Multielement polynomial chaos Kriging-based metamodelling for Bayesian inference of non-smooth systems

Mul ielemen polynomial chaos K iging-based me amodelling o Bayesian in e ence o non-smoo h sys ems J.C. Ga c´ ıa-Me inoa,∗, C. Cal o-Ju adoa, E. Ma ´ ınez-Pa˜ nedab, E. Ga c´ ıa-Mac´ ıasb,c aDepa men o Ma hema ics, School o Technology, 10003, C´ace es (Spain) bDepa men o Ci il and En i onmen al Enginee ing. Impe ial College London SW7 2AZ, (UK) cDepa men o S uc u al Mechanics and Hyd aulic Enginee ing, Campus Uni e si a io de Fuen enue a (Edi icio Poli ´ecnico) 18071 G anada (Spain) Abs ac This pape p esen s a su oga e modelling echnique based on domain pa i ioning o Bayesian pa ame e in e - ence o highly nonlinea enginee ing models. In o de o alle ia e he compu a ional bu den ypically in ol ed in Bayesian in e ence applica ions, a mul ielemen Polynomial Chaos Expansion based K iging me amodel is p o- posed. The de eloped su oga e model combines in a piecewise unc ion an a ay o local Polynomial Chaos based K iging me amodels cons uc ed on a ini e se o non-o e lapping subdomains o he s ochas ic inpu space. The ewi h, he p esence o non-smoo hness in he esponse o he o wa d model (e.g. nonlinea i ies and spa seness) can be ep oduced by he p oposed me amodel wi h minimum compu a ional cos s owing o i s local adap a ion capabili ies. The model pa ame e in e ence is conduc ed h ough a Ma ko chain Mon e Ca lo app oach comp ising adap i e explo a ion and delayed ejec ion. The e iciency and accu acy o he p oposed ap- p oach a e alida ed h ough wo case s udies, including an analy ical benchma k and a nume ical case s udy. The la e ela es he pa ial di e en ial equa ion go e ning he hyd ogen di usion phenomenon o me allic ma e ials in The mal Deso p ion Spec oscopy es s. Keywo ds: Bayesian In e ence, Model Calib a ion, Polynomial Chaos Expansion, K iging, Su oga e modelling, Hyd ogen emb i lemen , The mal Deso p ion Spec oscopy 1. In oduc ion The widesp ead use o high pe o mance compu ing (HPC) echnologies has enabled he inc easingly equen adop ion o compu a ionally in ensi e nume ical models in a my iad o disciplines in academia and indus y. Such high ideli y models allow conduc ing i ual es ing o enginee ing sys ems, a oiding he echnical limi a ions and minimizing he cos s associa ed wi h adi ional expe imen al es ing. Ne e heless, o midable challenges s ill a ise when implemen ing hese models in o compu a ionally demanding s udies such as op imiza ion [1], sensi i - i y analysis [2], model iden i ica ion and calib a ion [3], eliabili y analysis [4], o obus design [5]. As a solu ion, a a ie y o su oga e models o me amodels ha e been p oposed in he li e a u e in ecen yea s. Ne e heless, despi e conside able he ad ances in he ield, he e emain open esea ch challenges o hei ex ensi e use such as handling highly nonlinea model esponses, limi ed aining da ase s and, in gene al, he online implemen a ion o su oga e models o enable decision-making in enginee ing sys ems [6, 7]. In pa icula , he de elopmen o ∗Co esponding au ho . Depa men o Ma hema ics, School o Technology, 10003, C´ ace es (Spain) Email add ess: [email p o ec ed] (J.C. Ga c´ ıa-Me ino) P ep in submi ed o Applied Ma hema ical Modelling eal ime su oga e model-based pa ame e es ima ion app oaches d aws high in e es in on ie esea ch ields such as digi al wins de elopmen [8] and sma main enance o enginee ing sys ems [9]. In hei b oades sense, su oga e models a e compu a ionally ligh black box ep esen a ions o esou ce- in ensi e models. Su oga e modelling me hods can be gene ally classi ied in o h ee ca ego ies [10]: (i) p ojec ion- based me hods o educed-o de models (ROMs) [11]; (ii) mul i- ideli y me hods [12]; and (iii) da a-d i en o esponsi e su ace me hods (RSMs). P ojec ion-based echniques p ojec he go e ning equa ions o he o iginal model on o a low-dimensional subspace. Hence, al hough ROMs ha e he ad an age o e aining he physics unde lying he model, hese a e limi ed o si ua ions whe e access o he go e ning equa ions is g an ed, which is no he case in many p ac ical applica ions when using comme cial so wa e. Mul i- ideli y me hods a e buil by simpli ying he unde lying physics o educing he nume ical esolu ion. These me hods may also ende some di icul ies, being highly case-dependen and equi ing specialized expe ise o ind a sui able ade-o be ween p edic ion accu acy and compu a ional bu den. Such di icul ies ha e os e ed apid de elopmen s o RSMs in ecen yea s as a non-in usi e echnique wi h g ea lexibili y in a wide ange o applica ions. The key ad an age o hese me hods elies in he ac ha he o wa d model does no need o be modi ied and, he e o e, i can be essen ially ea ed as a black box [13]. Among he b oad a ie y o RSMs a ailable in he li e a u e, some o he mos popula ones a e K iging [14], adial basis unc ions (RBF) [15], suppo ec o eg ession (SVR) [16], a i i- cial neu al ne wo ks (ANN) [17], Gaussian p ocess (GP) eg ession [18], polynomial chaos expansions (PCE) [2] and Polynomial Chaos Expansion based K iging (PCK) [19]. These models a e ained by exploi ing a se o eal- iza ions o pa ame e s o in e es o he model, called he expe imen al design (ED), and he co esponding model e alua ions, also called quan i ies o in e es (QoI). The e o e, he accu acy o da a-d i en su oga e models is highly de e mined by he dimensions o he design space and he numbe and dis ibu ion o he aining samples in he ED [20]. In enginee ing p ac ice, ob aining he ED cons i u es he mos ime-consuming pa since i e- qui es he e alua ion o he compu a ionally in ensi e o wa d model a each sample poin . Choosing high quali y EDs is hus c i ical o achie e high accu acy in he su oga e model cons uc ion wi h he leas possible numbe o aining samples. Sampling app oaches can gene ally be di ided in o s a ic (one-sho ) and sequen ial me h- ods. One-sho sampling gene a es he aining sample poin s in one single s ep, and common app oaches include ac ional designs and o hogonal a ays [21]. While hese echniques o e easy implemen a ion and minimal compu a ional cos , he de e mina ion o he op imal sample size may be oublesome when he beha iou o he o wa d model is unknown. To minimize such di icul ies, a numbe o sequen ial sampling s a egies ha e been in oduced, including adap i e and space- illing sequen ial me hods [22]. On one hand, space- illing sequen ial designs such as La in Hype cube Sampling (LHS), Sobol, Hamme sley and Hal on sampling [23] gene a e sam- ples i e a i ely o a ain good co e age o he pa ame ic domain. On he o he hand, adap i e sampling echniques i e a i ely d aw new samples in egions o he pa ame ic space wi h la ge p edic ion e o s, enabling o accoun o local e inemen s in he ED ( e e o e e ences [22, 24] o a ho ough s a e-o - he-a e iew). A second majo challenge o non-in usi e su oga e models ega ds he di icul ies in ol ed in he i ing o non-smoo h models exhibi ing uns eadiness, spa seness o la ge pe u ba ions. In hese cases, adap i e solu ions accommoda ing local ele an e inemen s o he esponse su ace a e equi ed. A la ge olume o esea ch has been conduc ed in he las decade o add ess his issue, gi ing o igin o a numbe o ad anced su oga e modelling echniques such as mul i- esolu ion gene alized PCE [25], domain pa i ioning [26], spa se g id colloca ion [27], 2 Vo onoi essela ions [28], local sea ch a us egions [29], clus e ing-based pa i ioning [30], ensembles o su - oga e models [31], mul i-elemen gene alized polynomial chaos (ME-gPC) me hod [27] and mul i-elemen p ob- abilis ic colloca ion me hods (ME-PCM) [32]. In his ligh , di e en pa i ioning echniques can be ound in he li e a u e in he con ex o he pu e GPs [33, 34], PCE [35], and PCE-based mapping o likelihood unc ions in pa ame e in e ence applica ions [25, 36]. In hose wo ks, di e en domain pa i ion c i e ia we e p oposed based on dissimila i y measu emen s be ween egions [34], da a densi y [33], o maximum esidual di e ences [35, 36]. In gene al, hese app oaches usually gene a e se ies o subdomains whe e he non-smoo h esponse su ace can be assumed locally smoo h, hus enabling he de ini ion o local su oga e models con ibu ing o he global esponse in a piecewise ashion. The de elopmen o cos -e icien su oga e models opens as new oppo uni ies o eal- ime pa ame e es i- ma ion applica ions. P obabilis ic Bayesian app oaches a e pa icula ly a ac i e owing o hei abili y o assess he e ec s o unce ain ies on he model pa ame e s and he de i ed esponse p edic ions, as well as hei obus - ness o noise pollu ion and e iciency o deal wi h ill-condi ioning and ill-posedness. Such excellen ea u es ha e os e ed hei implemen a ion in mul iple ields such as s uc u al iden i ica ion [37], geo echnical p oblems [38], ma e ial cha ac e iza ion [39], and bioenginee ing [40], jus o men ion a ew. In gene al, Bayesian pa ame e es ima ion app oaches exploi expe imen al da a o in e he pos e io p obabili y dis ibu ion unc ions (PDFs) o ce ain unknown model pa ame e s h ough he Bayes’ heo em. None heless, he di ec e alua ion o he pos e- io PDFs equi es sol ing he possibly high-dimensional in eg al ela ed o he e idence o he model. The e o e, excep o some i ial cases, pos e io PDFs o en need o be app oxima ed nume ically. Ma ko chain me hods cons i u e he mos widesp ead se o echniques o ex ac se ies o samples o es ima e he pos e io PDFs, allow- ing o sample om a la ge class o high-dimensional dis ibu ions. Popula p ocedu es o Ma ko chain Mon e Ca lo (MCMC) sampling a e he Me opolis-Has ings [41] and Gibbs algo i hms [42]. The basic idea o hese echniques is o cons uc a Ma ko chain wi h a s a iona y dis ibu ion esembling he pos e io dis ibu ion, in such a way ha a sample o he join PDF o he model pa ame e s can be ob ained by collec ing he s a es o he chain. These me hods gua an ee asymp o ic con e gence o he exac PDF, al hough a conside ably la ge numbe o i e a ions a e ypically equi ed o achie e con e gence, which comp omises he compu a ional e iciency o he in e ence [43]. To alle ia e he compu a ional bu den in classical MCMC echniques, a a ie y o mo e e icien sampling algo i hms ha e been p oposed in ecen yea s, including Sequen ial Mon e Ca lo (SMC) [44], T ansi- ional MCMC [45], and Bayesian b oad lea ning [46]. No wi hs anding hese ad ances, hei ele a ed compu a- ional cos emains a c i ical limi a ion when high- ideli y models a e conside ed in he in e ence p oblem. He ein is whe e su oga e models o e an e icien solu ion o conduc cos -e icien Bayesian in e ence while e aining he accu acy o high- ideli y models. The eno mous po en ials o his app oach a e e idenced by he inc easing numbe o esea ch s udies epo ed in ecen yea s. I is wo h no ing he wo k by Schneide e al. [47] who p o- posed a Bayesian p ocedu e using a ional PCE me amodels o he esponse o dynamic sys ems in he equency domain, and demons a ed i s e ec i eness o he iden i ica ion o a c oss-lamina ed imbe pla e. Xing e al. [48] de eloped an addi i e GP model o mul i- ideli y su oga e modelling inse ed in o a non-pa ame ic Bayesian app oach wi h a closed- o m solu ion o he p edic i e pos e io PDF. The accu acy and lexibili y o hei ap- p oach we e alida ed wi h se e al benchma k p oblems, including he iden i ica ion o a solid oxide uel cell model and an elbow-shape pipe unde u bulen mixing low condi ions. Ie imon i and co-au ho s [49] p oposed a 3 K iging-based conjuga e Bayesian iden i ica ion me hodology o online damage iden i ica ion o an ins umen ed monumen al building, he Consoli Palace in Gubbio (I aly). Del Val e al. [50] implemen ed an MCMC app oach o in e he ca aly ic ecombina ion pa ame e s o eusable he mal p o ec ion ma e ials exploi ing plasma wind unnel measu emen s. In e es ingly, ins ead o bypassing a high- ideli y model, hose au ho s app oxima ed he likelihood unc ion in he Bayesian in e ence using a GP su oga e model o accele a e he pa ame e s es ima ion. In ligh o he li e a u e e iew abo e, i is appa en ha he ields o su oga e modelling and i s applica- ions o as pa ame e es ima ion ha e expe ienced conside able ad ances in ecen yea s. None heless, ligh me amodels capable o bypassing highly nonlinea models o online pa ame e es ima ion a e ye o be ully de eloped. In his con ex , he p esen wo k p oposes a no el mul ielemen su oga e model ex ending he PCK p oposed in [19] o online Bayesian pa ame e in e ence o non-smoo h models. Wi h he aim o accommoda ing nonlinea beha iou s in he o wa d model while e aining lexibili y and minimal compu a ional cos , a simple egula block pa i ioning app oach is implemen ed. On his basis, he space domain is pa i ioned in o dis inc sub egions wi h spli ing di ec ions chosen on he basis o a p elimina y sensi i i y analysis, p io i ising he di- ision o hose a iables wi h he highes sensi i i ies. In he p esen s udy, Sobol’s indices o e he ull domain ha e been conside ed o his pu pose, since hey can be eadily ob ained as a by-p oduc o PCE [51]. Wi h espec o he cons uc ion o he local PCK su oga e models he op imal o de o he polynomials in he PCE is au oma ically iden i ied by a model selec ion echnique o spa se linea models, he leas -angle eg ession (LAR) algo i hm se ou by E on e al. [52]. Then, he op imal PCE is inse ed in o a K iging p edic o as he end e m, while he s ochas ic e m is i ed h ough a gene ic algo i hm (GA) global op imiza ion app oach. In his ega d, he domain space is spli in o a disc e e numbe o subse s whe e local su oga e models a e cons uc ed. Then, he global model esponse is ob ained by combining he local su oga e models in a piecewise ashion. In o de o minimize he compu a ional bu den in he cons uc ion o he su oga e models, he ED is ob ained wi h he adap i e Mon e Ca lo-In e si e-p oj- h (MIPT) app oach [22]. Finally, he su oga e model is used o Bayesian pa ame e es ima ion using a cos -e icien adap i e MCMC wi h delayed ejec ion (DRAM) algo i hm de eloped by Haa io e al. [53]. The e ec i eness o he p oposed app oach is alida ed h ough wo benchma k case s ud- ies: (i) an analy ical benchma k; (ii) and a pa ial di e en ial equa ion (PDE) o he mal deso p ion spec oscopy (TDS) expe imen s o hyd ogen in me als. The la e ep esen s a o midable example o a model exhibi ing a non-smoo h beha iou in he shape o nonlinea i ies and uns eadiness. The p esen ed esul s demons a e he e - ec i eness o he p oposed app oach o conduc ing online Bayesian pa ame e s iden i ica ion, p o ing obus ness o he p esence o highly nonlinea beha iou s and mul imodali y in he pos e io dis ibu ions. The emainde o his pape is o ganized as ollows. Sec ion 2 ou lines he heo e ical o mula ion o he p oposed app oach. Sec ion 3 p esen s he nume ical esul s and discussion. In pa icula , wo case s udies a e in es iga ed, namely a benchma k analy ical model and a nume ical model o he TDS analysis o hyd ogen deso p ion in me als. Finally, Sec ion 4 discusses he con ibu ions o his wo k and p esen s he main concluding ema ks. 2. Theo e ical o mula ion The main pu pose o his sec ion is o p esen he heo e ical undamen als o he p oposed su oga e model- based Bayesian in e ence app oach. The gene al me hodology is ske ched in Fig. 1 and comp ises ou main s eps, 4 namely (i) domain pa i ioning; (ii) cons uc ion o he local su oga e models; (iii) su oga e models assemblage; and (i ) alida ion. No Domain pa i ioning Nume ical model (x) i- h Local Su oga e Model o i=1,...,d Sampling o he design space MIPT algo i hm Expe imen al design MCS Su oga e model Su oga e models assemblage Valida ion Compu a ional cos analysis Su oga e model (x) No Yes Yes Bayesian Pa am. In e ence Figu e 1: Flowcha o he p oposed PCK me amodel o pe o m Bayesian In e ence. 2.1. Su oga e modelling: Polynomial Chaos Expansion based K iging Le (Ω,Σ, µ) be a p obabili y space, wi h Ω⊂RMdeno ing he e en space equipped wi h a σ-algeb a Σo subse s o Ωand a p obabili y measu e µsuch ha µ(Ω)=1. Le M:Ω⊂RM→Rbe a compu a ional model mapping a ec o o inpu a iables x=[x1,...,xM]Tin o he ou pu a iable o quan i y o in e es y∈R. The goal o su oga e modelling is o app oxima e he o wa d model M, which is ypically compu a ionally in ensi e, by a compu a ionally inexpensi e unc ion ˆ M. In his pape , PCK me amodels combining PCE and K iging a e adop ed. In his ligh , PCE is used o app oxima e he global beha iou o he compu a ional model Mwhile he K iging me amodel cap u es i s local beha iou . 2.1.1. Polynomial Chaos Expansion Assume ha he inpu ec o x∈Ωis cons i u ed by Mindependen andom a iables componen s {xi}M i=1wi h p obabili y densi y unc ions µXi. Then, PCE ep esen s he ou pu esponse y∈Ras he in ini e expansion o M o e an o hono mal basis o mul i a ia e polynomials Ψαas: y=M(x)=X α∈NM aαΨα(x),(1) 5 whe e aα,α=(α1, . . . , αM), αi∈Na e he coe icien s o he expansion. O hono mal basis amilies associa ed wi h a a ie y o s anda d dis ibu ions can be ound in e e ence [54]. Mul i a ia e polynomials Ψαcan be exp essed in e ms o a amily o uni a ia e polynomials nψ(i) j,j∈Noas Ψα(x)=QM i=1ψ(i) αi(xi). Polynomials ψ(i) j a e also o hono mal wi h espec o he ma ginal dis ibu ion, ha is: Ehψ(i) j(xi), ψ(i) k(xi)i=Zψ(i) j(u)ψ(i) k(u)µxi(u) du=δj,k,(2) wi h δj,kbeing he Di ac Del a unc ion. Random a iable xinduces he p obabili y measu e µxon RM,BRM, wi h B(R)deno ing he Bo el σ-algeb a µx(B)=µ{ω∈Ω:x(ω)∈B},B∈ B. The PDF o xis gi en by he p obabili y measu e µxas µx(u)=µ(x≤u),u∈RM. No e ha , since he componen s o xa e independen , he join p obabili y unc ion µx(u) is gi en by he p oduc o he ma ginal PDFs o xi, i.e. µx(u)=QM i=1µxi(ui), u=(u1,...,uM)∈RM. The e o e, Eq. (2) can be w i en in a mo e compac o m as: EhΨα(x),Ψβ(x)i=ZΨα(u)Ψα(u)µx(u) du=δα,β,α,β∈NM.(3) Al hough he exp ession in Eq. (1) is exac o an in ini e numbe o e ms, in p ac ice only a ini e numbe can be compu ed and a ce ain unca ion scheme needs o be adop ed. One o he simples app oach consis s in selec ing all he polynomials whose o al deg ee |α|= M X i=1 αibelongs o he se : AM,p=nα∈NM: 0 ≤|α|≤po,wi h ca d AM,p= M+p p=(M+p)! M!p!.(4) Fo high-dimensional and non-linea p oblems, his unca ion p ocedu e usually leads o la ge numbe s o polynomial coe icien s and conside able compu a ional bu dens. Ne e heless, i is o en obse ed in many p ac- ical applica ions ha coe icien s co esponding o high in e ac ion e ms be ween he inpu a iables a e close o ze o, a phenomenon ha is also known as he spa si y-o -e ec p inciple [51]. To alle ia e his, an hype bolic unca ion scheme can be adop ed. This app oach selec s all mul i-indices wi h q-no m kαkq= M X i=1 αq i 1 q less han o equal o a ce ain model o de p, i.e. AM,p,q=nα∈NM:kαkq≤po. No e ha he expansion ends o only main ain uni a ia e polynomials as he q-no m dec eases, hus achie ing a educ ion in he compu a ional cos o he me amodel. None heless, he numbe o e ms may emain ele a ed since he po en ial spa seness in he coe - icien s is no being ac ually assessed. Following he wo k by Bla man and Sud e [51], he LAR algo i hm [52] is adop ed o u he educe he numbe o polynomial coe icien s. In he con ex o PCE, LAR cons uc s a se o expansions inco po a ing an inc easing numbe o basis polynomials Ψα, om 1 o P=ca d AM,p,q. The esul ing sequence o index se s is used o cons uc a amily o expansions wi h dec easing spa seness. Finally, a c oss- alida ion p ocedu e can be implemen ed o selec he bes me amodel among he ob ained amily o ex- pansions. In pa icula , he Bayesian In o ma ion C i e ion (BIC) is adop ed in his wo k. Once he op imal index se is selec ed, he expansion coe icien s a={aα,α∈ AM,p⊂NM}a e ob ained by minimizing he expec a ion o he leas squa ed e o : a=a g min a∈RP EM(x)−X α∈AM,p,q aαΨα(x) 2.(5) 6 In p ac ice, Eq. (5) is calib a ed on a se o N ealiza ions Ξ = nx(1),...,x(N)oo he inpu a iable x o ming he ED. In o de o ob ain a ep esen a i e ED, he adap i e MIPT sampling me hod [22] is adop ed in his wo k. Then, he expec a ion ope a o in Eq. (5) is eplaced by i s disc e ized e sion using he ED ealiza ions as ollows: a=a g min a∈RP 1 N N X i=1Mx(i)−X α∈AM,p,q aαΨαx(i) 2 .(6) Deno ing he ealiza ions o he ou pu a iable yby y={y(1) =M(x(1)),...,y(N)=M(x(N))}T, he solu ion o he op imiza ion p oblem in Eq. (6) eads: ˆ a=ΘTΘ−1ΘTy,Θ=(Θi j)=hψj(x(i))ij=1,...,P i=1,...,N,(7) whe e Θdeno es he in o ma ion ma ix calcula ed om he e alua ion o he basis polynomials on Ξ. Fo he leas -squa e minimiza ion p oblem in Eq. (6) o be well posed, he size o he ED is usually selec ed acco ding o he heu is ic ule N≈2·Po 3 ·P[51]. Once Eq. (6) has been sol ed, he p edic ions o he PCE su oga e model can be ob ained as: ˆy=ˆ MPCE (x)=X α∈A ˆ aαΨα(x).(8) 2.1.2. Polynomial Chaos Expansion based K iging (PCK) The K iging me hod assumes ha he esponse o a compu a ional model M(x) is modelled by he sum o a s ochas ic andom p ocess Z(x) and a eg ession model T(x), also called end, in he o m [22]: ˆ M(x)=T(x)+Z(x).(9) The s ochas ic componen in Eq. (9) is ully de e mined by he co a iance unc ion [55]: Co Z(x),Zx0=EZ(x)Z(x0)=σ2Rx−x0;θ,(10) wi h σ2being he p ocess a iance, and R(|x−x0|;θ)an au o-co ela ion unc ion [56] be ween wo inpu sample poin s xand x’ ha depends on ce ain hype -pa ame e s θ o be compu ed. In his wo k, he Gaussian co ela ion unc ion is adop ed as: Rx,x0,θ= M Y `=1 exp h−θ`x`−x0`2i.(11) The end e m o he K iging model in Eq. (9) in e pola es he o wa d model e alua ions a he ED, while he local a iabili y is cap u ed by he s ochas ic p ocess. Depending on he o m o he end, h ee di e en e sions o K iging a e ypically e e ed o in he li e a u e [22], including simple, o dina y and uni e sal K iging, which espec i ely co espond o polynomials o deg ees 0, 1 and N. In his wo k, wi h he aim o combining he excellen global app oxima ion capabili ies o he PCE p e iously in oduced in Sec ion 2.1.1, he spa se PC expansion ob ained by LAR is in oduced in he shape o he end e m in Eq. (9). The esul ing PCK me amodel eads: 7 ˆ MPCK (x)=ˆ MPCE (x)+Z(x)=X α∈A ˆ aαΨα(x)+Z(x).(12) The cons uc ion o he PCK me amodel in Eq. (12) consis s in wo s eps. Fi s ly, he op imal se o o hono - mal polynomials Ψα( o α∈ A he unca ion se ) is ob ained by LAR as indica ed in Sec ion 2.1.1. Secondly, he calcula ion o hype pa ame e s ˆ θand he polynomial coe icien s and he p ocess a iance {a(ˆ θ), σ2(ˆ θ)}a e ob ained. The op imal co ela ion pa ame e s ˆ θcan be de e mined by he Maximum-Likelihood-Es ima ion (ML) h ough he ollowing minimiza ion p oblem [57]: ˆ θ=a g min θ"1 N(y−Θa)TR−1(y−Θa) (de R)1/N#.(13) In o de o sol e he op imiza ion p oblem in Eq. (13), local op imiza ion algo i hms such as g adien -based me hods a e o en used. None heless, a majo d awback o hese echniques ela es he oublesome iden i ica ion o global maxima/minima, being possible o ge s uck in local maxima/minima. To a oid his, a global gene ic algo i hm op imiza ion p ocedu e is used in his wo k. Since he co ela ion ma ix is symme ic and posi i e de ini e, i s in e se in Eq. (13) is compu ed by Cholesky decomposi ion. Then, once ˆ θis compu ed, he polyno- mial coe icien s and he p ocess a iance {a(ˆ θ), σ2(ˆ θ)}a e calcula ed using he Empi ical Bes Linea Unbiased Es ima o (BLUE) as [58]: aˆ θ=ΘTR−1Θ−1ΘTR−1y, σ2ˆ θ=1 N(y−Θa)TR−1(y−Θa),(14) whe e Ri j =Rx(i)−x(j);ˆ θis he co ela ion ma ix and Θi j =ψjx(i) he in o ma ion ma ix e alua ed a all he samples o he ED. E ec i e explo a i e sampling. Wi h he aim o gene a ing ep esen a i e EDs, he MIPT algo i hm is adop ed as a compu a ionally e icien and easily implemen able adap i e sampling echnique. The main ad an age o his echnique compa ed o space- illing echniques such as LHS ega ds i s abili y o a oid local clus e ing o poin s which may consequen ly lead o nume ical ins abili ies in he in e se o he K iging co ela ion ma ix in Eq. (14). This explo a ion dis ance-based sampling me hod i e a i ely augmen s he ED by adding new sampling poin s wi h maximum dis ance wi h espec o he da a popula ion in he ED among a la ge se o Nc andom Mon e- Ca lo candida es. Speci ically, among he candida es se C={ξ(1),ξ(2),...,ξ(Nc)}, a new sample x(N+1) is chosen by sol ing he ollowing op imiza ion p oblem: x(N+1) =a g max ξ∗∈C minx(i),i=1,...,Nξ∗−x(i)2,(15) wi h k·k2 he euclidean no m, i.e. kxk2=PM i=1x2 i1/2. 2.1.3. Mul i-elemen su oga e model app oach The p e iously p esen ed PCK me amodel su e s om low con e gence a es when he o wa d model Mex- hibi s non-smoo hness [59]. Thus, conside ably la ge ED sizes a e o en equi ed o achie e accu a e p edic ions. This aspec unde mines he compu a ional e iciency o he PCE-based K iging model, which is domina ed by he O(N3) complexi y o he K iging p edic o . In u n, his implies long cons uc ion imes o e en memo y o e low 8 issues when sol ing he op imiza ion p oblem in Eq. (13). Mo eo e , he la ge he size o he ED, he slowe he e alua ion o he co esponding me amodel, which educes o anishes he ad an ages o he su oga e app oach. To add ess his issue, a mul i-elemen PCK model inspi ed by he ME-gPC me hod by Wan and Ka niadakis [60] is p oposed in his wo k. This app oach consis s in he pa i ioning o he andom inpu space in o a ini e se o non-o e lapping subdomains, he cons uc ion o a local PCK su oga e model in each one ollowing he o mu- la ion in Sec ion 2.1.2 and, inally, assembling hem in o a piecewise unc ion o ob ain a global me amodel, as ske ched in Fig. 1. In o de o add ess he di ec ion o he pa i ions, an app oach elying on sensi i i y analysis based on he Sobol’s indices has been adop ed in his wo k. No e ha he Sobol’s indices can be eadily compu ed as a by- p oduc o he PCE [51]. In his way, p io i y in he pa i ioning is gi en o he di ec ion o hose pa ame e s wi h highes sensi i i y, i.e., hose wi h he g ea es e ec on he a iabili y o he quan i y o in e es y. On he o he hand, in he ollowing analyses he numbe o di isions ha e been a p io i de e mined in o de o compa e models buil wi h he same amoun o in o ma ion. On his basis, o he andom a iable x:Ω→ Dx⊂RM, a decomposi ion is de ined as Dx=[ j∈J Dj,Dj∩Dj0=∅,i j,j0,(16) whe e χDj:Ω→Rdeno es he indica o andom a iable: χDj(x)=         1 i x∈ Dj 0 o he wise . In his way, he global model is de ined in a piecewise ashion as: ˆ MPCK (x)=X j∈J χDj(x)ˆ Mj(x).(17) As a o emen ioned, he numbe o pa i ions in his wo k is de ined a e a pa ame ic analysis. Ne e heless, he p e ious o mula ion may be eadily au oma ed as ollows. The spli ing c i e ion o he domain is de e mined by a ce ain use -de ined accu acy goal and a minimum numbe o samples Npe egion. A e wa ds, he spli ing p ocess is pe o med i e a i ely om a PCK model buil o e he ull pa ame e space Ω. In case he a ge accu- acy has no been eached, he space is spli in o wo egions and he ED is en iched in each o hese subdomains by he MIPT algo i hm un il he e a e Nsamples in each one. No e ha , gi en he sequen ial na u e o MIPT, he in o ma ion o he p e iously ex ac ed samples is no los . I he accu acy goal is no eached ye , a new di ision o he space and a new en ichmen o he ED a e pe o med 2.1.4. Su oga e model accu acy. Complexi y analysis o he algo i hm To e alua e accu acy o he de eloped me amodel, bo h local and global e o me ics a e conside ed. These me ics a e compu ed by conside ing a alida ion se (VS) Λ={ξ(1),...,ξ(K)},K∈N, o he pa ame e s space (in- dependen o he ED). Deno e by Υ={υ(1) =M(ξ(1)), . . . , υ(K)=M(ξ(K))}and ˆ Υ={ˆυ(1) =ˆ MPCK (ξ(1)),...,ˆυ(K)= ˆ MPCK (ξ(K))} he ou pu s o he VS es ima ed by he o wa d model and he me amodel, espec i ely. Then, he accu acy o he su oga e model can be assessed by using he e o me ics like hose collec ed in Table 1. In his able, ¯ Υand σΥ= PK i=1¯ Υ−υ(i)2/(K−1)deno e he a i hme ic mean and he quasi s anda d de ia ion o 9 01234 0 1 2 3 4 Fo wa d model M Su oga e ˆ M1 3 (a) ED = 1440, 1 pa i ion 01234 0 1 2 3 4 R2= 0.139 0 1 2 3 4 0 1 2 3 4 Fo wa d model M Su oga e ˆ M1 4 (b) ED = 2880, 1 pa i ion 0 1 2 3 4 0 1 2 3 4 R2= 0.999 0 1 2 3 4 0 1 2 3 4 Fo wa d model M Su oga e ˆ M3 4 (c) ED = 2880, 9 pa i ions 0 1 2 3 4 0 1 2 3 4 R2= 0.993 01234 0 1 2 3 4 Fo wa d model M Su oga e ˆ MSSE 4 (d) ED = 2880, SSE 01234 0 1 2 3 4 R2= 0.298 1 Figu e 3: Fo wa d model e alua ions e sus su oga e model p edic ions o he D op-Wa e unc ion. ˆ M1 3(a), ˆ M1 4(b), ˆ M3 4(c) and ˆ MS S E 4(d) (VS o 20 000 samples). 0500 1000 1500 2000 2500 3000 10−4 10−3 10−2 10−1 100 101 ED size No m. mean-squa e e o (NRMSE) 10−6 10−5 10−4 10−3 10−2 E alua ion ime ( e) [s] 1 egion ( 1 ) 4 egions ( 2 ) 9 egions ( 3) Figu e 4: Pe o mance assessmen o PCK su oga e models o he D op-Wa e unc ion (VS o 20 000 samples). 3.2. The mal Deso p ion Spec oscopy (TDS) o hyd ogen in me als This las sec ion epo s he use o he p oposed su oga e model o he Bayesian iden i ica ion o hyd ogen deso p ion and apping cha ac e is ics in me allic ma e ials. Hyd ogen emb i lemen (HE) e e s o he loss o duc ili y and oughness o me allic alloys induced by hyd ogen a oms deposi ed a la ice si es and mic o- s uc u al de ec s such as disloca ions, g ain bounda ies o acancies [69, 70]. Al hough his phenomenon has been ex ensi ely documen ed since he 19 h cen u y [71], he g owing end owa ds a hyd ogen-based economy as a means o mi iga ing CO2emissions and ossil uel dependency has gene a ed unp eceden ed in e es on HE esea ch. Mic o-s uc u al de ec s in me als ac as ‘ ap’ si es, which seques e hyd ogen and go e n he suscep ibili y o HE [72, 73]. Thei cha ac e iza ion is hus o pi o al impo ance o he unde s anding o HE and he design o HE- esis an alloys, and his is gene ally achie ed using TDS expe imen s [74]. The TDS es in ol es se e al s ages [75]: cha ging a sample wi h hyd ogen, hea ing he sample a a ixed a e, and de ec ing he lux o deso bing hyd ogen as a unc ion o empe a u e by using a mass spec ome e . The hyd ogen low cu e o deso bing hyd ogen as a unc ion o empe a u e de ines he TDS spec um, whose peaks can be associa ed wi h he p esence o di e se mic o-s uc u al de ec s. None heless, he o ma ion o peaks in he TDS spec um 16 may be induced by he combined ac ion o mani old hyd ogen aps, being necessa y o use simula ion models and in e se calib a ion o hei iden i ica ion. P e ious in es iga ions on he modelling o he TDS es e idenced he exis ence o non-smoo h ela ionships be ween he lux cu es and he pa ame e s cha ac e izing mic o-s uc u al de ec s (see e.g. [76]), making his applica ion a o midable benchma k case s udy o he o mula ion p esen ed in his wo k. In he emainde o his sec ion, he PDE go e ning he hyd ogen di usion in ma e ials es ed by TDS is in oduced in Sec ion 3.2.1. The cons uc ion o he su oga e model and i s pe o mance e alua ion is epo ed in Sec ion 3.2.2 and, inally, Sec ion 3.2.3 p esen s he Bayesian pa ame e iden i ica ion esul s. 3.2.1. TDS go e ning di usion equa ion Conside a one-dimensional specimen o leng h Las ske ched in Fig. 5 (a). The specimen is subjec ed o inc easing empe a u es T, s a ing om Toand inc easing a a cons an hea ing a e φ. Hyd ogen a oms occupy no mal in e s icial la ice si es (NILS) and addi ionally can eside a apping si es such as in e aces o disloca- ions. The kine ics o hyd ogen apping and de apping in me als is commonly desc ibed wi h a wo-le el sys em as ske ched in Fig. 5 (b) o he case o a single ap. The po en ial landscape in his igu e desc ibes he di usion pa h o hyd ogen in me als, he ap binding ene gy ∆Hbeing he di e ence be ween de apping and apping ene gies. Le us assume ha he numbe o hyd ogen aps in he specimen amoun s o N . Then, le CL(x, ) and CT,i(x, ), i=1,...,N , deno e he hyd ogen concen a ion in he la ice and in he i- h ap, espec i ely, wi h x∈[−L/2,L/2]and espec i ely deno ing space and ime. On his basis, he Fickian di usion equa ion needs o be en iched wi h sou ce and sink e ms as [76]: ∂CL ∂ + N X i=1 ∂CT,i ∂ =DL ∂2CL ∂x2,(27) L x θL(x)=θL 0 θL=0θL=0 (a) (c) (d) θL(x) 0 θL 1 -L/2 L/2 Tempe a u e T Flux J =0 1 2 3 (b) ap si e ap de ap ΔH H-a om Figu e 5: (a) A schema ic illus a ion o ini ial and bounda y condi ions in a TDS es . (b) Schema ic de ini ion o binding ene gy in a one-dimensional di usion pa h. (c) T ansien solu ion cu es o he no malised la ice occupancy ac ion θL/θ0 L a di e en imes along he specimen’s hickness. (d) A schema ic o ypical hyd ogen deso p ion lux e sus empe a u e cu es ob ained in a TDS es . wi h DL=Doexp (−Q/RT)being he la ice di usion coe icien , which is exp essed in e ms o he la ice ac i a ion ene gy Q, di usion p e-exponen ial ac o Do, and he uni e sal gas cons an R. I is con enien o in oduce he la ice and ap occupancy ac ions θLand θT,iθL, θT,i∈[0,1], espec i ely, by ew i ing he 17 co esponding concen a ions in he o m CL=θLβNLand CT,i=θT,iαNT,i. He e, βis he numbe o NILS pe uni olume, αis he numbe o a oms si es pe ap, NLis he numbe o la ice a oms pe uni olume, and NT,i is he numbe o ap si es pe uni olume. The e o e, Eq. (27) can be ew i en as: ∂θL ∂ + N X i=1 αNT,i βNL!∂θT,i ∂ =DL ∂2θL ∂x2.(28) The PDE in Eq. (28) needs o be complemen ed wi h ap kine ic equa ions. To his aim, he o mula ions by MacNabb and Fos e [77] and O iani [78] a e commonly adop ed. The la e ep esen s a simpli ica ion o he o me by assuming ha a local equilib ium exis s be ween he hyd ogen a oms a he la ice si es and he i- h ap such ha , o θL1, θT,i=KiθL (1+KiθL),(29) wi h Kibeing he local equilib ium cons an o he i- h ap: Ki=exp (−∆Hi RT ).(30) In oducing Eq. (29) in o (28), and ollowing he non-dimensional o mula ion de eloped by Raina e al. [76], he go e ning PDE desc ibing hyd ogen di usion in he TDS es can be ecas in a compac o m as: ∂θL ∂ 1+ N X i=1 KiNi 1+Kiθ0 LθL2+θL T2 N X i=1 KiNi∆Hiφ 1+Kiθ0 LθL2=DL ∂2θL ∂x2,(31) wi h θ0 Lbeing he ini ial la ice occupancy. The non-dimensional a iables employed a e lis ed in Table 4. Table 4: Non-dimensional a iables used in he hyd ogen di usion PDE employed o he TDS es s. Spa ial coo dina e x=x/LLa ice ac i a ion ene gy Q=Q/(RTo) Time coo dina e = Do/L2T ap binding ene gy ∆Hi= ∆Hi/(RTo) Hea ing a e φ=φL2/(ToDo)La ice di usion coe icien DL=DL/Do T ap densi y Ni=αNT,i/(βNL)Local equilib ium cons an K=exp −∆Hi T Tempe a u e T=T/ToF ac ional la ice occupancy θL=θL/θo L The ini ial and bounda y condi ions o he PDE in Eq. (31) a e schema ically p esen ed in Fig. 5 (a). A =0, i is assumed an ini ial uni o m la ice occupancy θLx, =0=1. The ea e , he hyd ogen la ice occupancy is assumed ze o a he bounda ies, ha is θLx=±1/2, >0=0. As empe a u e aises, he la ice occupancy e ol es spa ially and empo ally as ske ched in Fig. 5 (c), and he lux o hyd ogen a oms J( ) di using ou a bounda ies is measu ed as p esen ed in Fig. 5 (d). This lux can be ob ained in non-dimensional e ms a e sol ing Eq. (31) as [76]: J=−DLθo L ∂θL ∂x.(32) Gene ally, he magni udes o Q,D0and θ0 La e known, and he hea ing a e φis an inpu o he TDS sys em. The e o e, he TDS spec um can be used o map he mic os uc u al hyd ogen aps, as cha ac e ised by hei ap 18 densi ies (Ni) and binding ene gies (∆Hi). These can be ob ained o a gi en lux cu e Jby he in e se calib a ion o he PDE in Eq. (31). The su oga e modelling o he lux cu es ob ained a e sol ing Eq. (31) ep esen s a o midable p oblem due o he s ong nonlinea i ies o hese cu es. Speci ically, depending upon he hyd ogen ap con igu a ion, se e al di e en egimes can be obse ed, as p e iously discussed by Raina e al. [76]. Speci ically, hei esul s o he case o me als con aining a single ap showed ha no peak lux is a ained o low ap densi ies and binding ene gies. Al e na i ely, when a peak lux is ound, hose au ho s iden i ied wo dis inc egimes (I and II) o igina ed by wo ypes o mic os uc u al de ec s, e e ed o as shallow and deep aps. Shallow aps a e cha ac e ized by la ge ap densi ies, and gi e o igin o peak luxes ha a e highly sensi i e o bo h Nand ∆H. On he o he hand, deep aps a e cha ac e ized by low ap densi ies, esul ing in peak luxes ha a e insensi i e o he ap binding ene gy. The exis ence o hese di e en egimes u ns he cons uc ion o a su oga e model co e ing he whole domain o he aps in o an no ably challenging ask. No e ha a la ge numbe o high-o de polynomials and a dense ED need o be included in he PCE o accu a ely ep esen he whole global beha iou o he hyd ogen lux. Such la ge EDs may se e ely comp omise he compu a ional e iciency o he su oga e model since, as indica ed abo e, he complexi y o he Cholesky decomposi ion o he co ela ion ma ix Rin Eq. (13) is ON3. The me amodeling o TDS expe imen s hus ep esen s an excep ional case s udy o jus i y he use o he domain pa i ioning app oach p esen ed in Sec ion 2.1.3. 3.2.2. Su oga e modelling o TDS lux cu es o me als wi h wo aps The PDE in Eq. (31) is sol ed nume ically by using he pdepe sol e in MATLAB. A space disc e iza ion o 201 elemen s along xwas ound o p o ide mesh-independen esul s. To illus a e he beha iou o a e i ic s eel sample, ep esen a i e model pa ame e s om e e ence [76] ha e been adop ed he ein, including a la ice ac i a ion ene gy Q=6.7 kJmol−1, di usion p e-exponen ial ac o Do=2×10−7m2s−1, hea ing a e φ=0.1 and la ice densi y NL=8.46 ×1028 a oms m−3, wi h α=β=1. The ini ial empe a u e and occupancy ac ion a e chosen as To=293 K and θ0 L=10−6, espec i ely, and he hickness o he specimen is chosen as L=5 mm. The a ia ion o ap binding ene gies and ap densi ies a e selec ed as he physically meaning ul anges −40 ≤∆Hi≤ −10 and 10−7≤Ni≤10−2. In he p esen s udy, we limi o he modelling o me als wi h wo hyd ogen aps, i.e. N =2. The e o e, in he su oga e modelling, empe a u e and he ap densi ies and binding ene gies a e conside ed as inpu a iables, which amoun s o 5 design a iables, i.e. M(x)=Jwi h x=hT,∆H1,∆H2,log(N1),log(N2)iT⊂R5. 19 Table 5: Accu acy and compu a ional e iciency analysis o PCK me amodels de eloped o he su oga e modelling o hyd o- gen di usion lux cu es ob ained by TDS (VS=36 000). ˆ M1ˆ M2ˆ M3ˆ M4 72 subdomains 72 subdomains 108 subdomains 108 subdomains ED =72 000 ED =108 000 ED =72 036 ED =108 000 e[ms] 1.97 4.86 0.83 2.02 NAAE 1.0637E-02 7.540E-03 1.115E-02 7.736E-03 NMAE 2.517E-05 2.239E-05 2.363E-05 1.656E-05 NRMSE 9.452E-04 5.259E-04 1.208E-03 5.229E-04 R20.9991 0.9995 0.9988 0.9995 A e some p elimina y sensi i i y analyses, wo pa i ions o Dxha e been conside ed, namely P1,P2. Pa - i ion P1has been de ined by spli ing he empe a u e Tand ac i a ion ene gy domains (∆Hi,i=1,2) in wo, while h ee segmen s we e conside ed o he pa i ion o he domain o he ap densi ies (log(Ni), i=1,2). On he o he hand, he pa i ions in P2 emain iden ical excep o he empe a u e domain which is di ided in h ee sub-domains. In o de o de ine he op imal su oga e model, EDs o 1000 and 1500 sampling poin s pe - subdomain ha e been conside ed o P1, while EDs o 667 and 1000 poin s pe sub-domain ha e been de ined o P2. This amoun s o ou di e en su oga e models labelled wi h ˆ Mi,i=1,...,4. In o de make a ai com- pa ison be ween he di e en p oposals, models ˆ M1and ˆ M3a e ained wi h EDs o 72 000 and 72 036 samples espec i ely, while ˆ M2and ˆ M4a e ained wi h 108 000 samples. The compa ison o he me amodels in e ms o accu acy and compu a ional e iciency is epo ed in Table 5 o e a VS o 36 000 samples. Simila ly o he e- sul s in he p e ious case s udy, he conside a ion o domain pa i ioning leads o conside able compu a ional ime educ ions and mode a e educ ions in p edic ion accu acy. No e ha he e alua ion ime o he o wa d model is abou 280 ms, so all he me amodels achie e educ ions be ween 98.3%-99.7%. The compu a ion ime o he me amodel depends upon he size o he ED in each egion, which explains why models ˆ M3and ˆ M2a e he as es and slowes ones, espec i ely. On he o he hand, he accu acy o he me amodel inc eases as so does he size o he ED. Indeed, models ˆ M2and ˆ M4exhibi signi ican ly lowe e o s compa ed o models ˆ M1and ˆ M3. The e- o e, in iew o hese esul s, ˆ M4p o ides a good ade-o be ween compu a ional e iciency and accu acy, and i is selec ed in he subsequen Bayesian model pa ame e in e ence. To illus a e he e ec i eness o he su oga e model in ep esen ing he di e en s ages obse ed in he TDS es , Fig. 7 shows he compa ison o he o wa d model and he p edic ions by ˆ M4 o a a ie y o combina ions o aps, including he case o luxes wi hou peak, one single peak, and wo peaks. I is obse ed ha he p oposed PCK model can accu a ely ep oduce all he di e en egimes obse able in he TDS es . Only some mino e o s a e obse ed in he no lux egime, gi en he imposed limi a ion on he o de o he polynomials in he PCE o he sake o compu a ional e iciency. Finally, in o de o highligh he supe io pe o mance o he p oposed mul i-elemen PCK me amodel, Fig. 6 u nishes he compa ison o he p edic ions by s anda d LAR-PCE ( ained wi h 76 000 samples) and ˆ M4. These esul s clea ly e idence he supe io pe o mance o he p oposed app oach wi h espec o LAR-PCE, whose p edic ions e sus he o wa d model exhibi s a la ge sca e a ound he diagonal line wi h a low coe icien o de e mina ion o R2=0.46. 20 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Fo wa d model × 10−4  Su oga e model  ×10−4  (a) LAR 76000 samples -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Fo wa d model × 10−4  Su oga e model ×10−4 (b) ˆ M4 R2=0.46 Figu e 6: Sca e plo s o Hyd ogen lux cu es ob ained by he o wa d solu ion o he PDF o he TDS es e sus he p edic ions by s anda d LAR-PCE (a) and by he p oposed mul i-elemen PCK me amodel ˆ M4(b) (VS o 36 000 samples). 400 600 800 1000 1200 0 1E-7 2E-7 3E-7 4E-7 Temp. T H. Flux J (a) (−14,−11,−5.5,−6.5) 400 600 800 1000 1200 0 1E-7 2E-7 3E-7 Temp. T H. Flux J (b) (−18,−15,−6,−6) 400 600 800 1000 1200 0 1E-7 2E-7 3E-7 Temp. T H. Flux J (c) (−25,−15,−6.9,−6) 400 600 800 1000 1200 0 1E-7 2E-7 3E-7 Temp. T H. Flux J (d) (−15,−15,−6.9,−6) ˆ M4 M 400 600 800 1000 1200 0 1.5E-6 3E-6 4.5E-6 6E-6 Temp. T H. Flux J (e) (−35,−30,−4,−6) 400 600 800 1000 1200 0 3E-5 6E-5 9E-5 12E-5 Temp. T H. Flux J ( ) (−35,−25,−2.5,−3) 400 600 800 1000 1200 0 1E-5 2E-5 3E-5 4E-5 5E-5 Temp. T H. Flux J (g) (−15,−30,−6,−3) 400 600 800 1000 1200 0 1E-5 2E-5 3E-5 4E-5 5E-5 Temp. T H. Flux J (h) (−11,−30,−6.9,−3) 400 600 800 1000 1200 0 1.5E-5 3E-5 4.5E-5 6E-5 Temp. T H. Flux J (i) (−35,−15,−3,−3) 400 600 800 1000 1200 0 0.5E-4 1E-4 1.5E-4 2E-4 Temp. T H. Flux J (j) (−18,−35,−2.5,−3) 400 600 800 1000 1200 0 1E-5 2E-5 3E-5 4E-5 Temp. T H. Flux J (k) (−35,−15,−3,−4) 400 600 800 1000 1200 0 1.8E-6 3.6E-6 5.4E-6 7.2E-6 9E-6 Temp. T H. Flux J (l) (−20,−30,−4,−4) 1 Figu e 7: Su oga e modelling o he Hyd ogen lux cu es ob ained by TDS o me als wi h di e en alues o ap binding ene gies and concen a ions. Quan i ies in pa en hesis ep esen he pa ame e s o he aps ∆H1,∆H1,log(N1),log(N2). 3.2.3. Bayesian in e ence o he apping si es om a TDS expe imen In his las subsec ion, he p e ious su oga e model ˆ M4is used o conduc Bayesian pa ame e in e ence ollowing he MCMC algo i hm in Sec ion 2.2. The ap binding ene gies and densi ies o he wo ap sys- 21 em a e chosen as he in e ence pa ame e s θ=∆H1,∆H2,log(N1),log(N2)in Eq. (20). Wi h he pu pose o assessing he pe o mance o he implemen ed DRAM MCMC app oach o in e he pa ame e s o hyd ogen aps co e ing he wo di e en egions desc ibed in e e ence [76], wo di e en ap con igu a ions a e consid- e ed o gene a e syn he ic expe imen al da a om he o wa d model. A wo- ap sys em (EI) wi h p ope ies θ=(−25,−35,−3,−2.5)is conside ed i s . The second one (EII) ins ead is de ined by θ=(−15,−30,−6,−3). The lux cu es ob ained in EI and EII co espond o hose p e iously shown in Figs. 7 ( ) and (g), espec i ely. In addi ion, o e alua e he sensi i i y o he model pa ame e in e ence o he p esence o noise pollu ion in he expe imen , a second analysis o he EII expe imen was pe o med a e a ec ing he lux cu e wi h a ze o-mean Gaussian whi e noise wi h a s anda d de ia ion equal o 0.4 imes he mean alue o he unpollu ed lux cu e (no e la e in Fig. 11 ha such a noise le el ep esen s a conside ably low signal- o-noise a io). The expe imen EI was de ined o illus a e he po en ials o he implemen ed DRAM MCMC o d aw samples om a mul i-modal dis ibu ion. No e ha he PDE in Eq. (31) does no di e en ia e he o de o he aps, he eby he p oblem is ill- posed and he pos e io dis ibu ion is expec ed o exhibi wo modes co esponding o wo symme ic solu ions. Ins ead, he expe imen EII was designed o accoun o a ap (∆H1=−15, log(N1)=−6) in he egime wi h no lux as iden i ied by Raina and e al. [76], while he second ap (∆H2=−30, log(N2)=−3) ep esen s a deep ap. The e o e, he PDF in his case should be uni-modal. In he in e ence analyses, unin o ma i e uni o m p io s U(−40,−10) and U(−7,−2) a e selec ed o ∆Hiand log(Ni) (i=1,2), espec i ely. A o al numbe o 200 000 samples wi h a bu ning ime o 50 000 samples we e d awn by he p e iously in oduced Bayesian in e ence app oach o EI. The sampling o he pos e io PDF in Expe imen EII was mo e challenging gi en i s uni-modal na u e wi h la ge egions o low p obabili y, equi ing up o 480 000 samples wi h a bu ning pe iod o 160 000 samples o achie e con e gence. In e es ingly, his phe- nomenon a enua es when he lux cu e is a ec ed by noise, only equi ing a chain o 120 000 samples wi h a bu ning pe iod o 30 000 o a ain con e gence. This is expec able since he noise-induced lowe p obabili y con- cen a ion a ound he exac ue solu ions makes i easie o he chain o span om one solu ion o he symme ic one. The ini ial loca ion s a e was de ined as θ0=(−25,−25,−4.5,−4.5), while he p edic ion e o was se o σε=1E−9 and 1.6E−5 o he noise unpollu ed and pollu ed cases, espec i ely. A e some ini ial calib a ion by isual inspec ion o he chain aces, a diagonal co a iance ma ix wi h en ies equal (0.05 ·θ0)2was ini ially de ined o he Gaussian p oposal. In he AM s ep he p oposal dis ibu ion was scaled by a ac o sd=2.42/dand he non-adap a ion pe iod n0was se o 500 and 4000 o he EI and EII expe imen s, espec i ely. On he o he hand, in he DR s ep he p oposal is scaled down by a ac o o 0.2. The Ma ko chain and he join pos e io PDF ob ained o Expe imen EI a e p esen ed in Figs. 8 and 9, espec i ely. As an icipa ed, he p oblem is ill-posed and he e exis wo po en ial solu ions, namely θ= (−25,−35,−3,−2.5)and θ=(−35,−25,−2.5,−3). This mani es s in he ma ginal PDFs in Fig. 9. Speci ically, he PDFs co esponding o pa ame e s ∆H1and ∆H2ha e wo iden ical modes a −35 and −25, and pa ame e s log(N1) and log(N2) ha e wo modes a −3 and −2.5. I is obse ed in Fig. 9 ha , indeed, he implemen ed DRAM algo i hm is capable o explo ing he wo modes in he dis ibu ion, wi hou ge ing s uck a ound one o hem as i is usually he case when implemen ing s anda d MCMC me hods. Fo alida ion pu poses, he pos e io PDF has been also compu ed by di ec in eg a ion o he o wa d solu ion. To do so, he e idence o he model has been compu ed o e a mesh o 604elemen s. This equi ed o y i e hou s o pa allel compu a ion on en co es, while 22 HDR ∆H1∆H2log(N1) log(N2) 80% HDR (MCMC) (-38.004,-33.084) ∪ (-28.931,-22.606) (-37.833,-32.686) ∪ (-28.628,-23.134) (-3.075,-2.418) (-3.059,-2.736) ∪ (-2.683,-2.409) 80% HDR (Analy ical) (-38.515,-32.253) ∪ (-29.121,-22.729) (-38.179,-32.247) ∪ (-29.184,-22.991) (-3.069,-2.386) (-3.099,-2.389) 50% HDR (MCMC) (-36.726,-33.659) ∪ (-27.398,-24.842) (-36.546,-33.280) ∪ (-27.143,-25.164) (-2.925,-2.869) ∪ (-2.653,-2.427) (-2.898,-2.868) ∪ (-2.637,-2.416) 50% HDR (Analy ical) (-36.884,-32.840) ∪ (-27.295,-24.816) (-36.680,-32.899) ∪ (-27.424,-25.208) (-2.702,-2.406) (-2.703,-2.399) Table 6: HDR a 80% and 50% o he PDFs ob ained by di ec in eg a ion and by MCMC o expe imen EI. he MCMC app oach only equi ed abou ou hou s on a single co e. The Highes Densi y Regions (HDRs) a he 80% and 50% le el o bo h dis ibu ions a e epo ed in Table 6. The close i ings be ween he exac ma ginal PDFs and hose p edic ed by he su oga e model-based Bayesian in e ence in Fig. 9 demons a e he accu acy o he de eloped app oach, as i is also e iden om he compu ed HDRs in Table 6. Finally, he Ma ko chain, and he pos e io PDF ob ained o he TDS expe imen EII a e epo ed in Figs. 10 and 11, espec i ely, and he pos e io HDR alues a e epo ed in Table 7. In his case, he PDFs exhibi one single mode as p e iously an icipa ed. This co esponds o he shallow ap (∆H1=−30, log(N1)=−3), while he ap in he no- lux egime goes unno iced. F om a Bayesian pe spec i e, his ep esen s an obse abili y limi a ion o he expe imen , being he model o one single ap mo e likely o ep esen he ma e ial gi en he expe imen al e idence. Fu he mo e, i is no ed ha he p esence o measu emen noise does no subs an ially al e he in e ence ou come. The modes o he pos e io s o he ap densi ies pa ame e s ∆H1and ∆H2o he noise- ee scena io a e −29.966 and −30.002, while o he noisy scena io he alues −29.989 and −29.961 a e ob ained, which ep esen s a di e ence o 0.077% and 0.137%, espec i ely. On he o he hand, o pa ame e s log(N1) and log(N2) he modes o he pos e io s in he noise- ee case a e −3.009 and −3.016, whe eas in he noisy scena io hey ake alues −3.017 and −3.019, meaning a di e ence o 0.266% and 0.010%, espec i ely. This con i ms ha he p oposed app oach is obus o he p esence o measu emen noise. O e all, hese esul s illus a e he po en ial o he de eloped app oach o model selec ion and in o ma ion gain analysis o TDS esul s, which a e le o u u e de elopmen s. 23 0 0.2 0.40.60.811.2 1.41.61.8 2 −40 −35 −30 −25 −20 −15 −10 Sample No. ×105 ∆H1 0 0.2 0.40.60.811.2 1.41.61.8 2 −40 −35 −30 −25 −20 −15 −10 Sample No. ×105 ∆H2 0 0.2 0.40.60.811.2 1.41.61.8 2 −2 −3 −4 −5 −6 −7 Sample No. × 105  log  N1  0 0.2 0.40.60.811.2 1.41.61.8 2 −2 −3 −4 −5 −6 −7 Sample No. × 105  log N2 (a) (b) (c) (d) Bu ning pe iod Figu e 8: Ma ko chains gene a ed by DRAM MCMC o ap pa ame e s ∆H1,∆H2, log(N1), log(N2) o TDS Expe imen EI. 24 HDR ∆H1∆H2log(N1) log(N2) 80% HDR (noise- ee da a) (-39.872, -22.145) (-34.206,-25.128) (-5.887,-2.927) (-5.346,-2.920) 80% HDR (noisy da a) (-39.970,-26.221) (-39.314,-24.378) (-5.410,-2.900) (-5.733,-2.889) 50% HDR (noise- ee da a) (-31.526,-28.406) (-30.841,-29.199) (-3.886,-2.924) (-3.088,-2.946) 50% HDR (noisy da a) (-31.534,-28.698) (-31.576,-28.656) (-3.307,-2.896) (-3.286,-2.883) Table 7: HDR a 80% and 50% o he PDFs ob ained om noisy and noise- ee da a by MCMC o expe imen EII. −40 −30 −20 −10 ∆H2 -7 -6 -5 -4 -3 -2 log(N1) -40 -30 -20 -10 -7 -6 -5 -4 -3 -2 ∆H1 log(N2) -40 -30 -20 -10 ∆H2 -7 -6 -5 -4 -3 -2 log(N1) PDF MCMC Analy ic -7 -6 -5 -4 -3 -2 log(N2) 1 -15 0.02 0.04 0.06 0.08 0.10 0.12 -20 -25 -30 -35 ΔH2 π (ΔH1,ΔH2) ΔH1 -35 -30 -25 -20 -15 Figu e 9: Bayesian iden i ica ion esul s o he ap pa ame e s θ=∆H1,∆H2,log(N1),log(N2)o TDS Expe imen EI. The su ace plo in he op igh co ne co esponds o he ma ginal PDF o e ∆H1,∆H2ob ained by nume ical in eg a ion. 25 [47] F. Schneide , I. Papaioannou, D. S aub, C. Win e , G. M¨ ulle , Bayesian pa ame e upda ing in linea s uc- u al dynamics wi h equency ans o med da a using a ional su oga e models, Mech. Sys . Signal P ocess. 166 (2022) 108407. [48] W. W. Xing, A. A. Shah, P. Wang, S. Zhe, Q. Fu, R. M. Ki by, Residual Gaussian p ocess: A ac able nonpa ame ic Bayesian emula o o mul i- ideli y simula ions, Appl. Ma h. Model. 97 (2021) 36–56. [49] L. Ie imon i, N. Ca alagli, I. Venanzi, E. Ga c´ ıa-Mac´ ıas, F. Ube ini, A ans e bayesian lea ning me hod- ology o s uc u al heal h moni o ing o monumen al s uc u es, Eng. S uc . 247 (2021) 113089. [50] A. del Val, O. P. Le Maˆ ı e, T. E. Magin, O. Chazo , P. M. Congedo, A su oga e-based op imal likelihood unc ion o he bayesian calib a ion o ca aly ic ecombina ion in a mosphe ic en y p o ec ion ma e ials, Appl. Ma h. Model. 101 (2022) 791–810. [51] G. Bla man, B. Sud e , Adap i e spa se polynomial chaos expansion based on leas angle eg ession, J. Compu . Phys. 230 (6) (2011) 2345–2367. [52] B. E on, T. Has ie, I. Johns one, R. Tibshi ani, Leas angle eg ession, Ann. S a . 32 (2) (2004) 407–499. [53] H. Haa io, M. Laine, A. Mi a, E. Saksman, DRAM: e icien adap i e MCMC, S a . Compu . 16 (4) (2006) 339–354. [54] K. K. Phoon, J. Ching, Risk and eliabili y in geo echnical enginee ing, Boca Ra on, FL, USA: CRC P ess., 2015. [55] J. Sacks, W. J. Welch, T. J. Mi chell, H. P. Wynn, Design and analysis o compu e expe imen s, S a . Sci. 4 (4) (1989) 409–423. [56] C. Rasmussen, C. Williams, Gaussian P ocesses o Machine Lea ning, MIT P ess, 2006. [57] H. Cham, E. Reshe nyak, B. Rosen eld, W. B ei ba , Full in o ma ion maximum likelihood es ima ion o la en a iable in e ac ions wi h incomple e indica o s, Mul i a . Beha . Res. 52 (1) (2008) 12–30. [58] J. Kle e, J. N. K. Rao, Es ima ion o mean squa e e o o empi ical bes linea unbiased p edic o s unde a andom e o a iance linea model, J. Mul i a . Anal. 43 (1) (1992) 1–15. [59] G. Pelleg ini, Polynomial chaos expansion wi h applica ions o pdes, Ph. D. hesis, Uni e si y o Ve ona, 2014 (2014). [60] G. K. W. Xiaoliang, Mul i-elemen gene alized polynomial chaos o a bi a y p obabili y measu es, SIAM J. Sci. Compu . 28 (3) (2006) 901–928. [61] M. Mous apha, J. M. Bou ine , B. Guillaume, B. Sud e , Compa a i e s udy o K iging and suppo ec o eg ession o s uc u al enginee ing applica ions, ASCE-ASME J. Risk Unce ain. Eng. Sys . A: Ci . Eng. 4 (2) (2018) 04018005. [62] K. Ri e , A e age-Case Analysis o Nume ical P oblems, Sp inge , 2000. 32 [63] W. Wang, R. T., J. W., On p edic ion p ope ies o k iging: Uni o m e o bounds and obus ness, J. Am. S a . Assoc. 115 (530) (2020) 920–930. [64] A. Mi a, On Me opolis-Has ings algo i hms wi h delayed ejec ion, Me on 59 (3-4) (2001) 231–241. [65] H. Haa io, E. Saksman, J. Tamminen, An adap i e Me opolis algo i hm, Be noulli 7 (2) (2001) 223–242. [66] D. C. S enning, D. A. an Dyk, Bayesian S a is ical Me hods Fo As onomy Pa II: Ma ko Chain Mon e Ca lo, STATS As ophysics. EDP Sciences, 2021. [67] R. Salomon, Re-e alua ing gene ic algo i hm pe o mance unde coo dina e o a ion o benchma k unc ions. a su ey o some heo e ical and p ac ical aspec s o gene ic algo i hms, Biosys ems 39 (3) (1996) 263–278. [68] S. Ma elli, B. Sud e , Uqlab: A amewo k o unce ain y quan i ica ion in ma lab, The 2nd In e na ional Con e ence on Vulne abili y and Risk Analysis and Managemen (2014) 2554–2563. [69] R. P. Ganglo , Hyd ogen-assis ed C acking, in: I. Milne, R. Ri chie, B. Ka ihaloo (Eds.), Comp ehensi e S uc u al In eg i y Vol. 6, Else ie Science, New Yo k, NY, 2003, pp. 31–101. [70] S. K. Dwi edi, M. Vishwaka ma, Hyd ogen emb i lemen in di e en ma e ials: a e iew, In . J. Hyd og. Ene gy. 43 (46) (2018) 21603–21616. [71] W. H. Johnson, On Some Rema kable Changes P oduced in I on and S eel by he Ac ion o Hyd ogen and Acids, P oc. R. Soc. Lond. 23 (1875) 168–179. [72] R. Fe n´ andez-Sousa, C. Be eg´ on, E. Ma ´ ınez-Pa˜ neda, Analysis o he in luence o mic os uc u al aps on hyd ogen assis ed a igue, Ac a Ma e ialia 199 (2020) 253–263. [73] M. Is andbod, E. Ma ´ ınez-Pa˜ neda, A mechanism-based mul i- ap phase ield model o hyd ogen assis ed ac u e, In e na ional Jou nal o Plas ici y 144 (2021) 103044. [74] A. Za a, Z. Ha is, C. Sun, E. Ma inez-Pa˜ neda, Compa ison o hyd ogen di usi i ies measu ed by elec o- chemical pe mea ion and empe a u e-p og ammed deso p ion in cold- olled pu e i on, J. Na . Gas Sci. Eng. 98 (2022) 104365. [75] F. J. Cas o, G. Meye , The mal deso p ion spec oscopy (TDS) me hod o hyd ogen deso p ion cha ac e i- za ion (I): heo e ical aspec s, J. Alloys Compd. 330 (2002) 59–63. [76] A. Raina, V. S. Deshpande, N. A. Fleck, Analysis o he mal deso p ion o hyd ogen in me allic alloys, Ac a Ma e . 144 (2018) 777–785. [77] A. McNabb, P. K. Fos e , A new analysis o di usion o hyd ogen in i on and e i e, T ans. o he Me allic Soc. 227 (3) (1963) 618–627. [78] R. A. O iani, The di usion and apping o hyd ogen in s eel, Ac a Me all. 18 (1) (1970) 147–157. 33