{"paper":{"title":"A Parallel and Adaptive Mesh-Free method for Heterogeneous Porous Media","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Normalized radial basis functions with Shepard stabilization approximate discontinuous step functions to arbitrarily small L1 error.","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Kapil Chawla, Sanghyun Lee, Yeonjong Shin","submitted_at":"2026-05-15T19:07:29Z","abstract_excerpt":"Material properties such as permeability fields in heterogeneous porous media are often represented as discontinuous, piecewise constant data tied to a given spatial discretization. Such representations are inherently mesh-dependent, requiring interpolation or projection whenever they are transferred to a different discretization. In this work, we develop \\emph{Parallel and Adaptive Mesh-Free Approximation (PAM)}, a mesh-independent framework that approximates discontinuous data by a continuous, closed-form function. The resulting approximation can be evaluated consistently across different ge"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We provide a theoretical analysis showing that the proposed normalized RBF framework achieves arbitrarily small L^1 error in approximating discontinuous step functions.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"Shepard normalization stabilizes the RBF approximation near sharp interfaces so that sparse regression can produce robust continuous representations of the original discontinuous data.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"PAM is a mesh-independent RBF framework with Shepard normalization, sparse regression, adaptive refinement, and subdomain parallelism that approximates discontinuous data with arbitrarily small L1 error for step functions.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Normalized radial basis functions with Shepard stabilization approximate discontinuous step functions to arbitrarily small L1 error.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"d196087b47a9bcd1d0486f674dab80fec879e84d774bd99d0464ee2c65c74d67"},"source":{"id":"2605.16564","kind":"arxiv","version":1},"verdict":{"id":"f7bf2bcb-8c33-4003-a91c-8d33d510e7a8","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T21:21:44.954860Z","strongest_claim":"We provide a theoretical analysis showing that the proposed normalized RBF framework achieves arbitrarily small L^1 error in approximating discontinuous step functions.","one_line_summary":"PAM is a mesh-independent RBF framework with Shepard normalization, sparse regression, adaptive refinement, and subdomain parallelism that approximates discontinuous data with arbitrarily small L1 error for step functions.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"Shepard normalization stabilizes the RBF approximation near sharp interfaces so that sparse regression can produce robust continuous representations of the original discontinuous data.","pith_extraction_headline":"Normalized radial basis functions with Shepard stabilization approximate discontinuous step functions to arbitrarily small L1 error."},"integrity":{"clean":false,"summary":{"advisory":0,"critical":1,"by_detector":{"doi_compliance":{"total":1,"advisory":0,"critical":1,"informational":0}},"informational":0},"endpoint":"/pith/2605.16564/integrity.json","findings":[{"note":"DOI '10.1002/(sici' as printed in the bibliography is syntactically invalid and cannot resolve.","detector":"doi_compliance","severity":"critical","ref_index":20,"audited_at":"2026-05-19T21:31:16.459232Z","detected_doi":"10.1002/(sici","finding_type":"broken_identifier","verdict_class":"incontrovertible","detected_arxiv_id":null}],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T21:31:19.465084Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T21:31:16.459232Z","status":"completed","version":"1.0.0","findings_count":1},{"name":"claim_evidence","ran_at":"2026-05-19T19:21:56.878225Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T18:33:26.624449Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"4f88e30ba5d7e0b6d26e3323dee64f0c549793b5e03b2df01b5d468f1d049de7"},"references":{"count":31,"sample":[{"doi":"","year":1972,"title":"Bear, Dynamics of Fluids in Porous Media, Elsevier, 1972","work_id":"c44328e5-bb93-4441-b8c4-5721a024e001","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1989,"title":"Dagan, Flow and Transport in Porous Formations, Springer, 1989","work_id":"8f619652-f09c-4e3f-a9a1-e7d8ceb4c8a9","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1991,"title":"L. J. Durlofsky, Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media, Water Resources Research 27 (5) (1991) 699–708. 24","work_id":"7cab98e8-8abe-4717-b392-5bb7220cbfdd","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1997,"title":"T. Y . Hou, X.-H. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media, Journal of computational physics 134 (1) (1997) 169–189","work_id":"85cd1011-cd52-40dd-ba97-f3323f7824ae","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2009,"title":"Y . Efendiev, T. Y . 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