[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82125-en":3,"doc-seo-82125-105":29,"detail-sidebar-cat-0-en-105":90},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},82125,1099514067415,"Rowan","https://ap-avatar.wpscdn.com/avatar/100002539d78ffe74a7?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779092875211072502",8,"Research & Report","Another look at the static PDE approach to multidimensional extrapolation","A static PDE based extrapolation variant extends smooth fields across interfaces given implicitly as zero level sets. The method modifies fast sweeping by introducing relaxed upwinding finite differences. Efficiency and flexibility improve because only function values and normal derivatives are required on the first interior grid layer adjacent to the boundary. A boundary reconstruction step further reduces numerical error near the interface. Experiments across complex domain geometries, including high-curvature features, show improved performance.","arXiv :2607 .08980v1 [math .NA] 9 Jul 2026  \nAnother look at the static PDE approach to multidimensional  \nextrapolation  \nHwi Lee ∗  \nAbstract  \nWe present a variant of the static PDE based extrapolation method [1] for extending smooth fields across interfaces implicitly defined as zero level sets. Our approach introduces the idea of relaxed upwinding finite differences as a key modification within the fast sweeping method. Unlike existing approaches, we require function values and their normal derivatives only on the first grid layer inside the domain adjacent to the boundary, thereby improving computational efficiency and flexibility. To further improve accuracy, we introduce a simple boundary reconstruction technique that significantly reduces the numerical error in the extrapolated solution near the boundary. Numerical experiments indicate enhanced performance of our approach across a range of domain geometries, including those with high curvature features.  \n1 Introduction  \nIn numerous scientific computing applications, the importance of accurately and efficiently extrapolating known function values outside their known regions is well documented. Specific instances include multiphase flows [9, 11], image processing [22], electrodiffusion [13], self-assembly of polymers [17], to name only a few. In some nonlocal models such as peridynamics [20], local Taylor series based extrapolation remains a common modeling choice for imposing boundary conditions, despite the nonlocal models being viewed as alternatives to partial differential equations based models for singular solutions. In addition, as a numerical subroutine, extrapolation is essential to global numerical PDE solvers such as [10, 18] where the closest point mapping relies on constant extrapolation.  \nOur present work is in the context of level set methods [16], which extensively employ extrapolation to construct so-called ghost point values [9] . The flexibility of the level set framework for interface problems depends on the construction of ghost values that are sufficiently smooth and consistent with the prescribed boundary conditions. Assuming a smooth interface, this can be achieved by extrapolating along normal directions, which effectively reduces the construction to a one dimensional process. To this end, Aslam [2] proposed one of the most frequently used numerical methods, in which a sequence of linear transport equations is solved until steady state is reached. A variant of this time dependent approach is demonstrated in [4] to handle interfaces with kinks and high curvature. As computationally efficient alternatives, static PDE based approaches have been studied in [1, 12], where the steady state equations are solved iteratively using the fast sweeping [25] and fast marching methods [19] . We also note an implicit static approach in [14], based on solutions of biharmonic equations.  \nWe adopt the fast sweeping method approach here due to its simplicity and robustness. Our contributions consist of modifications designed to further improve efficiency and flexibility. First, we introduce the idea of relaxed upwinding finite differences which remain compatible with the fast sweeping method,  \nKey words: static PDE, extrapolation, fast sweeping method  \n∗ Department of Mathematics, New York Institute of Technology, Old Westbury, NY 11568 (E-mail: [hlee50@nyit.edu](hlee50@nyit.edu))  \nalthough they permit some downwind differences. A key feature of our approach is its reduced data requirement: function values and normal derivatives need to be available only on the first grid layer inside the domain adjacent to the boundary. This is consistent with the structure of the analytic normal extension, where boundary data alone determines the smooth extension of function values off the interface. Beyond computational efficiency, limited data availability is intrinsic to some problem settings, for example, numerical solutions of PDEs on surfaces of co-dimesion one. We f","cbCailRxNenVR3IB","https://ap.wps.com/l/cbCailRxNenVR3IB","pdf",1147470,1,14,"English","en",105,"# Introduction\n## Level set extrapolation and ghost values\n## Fast sweeping and static PDE alternatives","[{\"question\":\"What interface representation does the method assume for extrapolation?\",\"answer\":\"Interfaces are represented implicitly as zero level sets. Extrapolation extends smooth fields across these level-set-defined boundaries.\"},{\"question\":\"How does relaxed upwinding change the fast sweeping approach?\",\"answer\":\"The method introduces relaxed upwinding finite differences that remain compatible with fast sweeping while allowing some downwind differences. This enables a second-order local discretization near the boundary.\"},{\"question\":\"What data is required to perform the extrapolation efficiently?\",\"answer\":\"Function values and normal derivatives are needed only on the first grid layer inside the domain adjacent to the boundary. This reduces computational cost and supports settings with limited boundary-adjacent data.\"}]",1784178344,35,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":27},"another-look-at-the-static-pde-approach-to-multidimensional-extrapolation","",{"@graph":35,"@context":84},[36,53,67],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/another-look-at-the-static-pde-approach-to-multidimensional-extrapolation/82125/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What interface representation does the method assume for extrapolation?","Question",{"text":74,"@type":75},"Interfaces are represented implicitly as zero level sets. Extrapolation extends smooth fields across these level-set-defined boundaries.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How does relaxed upwinding change the fast sweeping approach?",{"text":79,"@type":75},"The method introduces relaxed upwinding finite differences that remain compatible with fast sweeping while allowing some downwind differences. This enables a second-order local discretization near the boundary.",{"name":81,"@type":72,"acceptedAnswer":82},"What data is required to perform the extrapolation efficiently?",{"text":83,"@type":75},"Function values and normal derivatives are needed only on the first grid layer inside the domain adjacent to the boundary. 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