[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86354-en":3,"doc-seo-86354-105":30,"detail-sidebar-cat-0-en-105":92},{"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},86354,1099513958762,"Logic","https://ap-avatar.wpscdn.com/avatar/1000023916a998db790?x-image-process=image/resize,m_fixed,w_180,h_180&k=1784791008015729253",8,"Research & Report","Goal-Oriented Adaptive Finite Element Multilevel Quasi-Monte Carlo","Efficiently approximating quantities of interest from PDEs with lognormal diffusivity is a central uncertainty-quantification challenge. This work addresses a difficult class combining geometric boundary singularities, lognormal coefficients without deterministic positive bounds, parameter-dependent mesh adaptation causing discontinuities, and infinitely many discontinuity locations that block classical smoothing. A multilevel quasi-Monte Carlo framework approximates deterministic bounded linear functionals of solutions to linear elliptic PDEs with random coefficients, using parametric regularity analysis, adaptive meshes, and variance reduction.","arXiv :2508 .02925v2 [math .NA] 12 Jul 2026  \nGoal-Oriented Adaptive Finite Element Multilevel Quasi-Monte Carlo  \nJoakim Beck a , Yang Liu b,1,∗, Erik von Schwerin b,1 , Raúl Tempone b,1,2 a College of Petroleum Engineering & Geosciences, Center for Integrative Petroleum  \nResearch, King Fahd University of Petroleum and Minerals, Dhahran 31261, Kingdom of  \nSaudi Arabia  \nb Computer, Electrical and Mathematical Sciences and Engineering,  \n4700 King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Kingdom of Saudi Arabia  \nAbstract  \nThe efficient approximation of quantities of interest derived from PDEs with lognormal diffusivity is a central challenge in uncertainty quantification. This paper targets a problem class that combines four analytical difficulties: a geometric boundary singularity, a lognormal coefficient field without a deterministic positive lower bound, sample-dependent mesh selection that introduces parameter-space discontinuities, and infinitely many discontinuity locations that preclude classical pre-integration smoothing. In this study, we propose a multilevel quasi-Monte Carlo framework to approximate deterministic, real-valued, bounded linear functionals that depend on the solution of a linear elliptic PDE with a lognormal diffusivity coefficient parameterized by a multi-dimensional Gaussian random vector and deterministic geometric singularities in bounded domains of Rd. We analyze the parametric regularity and develop the multilevel implementation based on a sequence of adaptive meshes, developed in our earlier work “Goal-oriented adaptive finite element multilevel Monte Carlo with convergence rates”, CMAME, 402 (2022), p. 115582. For further variance reduc-  \n∗ Corresponding author  \nEmail address: [yang.liu.3@kaust.edu.sa](yang.liu.3@kaust.edu.sa) (Yang Liu )  \n1 KAUST SRI Center for Uncertainty Quantification in Computational Science and Engineering  \n2 Alexander von Humboldt Professor in Mathematics for Uncertainty Quantification, RWTH Aachen University, 52062 Aachen, Germany.  \ntion, we incorporate importance sampling and introduce a level-0 control variate within the multilevel hierarchy. Introducing such a control variate can alter the optimal choice for the initial mesh, further highlighting the advantages of adaptive meshes. On a 2-D slit benchmark discretized with bilinear, quadrilateral Q1-FEM, numerical experiments show that, in the parameter range explored, the proposed adaptive MLQMC algorithm achieves a prescribed accuracy at markedly lower computational cost than a standard multilevel Monte Carlo estimator on the same mesh hierarchy.  \nKeywords: Multilevel Quasi-Monte Carlo, Goal-oriented adaptivity, Computational complexity, Finite elements, Partial differential equations with random data, Lognormal diffusion  \n2020 MSC: 65C05, 65N50, 65N22, 35R60  \n1. Introduction  \nWe consider a physical system subject to uncertainty and modeled by a random partial differential equation (RPDE) [10, 37, 31], and a given scalar quantity of interest (QoI), depending on the solution to a boundary value problem for the RPDE. The topic of this work is adaptive computation and error control for QoI expectations of the form E [Q(u)], where Q is a deterministic, real-valued, bounded linear functional of u which almost surely solves the boundary value problem of a linear elliptic partial differential equation (PDE) with random coefficients:  \n−∇ · (a(x;ω)∇u(x;ω)) = f(x) u (x;ω) = 0  \n∂nu (x;ω) = 0  \nfor x ∈ D ,  \nfor x ∈ ∂D1 ,  \nfor x ∈ ∂D − ∂D1 .  \n(1a)  \n(1b)  \n(1c)  \nThe variable ω corresponds to an outcome associated with a complete probability space (Ω , F, P) and the variable x belongs to an open and bounded polygonal/polyhedral domain D in Rd , where d ≥ 2. The divergence and gradient operators ∇ · and ∇ are applied with respect to the spatial variable x. The  \nrandomness in the stochastic diffusivity coefficient field a(x;ω) in general causes the solution u to be stochastic.  \nT","cbCaieNHsuzGXZTk","https://ap.wps.com/l/cbCaieNHsuzGXZTk","pdf",2338356,5,1,49,"English","en",105,"# Introduction\n## Uncertain PDEs and quantity of interest expectations\n## Boundary conditions and geometric singularities\n## Lognormal coefficient fields and parameterization","[{\"question\":\"What quantity of interest does the method approximate?\",\"answer\":\"It approximates expectations of the form E[Q(u)], where Q is a deterministic, bounded linear functional of the PDE solution u.\"},{\"question\":\"Why do adaptive meshes create additional difficulty in this problem class?\",\"answer\":\"Sample-dependent mesh selection introduces discontinuities in parameter space, which complicates standard variance-reduction and smoothing arguments.\"},{\"question\":\"What variance reduction techniques are incorporated beyond multilevel quasi-Monte Carlo?\",\"answer\":\"The approach adds importance sampling and introduces a level-0 control variate within the multilevel hierarchy.\"}]",1784210799,123,{"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":87,"head_meta":89,"extra_data":91,"updated_unix":28},"goal-oriented-adaptive-finite-element-multilevel-quasi-monte-carlo","",{"@graph":36,"@context":86},[37,54,69],{"@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":53},"https://docshare.wps.com/document/goal-oriented-adaptive-finite-element-multilevel-quasi-monte-carlo/86354/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-25","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What quantity of interest does the method approximate?","Question",{"text":76,"@type":77},"It approximates expectations of the form E[Q(u)], where Q is a deterministic, bounded linear functional of the PDE solution u.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"Why do adaptive meshes create additional difficulty in this problem class?",{"text":81,"@type":77},"Sample-dependent mesh selection introduces discontinuities in parameter space, which complicates standard variance-reduction and smoothing arguments.",{"name":83,"@type":74,"acceptedAnswer":84},"What variance reduction techniques are incorporated beyond multilevel quasi-Monte Carlo?",{"text":85,"@type":77},"The approach adds importance sampling and introduces a level-0 control variate within the multilevel 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