{"paper":{"title":"Walk on spheres and Array-RQMC","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Array-RQMC sampling in the walk on spheres algorithm reduces Monte Carlo variance by factors of 57 to 2290 at sample size 2 to the 17.","cross_cats":["cs.NA","stat.CO"],"primary_cat":"math.NA","authors_text":"Art B. Owen, Valerie N. P. Ho","submitted_at":"2026-05-13T00:42:15Z","abstract_excerpt":"We use Array-RQMC sampling in a walk on spheres (WOS) algorithm for Dirichlet boundary value problems. On a collection of problems, we find that Array-RQMC-WOS reduces the Monte Carlo variance by factors ranging from $57$-fold to $2290$-fold at $n=2^{17}$ trajectories. The variance is known to be $o(1/n)$ but attains empirical rates between $n^{-1.4}$ and $n^{-1.8}$ in our examples. A simpler RQMC-WOS algorithm studied in Ho and Owen (2026) has more theoretical support but only reduced variance by 1.8 to 10.7-fold on the same set of examples. In order to explain this improvement, we introduce "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Array-RQMC-WOS reduces the Monte Carlo variance by factors ranging from 57-fold to 2290-fold at n=2^{17} trajectories, attaining empirical rates between n^{-1.4} and n^{-1.8}.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the observed variance reductions and higher column-wise mean dimension will persist for problems outside the tested collection and that the Sobol'-index-based mean dimension correctly isolates the source of the improvement.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Array-RQMC-WOS cuts Monte Carlo variance by 57-2290 times with empirical rates n^{-1.4} to n^{-1.8} and introduces a column-wise mean dimension to explain the gain.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Array-RQMC sampling in the walk on spheres algorithm reduces Monte Carlo variance by factors of 57 to 2290 at sample size 2 to the 17.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"35685072ccd70fdf3c4214308f66b22801743b5addb914a20a5b5f3b36e33592"},"source":{"id":"2605.12844","kind":"arxiv","version":1},"verdict":{"id":"251d11ca-24ab-4457-85e8-a7350c965cc6","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T19:14:02.795121Z","strongest_claim":"Array-RQMC-WOS reduces the Monte Carlo variance by factors ranging from 57-fold to 2290-fold at n=2^{17} trajectories, attaining empirical rates between n^{-1.4} and n^{-1.8}.","one_line_summary":"Array-RQMC-WOS cuts Monte Carlo variance by 57-2290 times with empirical rates n^{-1.4} to n^{-1.8} and introduces a column-wise mean dimension to explain the gain.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the observed variance reductions and higher column-wise mean dimension will persist for problems outside the tested collection and that the Sobol'-index-based mean dimension correctly isolates the source of the improvement.","pith_extraction_headline":"Array-RQMC sampling in the walk on spheres algorithm reduces Monte Carlo variance by factors of 57 to 2290 at sample size 2 to the 17."},"references":{"count":45,"sample":[{"doi":"","year":2012,"title":"Bilyk, D., V. N. Temlyakov, and R. Yu (2012). Fibonacci sets and symmetrization in discrepancy theory. Journal of Complexity\\/ 28\\/ (1), 18--36","work_id":"df35b7f1-b002-4960-af42-19c1fdcc1a65","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2012,"title":"Binder, I. and M. Braverman (2012). The rate of convergence of the walk on spheres algorithm. Geometric and Functional Analysis\\/ 22\\/ (3), 558--587","work_id":"d99261ef-b4d8-4e20-b294-e12f5604290f","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1997,"title":"Caflisch, R. E., W. Morokoff, and A. B. Owen (1997). Valuation of mortgage backed securities using Brownian bridges to reduce effective dimension. Journal of Computational Finance\\/ 1 , 27--46","work_id":"a01ad732-a763-4e60-ba1d-5d3af5dffb1e","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2026,"title":"Choi, S.-C. T., F. J. Hickernell, M. McCourt, J. Rathinavel, and A. G. Sorokin (2026). QMCPy : A Q uasi- M onte C arlo P ython L ibrary","work_id":"96af03e1-16f0-43f1-a797-7d6d7cfb7e8a","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2006,"title":"Cools, R., F. Y. Kuo, and D. Nuyens (2006). Constructing embedded lattice rules for multivariate integration. SIAM Journal on Scientific Computing\\/ 28\\/ (6), 2162--2188","work_id":"7c655eff-50ef-4454-9545-5fc5565a4fb1","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":45,"snapshot_sha256":"3ce7c1cd9919bc09dc8ebc7e2defa68f9f7ba4597e3b0f7845dde3874a7f790c","internal_anchors":1},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}