [v4] Sun, 2 Aug 2026 19:31:05 UTC (3,045 KB) [v5] Fri, 7 Aug 2026 20:56:58 UTC (3,044 KB) [v6] Sun, 16 Aug 2026 04:54:13 UTC (3,044 KB) [v7] Thu, 27 Aug 2026 08:12:33 UTC (4,280 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: eess.SY < prev | next > new | recent | 2026-05 Change to browse by: cs cs.SY eess References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. 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