[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-117234-en":3,"doc-seo-117234-105":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":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":27,"seo_description":14,"update_tm":28,"read_time":29},117234,2336464648322,"Aria","https://ap-avatar.wpscdn.com/avatar/2200025388227c56fec?_k=1778556882303663488",8,"Research & Report","ACCELERATING MULTILEVEL MARKOV CHAIN MONTE CARLO USING MACHINE LEARNING MODELS - A Preprint","This work presents an efficient approach for accelerating multilevel Markov Chain Monte Carlo (MCMC) sampling for large-scale problems using low-fidelity machine learning models. Instead of replacing high-fidelity models entirely and incurring approximation error, the method augments high-fidelity models with low-fidelity ones in a hierarchical framework. The low-fidelity model cheaply evaluates proposed samples to improve acceptance by the high-fidelity model, with hierarchy inspired by geometric multigrid. The paper includes proofs of detailed balance, shows consistency of the resulting algorithm, and derives accuracy conditions for the learning model. The method is tested on a groundwater flow benchmark using a four-level scheme, achieving a twofold acceleration with comparable accuracy.","ACCELERATING MULTILEVEL MARKOV CHAIN MONTE CARLO USING MACHINE LEARNING MODELS  \nA PREPRINT  \narXiv :2405 . 11179v1 [ stat .ML] 18 May 2024  \n Sohail Reddy  \nLawrence Livermore National Laboratory Liveremore, CA 94550 [reddy6@llnl.gov](reddy6@llnl.gov)  \n Hillary Fairbanks  \nLawrence Livermore National Laboratory Liveremore, CA 94550 [fairbanks5@llnl.gov](fairbanks5@llnl.gov)  \nMay 21, 2024  \nABSTRACT  \nThis work presents an efficient approach for accelerating multilevel Markov Chain Monte Carlo  \n(MCMC) sampling for large-scale problems using low-fidelity machine learning models. While conventional techniques for large-scale Bayesian inference often substitute computationally expensive high-fidelity models with machine learning models, thereby introducing approximation errors, our approach offers a computationally efficient alternative by augmenting high-fidelity models with low-fidelity ones within a hierarchical framework. The multilevel approach utilizes the low-fidelity machine learning model (MLM) for inexpensive evaluation of proposed samples thereby improving the acceptance of samples by the high-fidelity model. The hierarchy in our multilevel algorithm is derived from geometric multigrid hierarchy. We utilize an MLM to acclerate the coarse level sampling.  \nTraining machine learning model for the coarsest level significantly reduces the computational cost associated with generating training data and training the model. We present an MCMC algorithm to accelerate the coarsest level sampling using MLM and account for the approximation error introduced.  \nWe provide theoretical proofs of detailed balance and demonstrate that our multilevel approach constitutes a consistent MCMC algorithm. Additionally, we derive conditions on the accuracy of the machine learning model to facilitate more efficient hierarchical sampling. Our technique is demonstrated on a standard benchmark inference problem in groundwater flow, where we estimate the probability density of a quantity of interest using a four-level MCMC algorithm. Our proposed algorithm accelerates multilevel sampling by a factor of two while achieving similar accuracy compared to sampling using the standard multilevel algorithm.  \nKeywords Bayesian Inference, Machine Learning, Multilevel Markov Chain Monte Carlo, Stochastic PDE  \n1 Introduction  \nIn the realm of statistical inference, Bayesian methods offer a powerful framework for decision making under uncertainty by integrating prior knowledge with observational data. However, the computational complexity of exact Bayesian inference often becomes prohibitive, particularly for models with high-dimensional parameter spaces or complex likelihood functions. To address this challenge, Markov Chain Monte Carlo (MCMC) methods have emerged as indispensable tools [1] . MCMC algorithms provide a means to approximate posterior distributions by generating samples from them, circumventing the need for analytical solutions which may be intractable or computationally expensive. This has led to widespread use across many disciplines. Although MCMC sampling has alleviated the challenges of analytically computing the posterior distribution, the method’s accuracy and sampling efficiency is still directly governed by the likelihood and the forward model. Employing numerical models for likelihood computations requires high-fidelity solutions, often on a finely resolved space-time discretization, which greatly increases the computational cost of the MCMC sampling.  \nTo enable MCMC sampling with computationally expensive forward maps, reduced order or surrogate models have been extensively employed [2] . These surrogate models, in the form of Gaussian processes [3, 4], deep neural networks [5, 6] and radial basis functions [7], have been employed to reduce the computational cost of the forward model at the expense  \n2.1 Hierarchical Sampling of Gaussian Random Fields  \nGaussian random fields have found application in several fields including st","cbCaisM1n02KpNSz","https://ap.wps.com/l/cbCaisM1n02KpNSz","pdf",888111,1,13,"English","en",105,"# ABSTRACT\n# 1 Introduction\n## 2.1 Hierarchical Sampling of Gaussian Random Fields","[{\"question\":\"How does the proposed multilevel MCMC method use low-fidelity machine learning models?\",\"answer\":\"It augments high-fidelity models with low-fidelity ones within a hierarchical framework. The low-fidelity model is used for inexpensive evaluation of proposed samples to improve acceptance by the high-fidelity model.\"},{\"question\":\"What theoretical guarantees does the paper provide for the multilevel algorithm?\",\"answer\":\"The work provides theoretical proofs of detailed balance and demonstrates that the multilevel approach constitutes a consistent MCMC algorithm.\"},{\"question\":\"Where is the method demonstrated and what performance improvement is reported?\",\"answer\":\"It is demonstrated on a standard groundwater flow benchmark using a four-level MCMC algorithm. The proposed approach accelerates multilevel sampling by a factor of two while achieving similar accuracy to the standard multilevel algorithm.\"}]","ACCELERATING MULTILEVEL MARKOV CHAIN MONTE CARLO USING MACHINE LEARNING MODELS - A Preprint | PDF",1785674596,33,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"accelerating-multilevel-markov-chain-monte-carlo-using-machine-learning-models-a-preprint","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"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/accelerating-multilevel-markov-chain-monte-carlo-using-machine-learning-models-a-preprint/117234/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-02",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"How does the proposed multilevel MCMC method use low-fidelity machine learning models?","Question",{"text":75,"@type":76},"It augments high-fidelity models with low-fidelity ones within a hierarchical framework. The low-fidelity model is used for inexpensive evaluation of proposed samples to improve acceptance by the high-fidelity model.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What theoretical guarantees does the paper provide for the multilevel algorithm?",{"text":80,"@type":76},"The work provides theoretical proofs of detailed balance and demonstrates that the multilevel approach constitutes a consistent MCMC algorithm.",{"name":82,"@type":73,"acceptedAnswer":83},"Where is the method demonstrated and what performance improvement is reported?",{"text":84,"@type":76},"It is demonstrated on a standard groundwater flow benchmark using a four-level MCMC algorithm. The proposed approach accelerates multilevel sampling by a factor of two while achieving similar accuracy to the standard multilevel algorithm.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]