{"id":"https://openalex.org/W2161241898","doi":"https://doi.org/10.1137/120899959","title":"A Square Root Approximation of Transition Rates for a Markov State Model","display_name":"A Square Root Approximation of Transition Rates for a Markov State Model","publication_year":2013,"publication_date":"2013-01-01","ids":{"openalex":"https://openalex.org/W2161241898","doi":"https://doi.org/10.1137/120899959","mag":"2161241898"},"language":"en","primary_location":{"id":"doi:10.1137/120899959","is_oa":false,"landing_page_url":"https://doi.org/10.1137/120899959","pdf_url":null,"source":{"id":"https://openalex.org/S16958353","display_name":"SIAM Journal on Matrix Analysis and Applications","issn_l":"0895-4798","issn":["0895-4798","1095-7162"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Matrix Analysis and Applications","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5052637565","display_name":"Han Cheng Lie","orcid":"https://orcid.org/0000-0002-6905-9903"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Han Cheng Lie","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5040304386","display_name":"Konstantin Fackeldey","orcid":"https://orcid.org/0000-0002-4373-3672"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Konstantin Fackeldey","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5026501395","display_name":"Marcus Weber","orcid":"https://orcid.org/0000-0003-3939-410X"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Marcus Weber","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.295,"has_fulltext":false,"cited_by_count":32,"citation_normalized_percentile":{"value":0.62775851,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"34","issue":"2","first_page":"738","last_page":"756"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10044","display_name":"Protein Structure and Dynamics","score":0.9980000257492065,"subfield":{"id":"https://openalex.org/subfields/1312","display_name":"Molecular Biology"},"field":{"id":"https://openalex.org/fields/13","display_name":"Biochemistry, Genetics and Molecular Biology"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}},"topics":[{"id":"https://openalex.org/T10044","display_name":"Protein Structure and Dynamics","score":0.9980000257492065,"subfield":{"id":"https://openalex.org/subfields/1312","display_name":"Molecular Biology"},"field":{"id":"https://openalex.org/fields/13","display_name":"Biochemistry, Genetics and Molecular Biology"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}},{"id":"https://openalex.org/T11948","display_name":"Machine Learning in Materials Science","score":0.9979000091552734,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12056","display_name":"Markov Chains and Monte Carlo Methods","score":0.994700014591217,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.753109872341156},{"id":"https://openalex.org/keywords/curse-of-dimensionality","display_name":"Curse of dimensionality","score":0.710211992263794},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6799883842468262},{"id":"https://openalex.org/keywords/configuration-space","display_name":"Configuration space","score":0.6339588165283203},{"id":"https://openalex.org/keywords/markov-chain","display_name":"Markov chain","score":0.6029788255691528},{"id":"https://openalex.org/keywords/stochastic-matrix","display_name":"Stochastic matrix","score":0.5754715204238892},{"id":"https://openalex.org/keywords/transition-rate-matrix","display_name":"Transition rate matrix","score":0.5720528364181519},{"id":"https://openalex.org/keywords/energy-landscape","display_name":"Energy landscape","score":0.5683514475822449},{"id":"https://openalex.org/keywords/metastability","display_name":"Metastability","score":0.5437371730804443},{"id":"https://openalex.org/keywords/statistical-physics","display_name":"Statistical physics","score":0.5313019752502441},{"id":"https://openalex.org/keywords/state-space","display_name":"State space","score":0.4324783682823181},{"id":"https://openalex.org/keywords/granularity","display_name":"Granularity","score":0.4212179183959961},{"id":"https://openalex.org/keywords/monte-carlo-method","display_name":"Monte Carlo method","score":0.41151151061058044},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4003893733024597},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.22145789861679077},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.16079437732696533},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.1255168616771698},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.10415437817573547},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.08193257451057434}],"concepts":[{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.753109872341156},{"id":"https://openalex.org/C111030470","wikidata":"https://www.wikidata.org/wiki/Q1430460","display_name":"Curse of dimensionality","level":2,"score":0.710211992263794},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6799883842468262},{"id":"https://openalex.org/C90738871","wikidata":"https://www.wikidata.org/wiki/Q41642869","display_name":"Configuration space","level":2,"score":0.6339588165283203},{"id":"https://openalex.org/C98763669","wikidata":"https://www.wikidata.org/wiki/Q176645","display_name":"Markov chain","level":2,"score":0.6029788255691528},{"id":"https://openalex.org/C49555168","wikidata":"https://www.wikidata.org/wiki/Q176583","display_name":"Stochastic matrix","level":3,"score":0.5754715204238892},{"id":"https://openalex.org/C70557757","wikidata":"https://www.wikidata.org/wiki/Q7834432","display_name":"Transition rate matrix","level":2,"score":0.5720528364181519},{"id":"https://openalex.org/C119621388","wikidata":"https://www.wikidata.org/wiki/Q5377166","display_name":"Energy landscape","level":2,"score":0.5683514475822449},{"id":"https://openalex.org/C89464430","wikidata":"https://www.wikidata.org/wiki/Q849516","display_name":"Metastability","level":2,"score":0.5437371730804443},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.5313019752502441},{"id":"https://openalex.org/C72434380","wikidata":"https://www.wikidata.org/wiki/Q230930","display_name":"State space","level":2,"score":0.4324783682823181},{"id":"https://openalex.org/C177774035","wikidata":"https://www.wikidata.org/wiki/Q1246948","display_name":"Granularity","level":2,"score":0.4212179183959961},{"id":"https://openalex.org/C19499675","wikidata":"https://www.wikidata.org/wiki/Q232207","display_name":"Monte Carlo method","level":2,"score":0.41151151061058044},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4003893733024597},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.22145789861679077},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.16079437732696533},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.1255168616771698},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.10415437817573547},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.08193257451057434},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/120899959","is_oa":false,"landing_page_url":"https://doi.org/10.1137/120899959","pdf_url":null,"source":{"id":"https://openalex.org/S16958353","display_name":"SIAM Journal on Matrix Analysis and Applications","issn_l":"0895-4798","issn":["0895-4798","1095-7162"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Matrix Analysis and Applications","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/7","display_name":"Affordable and clean energy","score":0.8299999833106995}],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":39,"referenced_works":["https://openalex.org/W147634160","https://openalex.org/W287611495","https://openalex.org/W1539138075","https://openalex.org/W1584991990","https://openalex.org/W1592120838","https://openalex.org/W1592998450","https://openalex.org/W1967722756","https://openalex.org/W1984414107","https://openalex.org/W1986155795","https://openalex.org/W1988220126","https://openalex.org/W1991901383","https://openalex.org/W2012029966","https://openalex.org/W2012926794","https://openalex.org/W2021751389","https://openalex.org/W2033040247","https://openalex.org/W2051752778","https://openalex.org/W2085213650","https://openalex.org/W2088941628","https://openalex.org/W2091707670","https://openalex.org/W2092924973","https://openalex.org/W2096223127","https://openalex.org/W2114710208","https://openalex.org/W2120595908","https://openalex.org/W2133908052","https://openalex.org/W2139230981","https://openalex.org/W2147848673","https://openalex.org/W2149906774","https://openalex.org/W2153504150","https://openalex.org/W2178586385","https://openalex.org/W2181864609","https://openalex.org/W2215229519","https://openalex.org/W2246461128","https://openalex.org/W2261128835","https://openalex.org/W2313374938","https://openalex.org/W2341171179","https://openalex.org/W2400113642","https://openalex.org/W4212965029","https://openalex.org/W4232095703","https://openalex.org/W4250277195"],"related_works":["https://openalex.org/W2181956924","https://openalex.org/W2146188408","https://openalex.org/W2098152723","https://openalex.org/W2025483324","https://openalex.org/W2166796864","https://openalex.org/W2156339635","https://openalex.org/W2468306076","https://openalex.org/W2339383482","https://openalex.org/W2518587987","https://openalex.org/W4244281465"],"abstract_inverted_index":{"Trajectory-":[0],"or":[1],"mesh-based":[2],"methods":[3],"for":[4,155],"analyzing":[5],"the":[6,18,27,30,38,66,75,88,105,127,131,159,167,177,180],"dynamical":[7],"behavior":[8],"of":[9,20,29,46,74,104,135,150,161,170,179,185],"large":[10],"molecules":[11],"tend":[12],"to":[13,17,36,118,138],"be":[14,194],"impractical":[15],"due":[16],"curse":[19,39],"dimensionality---their":[21],"computational":[22,168],"cost":[23,169],"increases":[24],"exponentially":[25],"with":[26,176],"size":[28,178],"molecule.":[31,181],"We":[32,108,164],"propose":[33],"a":[34,41,53,85,100,139,144,197],"method":[35,137,172],"break":[37],"by":[40],"novel":[42],"square":[43],"root":[44],"approximation":[45],"transition":[47,89,96,128],"rates,":[48],"Monte":[49],"Carlo":[50],"quadrature":[51],"and":[52,70,113,125,152],"discretization":[54,93],"approach":[55],"based":[56],"on":[57,65,143],"solving":[58],"linear":[59],"programs.":[60],"With":[61],"randomly":[62,71],"sampled":[63],"points":[64],"molecular":[67,76,106],"energy":[68,141],"landscape":[69,142],"generated":[72],"discretizations":[73,184],"configuration":[77,123,146,187],"space":[78,124,147],"as":[79],"our":[80,136,171],"initial":[81],"data,":[82],"we":[83,157],"construct":[84],"matrix":[86,98],"describing":[87],"rates":[90,129],"between":[91,130],"adjacent":[92],"regions.":[94],"This":[95],"rate":[97],"yields":[99],"Markov":[101],"state":[102],"model":[103],"dynamics.":[107],"use":[109],"Perron":[110],"cluster":[111],"analysis":[112],"coarse-graining":[114],"techniques":[115],"in":[116,122,189],"order":[117],"identify":[119],"metastable":[120,132,191],"sets":[121,192],"approximate":[126],"sets.":[133],"Application":[134],"simple":[140],"two-dimensional":[145],"provides":[148],"proof":[149],"concept":[151],"an":[153],"example":[154],"which":[156,190],"compare":[158],"performance":[160],"different":[162],"discretizations.":[163],"show":[165],"that":[166],"grows":[173],"only":[174],"polynomially":[175],"However,":[182],"finding":[183],"higher-dimensional":[186],"spaces":[188],"can":[193],"identified":[195],"remains":[196],"challenge.":[198]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":5},{"year":2023,"cited_by_count":5},{"year":2022,"cited_by_count":4},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":4},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":4},{"year":2017,"cited_by_count":1},{"year":2015,"cited_by_count":1},{"year":2014,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
