[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83715-en":3,"doc-seo-83715-105":30,"detail-sidebar-cat-0-en-105":91},{"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":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83715,4398048949847,"Eliana","https://ap-avatar.wpscdn.com/avatar/400002536579ef2da7f?_k=1778318612642679267",8,"Research & Report","Sketch Low-Rank Dynamics: Orthogonal vs. Oblique Projections","Study of sketching methods from randomized numerical linear algebra within dynamical low-rank approximation (DLRA) for large-scale matrix differential equations. The work sketches the Galerkin condition, yielding an oblique tangent-space projection that matches standard DLRA only under restrictive assumptions and degrades when perpendicular residuals are large. An orthogonal sketch DLRA is proposed, evolving sketch-orthogonal bases and using standard orthogonal projections for dynamics. The method preserves DLRA geometry, is numerically stable, and uses randomized Gram–Schmidt or randomized Cholesky QR for efficient basis updates. Sketch versions of projector-splitting and BUG integrators are derived and validated on Allen–Cahn, Fokker–Planck, and Vlasov–Poisson.","arXiv :2607 .03402v1 [math .NA] 3 Jul 2026  \nSketch low-rank dynamics: orthogonal vs. oblique projections  \nBenjamin Carrel ∗, Laura Grigori†  \nJuly 2026  \nAbstract  \nWe study how sketching techniques from randomized numerical linear algebra can be incorporated into the dynamical low-rank approximation (DLRA) of large-scale matrix differential equations. A natural approach is to sketch the Galerkin condition that defines the DLRA, which leads to an oblique tangent space projection. We show that this oblique projection approximately reproduces the standard DLRA only under restrictive conditions on the vector field, and that it fails on problems with a large perpendicular residual. As an alternative, we propose an orthogonal sketch DLRA that evolves sketch-orthogonal bases while using standard orthogonal projections for the dynamics. This approach preserves the geometric structure of the classical DLRA and is numerically stable. The computational advantage of randomized Gram–Schmidt over Householder QR lies in fewer global synchronizations on a row-distributed basis, at a comparable flop count; when the basis is well conditioned, randomized Gram–Schmidt can be replaced by randomized Cholesky QR, which additionally shifts the basis update from BLAS-2 to BLAS-3 kernels, making it well-suited to modern accelerators. We derive sketch versions of the projector-splitting and BUG integrators, and demonstrate the approach on the Allen–Cahn, Fokker–Planck, and Vlasov–Poisson equations.  \n1 Introduction  \nLarge-scale matrix differential equations arise in a variety of scientific computing applications. In kinetic theory, the Vlasov–Poisson equation describes the evolution of particle distributions in collisionless plasmas. In materials science, the Allen–Cahn equation models phase separation processes. In stochastic analysis, the Fokker–Planck equation governs the evolution of probability density functions. When discretized on tensor-product grids, these equations lead to matrix differential equations of the form  \nA˙(t) = F(t, A(t)), A(0) = A0 ∈ Rm ×n , (1)  \nwhere m and n can be very large. We refer to (1) as the full order model (FOM) . In the rest of the paper, we will consider autonomous problems, without loss of generality, in order to keep the notation light.  \nIn the last decade, various techniques for compressing the dynamics have been proposed. One of the most popular and well-studied techniques is the dynamical low-rank approximation (DLRA) (Koch and Lubich, 2007), which has since been applied to kinetic equations (Einkemmer and Joseph, 2021; Einkemmer and Lubich, 2019), weakly compressible flows (Einkemmer, 2019), and high-dimensional nonlinear PDEs (Dektoret al. , 2021) . The key idea is to evolve the lower-dimensional dynamics by projecting the vector field onto the tangent space of the manifold Mr of fixed rank-r matrices. This yields a system of coupled differential equations for the low-rank factors, reducing the complexity from O (mn) to O((m + n)r) . The DLRA is defined by the Dirac–Frenkel variational principle:  \n Y˙(t) = PY (t)F (Y(t)), Y (0) = Y0 ∈ Mr , (2)  \n∗ The work of this author was performed while at PSI Center for Scientific Computing, Theory and Data, Paul Scherrer Institute, Villigen PSI, Switzerland  \n†PSI Center for Scientific Computing, Theory and Data, Paul Scherrer Institute, Villigen PSI, and Institute of Mathematics, EPFL, Switzerland  \nwhere PY denotes the ℓ2-orthogonal projection onto the tangent space TY Mr .  \nPractical numerical integrators for the DLRA include the projector-splitting integrator (Lubich and Oseledets, 2014) and the BUG integrator (Ceruti and Lubich, 2022), together with its higher-order and rank-adaptive variants (Ceruti et al. , 2022 , 2024) and related projection methods (Carrel and Vandereycken, 2023; Kieri and Vandereycken, 2019), as well as interpolatory low-rank integrators (Carrel et al. , 2025) . These methods always require QR decompositions at each time step in order ","cbCaitXRoWjWntge","https://ap.wps.com/l/cbCaitXRoWjWntge","pdf",732635,4,1,22,"English","en",105,"# Introduction\n## Dynamical low-rank approximation (DLRA)\n## Sketch-based orthogonalization via RNLA\n## Time integration and related randomized methods","[{\"question\":\"How does the proposed oblique sketch relate to the DLRA Galerkin condition?\",\"answer\":\"The paper sketches the Galerkin condition defining DLRA, which results in an oblique tangent-space projection for the reduced dynamics.\"},{\"question\":\"When does the oblique projection fail to reproduce standard DLRA behavior?\",\"answer\":\"It approximately reproduces standard DLRA only under restrictive conditions on the vector field and fails on problems with a large perpendicular residual.\"},{\"question\":\"What is the key idea behind the orthogonal sketch DLRA?\",\"answer\":\"It evolves sketch-orthogonal bases while using standard orthogonal projections for the dynamics, preserving the geometric structure of classical DLRA and improving numerical stability.\"}]",1784189940,55,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"sketch-low-rank-dynamics-orthogonal-vs-oblique-projections","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/sketch-low-rank-dynamics-orthogonal-vs-oblique-projections/83715/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-25","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"How does the proposed oblique sketch relate to the DLRA Galerkin condition?","Question",{"text":75,"@type":76},"The paper sketches the Galerkin condition defining DLRA, which results in an oblique tangent-space projection for the reduced dynamics.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"When does the oblique projection fail to reproduce standard DLRA behavior?",{"text":80,"@type":76},"It approximately reproduces standard DLRA only under restrictive conditions on the vector field and fails on problems with a large perpendicular residual.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the key idea behind the orthogonal sketch DLRA?",{"text":84,"@type":76},"It evolves sketch-orthogonal bases while using standard orthogonal projections for the dynamics, preserving the geometric structure of classical DLRA and improving numerical 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