[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83318-en":3,"doc-seo-83318-105":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},83318,1374391974585,"Genevieve","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Multi-type Sensor Placement for PDE-based Bayesian Inverse Problems","Multi-type sensor placement for Bayesian inverse problems governed by partial differential equations (PDEs) is studied through an optimal experimental design framework. Sensors with different accuracies and observation types are incorporated by formulating the OED task as a knapsack-constrained binary optimization problem maximizing expected information gain (EIG). A stochastic cost-benefit greedy algorithm provides theoretical guarantees via monotone submodularity. Linear Gaussian cases use a closed-form EIG, while nonlinear problems use a Bayesian approximation error-based, error-corrected global linear surrogate with a provable lower bound.","arXiv :2607 .08074v1 [math .NA] 9 Jul 2026  \nMulti-type Sensor Placement for PDE-based Bayesian Inverse Problems  \nSteven Maio∗ , Alen Alexanderian∗ , Karina Koval†, and Ruanui Nicholson‡  \nAbstract. We address optimal placement of multi-type sensors for Bayesian inverse problems governed by partial differential equations (PDEs) . The proposed framework allows for sensors with different accuracies and observation types. We formulate the optimal experimental design (OED) problem as a knapsack-constrained binary optimization problem for maximizing expected information gain (EIG) . To approximately solve the resulting optimization problems, we propose a stochastic cost-benefit greedy algorithm, which admits theoretical guarantees for monotone submodular set functions. Specifically, these guarantees apply in the case of linear Gaussian inverse problems with uncorrelated measurement errors, where the EIG admits a convenient closed-form expression. For nonlinear inverse problems, we develop a non-intrusive approach that uses the Bayesian approximation error framework to define an observation model with an error-corrected global linear model. We show that the corresponding approximate EIG is a lower bound for the exact EIG and thus provides a principled surrogate objective for the OED problem. The effectiveness of the proposed methods is demonstrated in two model inverse problems governed by PDEs.  \nKey words. Inverse problems, Bayesian inference, optimal sensor placement, multi-type sensors, submodularity, knapsack constraints, expected information gain, Bayesian approximation error.  \nMSC codes. 65C20, 62K05, 35R30, 62F15, 90C27 .  \n1. Introduction. We consider infinite-dimensional Bayesian inverse problems governed by partial differential equations (PDEs), with data collected from a set of sensors. In such problems, the statistical quality of the estimated parameters depends strongly on the design of the sensor network, making optimal sensor placement crucial. We consider problems in which one has access to multiple sensor types. In this setting, we develop efficient methods for optimal placement of multi-type sensors for linear and nonlinear Bayesian inverse problems. The search for an optimal sensor placement can be formulated as an optimal experimental design (OED) problem [49] . OED for infinite-dimensional inverse problems governed by PDEs is challenging. Specifically, upon discretization, one obtains a difficult optimization problem with an expensive-to-evaluate objective that quantifies the uncertainty or information gain about a high-dimensional inversion parameter [2] . We focus on finding sensor placements that maximize the expected information gain (EIG), defined as the Kullback–Leibler divergence from the posterior to the prior. For such problems, it is common to consider the case where the sensors have equal deployment costs and measure the same quantity. A typical problem formulation is a cardinality-constrained binary optimization problem that seeks to identify an  \nFunding: The work of SM and AA was supported in part by U.S. Department of Energy, Office of Advanced Scientific Computing Research Field Work Proposal Number 23-02526 . The work of KK was partially funded by the Carl Zeiss Stiftung through the project “Model-Based AI: Physical Models and Deep Learning for Imaging and Cancer Treatment”.The work of RN was partially funded by Royal Society of New Zealand Te Ap¯arangi (Marsden Fund Council) Grant MFP-24-UOA-279  \n∗ Department of Mathematics, North Carolina State University, Raleigh, NC, USA  \n† Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University, Heidelberg, Germany ‡ Department of Engineering Science, University of Auckland, Auckland, New Zealand  \noptimal subset from a set of candidate sensor locations; see Section 2.2.  \nHowever, in some applications, sensor placement is not limited to deploying identical sensors throughout the domain. For example, in groundwater flow applicati","cbCaiuFvLNNW3Znu","https://ap.wps.com/l/cbCaiuFvLNNW3Znu","pdf",1968502,3,1,30,"English","en",105,"# Introduction\n## Problem formulation and motivation\n## Related work","[{\"question\":\"What objective does the framework optimize for sensor placement?\",\"answer\":\"It maximizes expected information gain (EIG), defined as the Kullback–Leibler divergence from the posterior to the prior for the inverse parameters.\"},{\"question\":\"How are different sensor types and varying costs handled?\",\"answer\":\"Different accuracies, observation types, and nonuniform deployment costs are modeled, leading to a knapsack-constrained binary optimization formulation rather than a simple equal-cost cardinality constraint.\"},{\"question\":\"How does the method address nonlinear PDE-based inverse problems?\",\"answer\":\"It uses a non-intrusive approach based on the Bayesian approximation error framework to build an error-corrected global linear model, yielding an approximate EIG that is a lower bound for the exact 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objective does the framework optimize for sensor placement?","Question",{"text":75,"@type":76},"It maximizes expected information gain (EIG), defined as the Kullback–Leibler divergence from the posterior to the prior for the inverse parameters.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are different sensor types and varying costs handled?",{"text":80,"@type":76},"Different accuracies, observation types, and nonuniform deployment costs are modeled, leading to a knapsack-constrained binary optimization formulation rather than a simple equal-cost cardinality constraint.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the method address nonlinear PDE-based inverse problems?",{"text":84,"@type":76},"It uses a non-intrusive approach based on the Bayesian approximation error framework to build an error-corrected global linear model, yielding an approximate EIG that is a lower bound for the exact 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