TN-01_ReportSuitabilityPotentialRomFusionANonIntrusiveRomSolversHighDimensionalOutputs
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Report,on,suitability,and,potential,of,ROM,to,fusion,models,A,Non-intrusive,ROM,for,Solvers,with,High-dimensional,Outputs,Deyu,Ming,and,Serge,Guillas,University,College,London,Progress,report,with,preliminary,findings,UKAEA,Report:,2047352,2-TN-01,D1.1,April,27,,2021,1,Introduction,Many,modern,physical,computer,models,involve,solving,PDEs,with,numerical,solvers,,such,as,finite,element,methods,(FEM),,which,can,be,computationally,expensive,due,to,•,ever,more,complex,and,larger-scale,models;,•,high-dimensional,input,and,output;,•,large,demands,on,computational,resources.,These,create,challenges,to,efficient,uncertainty,quantification,of,computer,models,,such,as,the,fusion,models,,as,we,often,need,to,run,the,models,in,a,large,number,of,times,for,tasks,such,as,sensitivity,analysis,,uncertainty,propagation,and,model,calibration.,To,tackle,these,challenges,,reduced,order,models,(ROM),are,needed,to,•,serve,as,low-dimensional,replacements,with,comparable,accuracy;,•,reduce,evaluation,time,of,original,solvers;,•,save,storage,,e.g.,,for,high-dimensional,output.,Traditional,reduced,order,models,,also,known,as,intrusive,reduced,order,models,,often,are,constructed,using,reduced,basis,methods,[16],,among,which,the,Proper,Orthogonal,Decomposition,(POD),is,perhaps,the,most,popular,technique.,The,intrusive,reduced,order,models,for,original,high-fidelity,models,with,high-dimensional,output,are,typically,built,using,a,two-phase,procedure,called,offline-online,decomposi-,tion:,•,offline,phase:,high-fidelity,solutions/outputs,are,obtained,and,reduced,basis,is,calculated;,•,online,phase:,the,original,problems,are,projected,onto,the,reduced,space,for,efficient,computation,of,solutions,at,new,inputs.,However,,the,online,phase,of,the,intrusive,reduced,order,modelling,is,challenging,in,practice,because:,•,expertise,and,domain,knowledge,are,required,to,project,the,equations,and,physics,of,the,original,high-fidelity,problems,to,constructed,reduced,space;,•,dimensionality,reduction,techniques,are,largely,constrained,by,the,problem,formulation;,•,uncertainty,is,not,incorporated.,For,these,reasons,,in,this,report,we,focus,on,non-intrusive,reduced,order,models,for,problems,with,high,dimensional,outputs,,utilising,the,family,of,Gaussian,process,(GP),surrogates.,These,have,been,successfully,implemented,for,dimension,reduction,of,either,outputs,or,inputs.,For,instance:,•,[9],used,Functional,Principal,Components,Analysis,(FPCA),as,an,equivalent,approach,to,POD,for,time,series,outputs,of,tsunami,waves,,and,[2],used,Spherical,Harmonics,and,Gaussian,Markov,Random,Fields,for,optimal,reduction,of,surfaces,outputs.,•,For,inputs,,[13],employed,a,kernel-based,approach,to,extract,the,few,input,field,directions,of,most,influence,for,the,outputs,in,order,to,build,GPs,with,few,input,dimensions,(orders,of,magnitude,gain,in,dimension).,The,report,is,organised,as,follows.,In,Section,2,,a,non-intrusive,ROM,with,GP,surrogates,and,POD,is,described.,The,method,is,then,applied,in,Section,3,for,a,anisotropic,heat,transport,problem.,Future,directions,are,discussed,in,Section,4.,1,2,Non-intrusive,ROM,with,Gaussian,Process,Surrogates,The,non-intrusive,reduced,order,modelling,is,a,data-driven,approach,that,uses,a,statistical,surrogate,model,to,mimic,the,functional,relations,between,the,model,input,and,constructed,reduced,output,space,in,the,online,phase,of,the,offline-online,decomposition.,The,utilisation,of,statistical,surrogates,alleviates,the,difficulties,involved,in,reformulating,the,original,high-fidelity,problems,under,the,intrusive,reduced,In,particular,,with,GP,surrogates,we,are,able,to,quantify,uncertainty,of,the,high-,order,modelling.,dimensional,outputs,predicted,at,unobserved,input,positions.,Let,X,∈,RN,×D,contain,N,sets,of,D,dimensional,input,of,a,computer,model,,which,produces,N,corresponding,sets,of,K,dimensional,output,Y,∈,RN,×K,accordingly.,Then,,one,can,mimic,the,functional,relationships,between,the,input,X,and,each,output,dimension,Yk,∈,RN,×1,by,a,GP,surrogate,GP,k,independently,for,k,=,1,,.,.,.,,,K,without,considering,the,dependence,between,output,dimensions,[6].,Ignoring,the,potential,cross-dependence,does,not,pose,a,serious,issue,unless,we,are,interested,in,the,joint,distribution,of,the,output,,and,it,can,be,shown,[11],that,the,independently,constructed,GP,surrogates,correspond,to,the,marginal,GPs,of,a,joint,GP,surrogate,under,certain,dependence,structures.,The,GP,surrogate,GP,k,is,formally,defined,as,a,multivariate,normal,distribution,with,respect,to,Yk:,Yk,∼,N,(µk(X),,σ2,kRk(X)),,in,which,the,i-th,element,of,µk(X),∈,RN,×1,is,often,specified,by,a,trend,function,fk(Xi),with,Xi,∈,R1×D,being,the,i-th,row,of,X,,and,the,ij-th,element,of,Rk(X),∈,RN,×N,is,given,by,ck(Xi,,Xj),,where,ck,is,a,given,kernel,function.,The,trend,function,fk,can,be,formulated,as,a,linear,combination,of,a,set,of,basis,functions,of,Xi,and,we,assume,a,constant,trend,function,fk(Xi),=,bk,in,this,report.,There,are,various,choices,for,ck,(see,[17]).,In,this,report,,we,use,the,separable,kernel,function:,ck(Xi,,Xj),=,D,(cid:89),d=1,ck,d(Xid,,Xjd),,where,ck,d,is,a,one-dimensional,kernel,function.,A,typical,choice,for,ck,d,in,computer,model,emulation,is,the,squared,exponential,(SExp),kernel:,ck,d(Xid,,Xjd),=,exp,−,(cid:40),(Xid,−,Xjd)2,γ2,k,d,(cid:41),,,where,γk,d,>,0,is,the,range,parameter.,However,,the,SExp,kernel,has,been,criticised,for,its,over-,smoothness,[20],for,physical,problems,as,well,as,its,associated,ill-conditioned,problems,[3,,8].,Another,popular,kernel,choice,is,the,Mat´ern,kernel,[17]:,ck,d(Xid,,Xjd),=,exp,−,√,(cid:18),2p,+,1,rij,d,γk,d,(cid:19),p!,(2p)!,p,(cid:88),i=0,(p,+,i)!,i!(p,−,i)!,√,(cid:18),2rij,d,2p,+,1,(cid:19)p−i,γk,d,,,where,rij,d,=,Xid,−,Xjd.,The,Mat´ern,kernel,is,known,to,be,less,prone,to,ill-conditioning,issues,and,provides,a,reasonably,adequate,smoothness,to,the,GP,surrogates.,In,particular,,the,Mat´ern-2.5,kernel,,which,is,defined,as,the,Mat´ern,kernel,with,p,=,2:,(cid:32),√,ck,d(Xid,,Xjd),=,1,+,5|Xid,−,Xjd|,γk,d,+,5(Xid,−,Xjd)2,3γ2,k,d,(cid:33),(cid:40),√,exp,−,5|Xid,−,Xjd|,γk,d,(cid:41),,,is,the,default,kernel,choice,for,many,computer,model,emulation,packages,,such,as,DiceKriging,[19],and,RobustGaSP,[7].,Therefore,,we,employ,the,Mat´ern-2.5,kernel,in,this,report.,The,posterior,predictive,distribution,N,((cid:98)µk(x∗),,(cid:98)σ2,k,(x∗),at,an,unobserved,input,position,x∗,is,given,in,different,analytical,forms,depending,on,how,the,model,parameters,bk,,σ2,k,and,{γk,d}d=1,...,D,are,estimated.,Different,maximum-likelihood-based,estimation,approaches,and,the,corresponding,expressions,for,(cid:98)µk(x∗),and,(cid:98)σ2,k(x∗)),of,GP,k,with,respect,to,the,output,Y,∗,k(x∗),are,discussed,in,[19,,8].,The,main,computational,bottlenecks,of,the,GP,surrogate,construction,are,the,number,of,data,points,N,and,the,dimension,K,of,the,output,of,a,computer,model.,Since,the,inference,of,GP,surrogates,involve,inversions,of,the,correlation,matrix,Rk,∈,RN,×N,with,computational,complexity,of,O(N,3),,it,soon,becomes,computationally,prohibitive,to,build,GP,surrogates,in,practice,when,N,is,more,than,several,thousands.,In,such,a,case,,one,may,need,sparse,approximations,[12],to,the,GP,to,reduce,the,computational,complexity,induced,by,the,big,data.,2,In,computer,model,experiments,,one,often,does,not,have,big,data,(i.e.,,realisations,from,the,underlying,computer,model),due,to,the,limited,computational,budget.,However,,if,the,input,dimension,D,is,large,,then,small,data,are,insufficient,to,explore,adequately,the,whole,input,domain,and,thus,the,resulting,GP,surrogates,can,be,inaccurate.,High,input,dimension,also,causes,challenges,to,the,model,estimation,because,a,large,number,of,range,parameters,{γk,d}d=1,...,D,need,to,be,estimated,for,each,output,dimension.,To,alleviate,this,issue,,one,can,reduce,the,input,dimension,D,to,P,such,that,P,(cid:28),D,by,dimension,reduction,techniques,such,as,POD,,kernel,dimension,reduction,[13],,and,active,subspace,[21].,A,high,output,dimension,K,creates,the,issue,that,it,can,be,computational,burdensome,to,build,K,independent,GP,surrogates:,without,parallel,implementation,the,training,and,validation,of,a,huge,amount,of,GP,surrogates,are,practically,infeasible.,This,report,tackles,the,latter,issue,on,high-dimensional,outputs,(e.g.,,a,snapshot,where,each,point,on,the,snapshot,represents,a,FE,solution,and,contributes,to,the,output,dimensionality),produced,by,computer,models.,Perhaps,the,most,straightforward,approach,to,address,the,issue,is,to,reduce,the,output,dimension,K,to,L,such,that,L,(cid:28),K,by,POD.,The,POD,of,Y,∈,RN,×K,can,be,done,with,following,steps:,1.,Compute,the,sample,mean,µY,∈,R1×K,of,Y,and,obtain,the,centred,output,matrix,Yc,=,Y,−,µY;,2.,Implement,the,eigendecomposition,of,G,=,1,c,such,that,G,=,VΛV(cid:62),,where,the,columns,of,N,YcY(cid:62),V,∈,RN,×N,contains,the,eigenvectors,of,G,and,the,diagonal,of,Λ,∈,RN,×N,contains,the,correspond-,ing,eigenvalues,(λ1,,.,.,.,,,λN,),in,descending,order;,3.,Compute,˜V,=,Y(cid:62),c,Yc;,N,Y(cid:62),1,c,V,∈,RK×N,,,which,contains,the,eigenvectors,of,sample,covariance,matrix,C,=,4.,Choose,L,≤,N,and,obtain,the,low,dimensional,output,(cid:98)Y,=,Yc,˜VL,∈,RN,×L,,where,˜VL,∈,RK×L,contains,the,first,L,eigenvectors,included,in,˜V.,One,can,also,obtain,˜V,by,performing,the,singular,value,decomposition,(SVD),of,Yc,that,is,implemented,,e.g.,,in,the,PCA,function,of,Python,package,scikit-learn,[15].,After,obtaining,the,low,dimensional,data,(cid:98)Y,,we,then,construct,L,independent,GP,surrogates,of,each,of,L,dimensions,of,(cid:98)Y.,Let,N,((cid:98)µl(x∗),,(cid:98)σ2,l,(x∗)),be,the,posterior,predictive,distribution,of,(cid:98)Y,∗,l,(x∗),,the,l-th,dimension,of,the,low,dimensional,output,,pre-,dicted,at,an,unobserved,input,position,x∗.,Then,the,posterior,predictive,distribution,of,the,corresponding,high,dimensional,output,Y∗(x∗),∈,R1×K,is,given,by,(cid:16),N,(cid:98)µ(x∗),˜V(cid:62),L,+,µY,,˜VL,(cid:98)Σ(x∗),˜V(cid:62),L,(cid:17),,,where,(cid:98)µ(x∗),=,((cid:98)µ1(x∗),,.,.,.,,,(cid:98)µL(x∗)),and,(cid:98)Σ(x∗),=,diag((cid:98)σ2,1(x∗),,.,.,.,,,(cid:98)σ2,L(x∗)).,Figure,1,demonstrates,the,procedure,to,build,non-intrusive,reduced,order,model,with,GP,surro-,gates.,In,the,offline,phase,,dimension-reduction,techniques,,e.g.,,POD,,are,applied,to,reduce,the,high-,dimensional,output,to,a,low-dimensional,space.,Then,in,the,online,phase,,GP,surrogates,are,constructed,independently,on,each,reduced,dimension.,Using,the,constructed,GP,surrogate,and,reduced,basis,,one,can,obtain,the,predicted,low-dimensional,and,in,turn,the,high-dimensional,output,at,new,input,positions,with,little,computational,efforts.,Input,Solver,High-dim,Output,New,Input,GP,Surrogate,Low-dim,Output,Figure,1:,The,workflow,to,construct,non-intrusive,ROM,with,GP.,The,black,arrows,represent,the,offline,phase;,the,blue,arrows,represent,the,online,phase;,the,red,arrows,represent,the,prediction,procedure,using,the,constructed,non-intrusive,ROM,with,GP.,3,3,2-D,model,of,anisotropic,heat,transport,In,this,section,,we,explore,the,non-intrusive,ROM,with,GP,to,mimic,the,FE,solver,to,the,2-D,problem,“Open,field,lines,with,oscillating,anisotropy,directions”,in,[5].,The,problem,has,two,key,inputs,m,and,α,that,control,the,anisotropy,of,the,solution,field,,i.e.,,the,anisotropy,direction,is,defined,by,b,=,B,|B|,,,B,=,(cid:18)α(2y,−,1),cos(mπx),+,π,παm(y2,−,y),sin(mπx),(cid:19),,,where,m/2,is,the,number,of,oscillation,periods,in,the,computational,domain,and,α,is,the,amplitude.,The,output,is,a,high-dimensional,2-D,field,defined,on,the,square,computational,domain,[0,,1],×,[0,,1],and,allows,a,closed,form,solution:,.,3.1,Experimental,Design,To,construct,the,reduced,basis,via,the,POD,and,the,GP,surrogate,,N=40,samples,are,arranged,in,a,Latin,hypercube,over,m,∈,[0,,12],and,α,∈,[0,,3],(see,the,left,plot,in,Figure,2).,We,then,run,the,FE,solver,(implemented,in,Firedrake,[18]),of,the,toy,problem,to,obtain,the,corresponding,2-D,outputs,,each,of,which,contains,FE,solutions,on,K,=,78961,nodes.,These,40,×,78961,high-dimensional,outputs,are,then,reduced,to,40,low-dimensional,outputs,(40×25),using,POD,by,retaining,the,first,25,principal,components,out,of,the,total,40,components,,see,the,right,plot,in,Figure,2,,where,the,cumulative,explained,variance,is,defined,as,with,L,be,the,number,of,components.,(cid:80)L,(cid:80)N,i=1,λi,i=1,λi,Figure,2:,(Left):,Training,and,designing,points,generated,for,the,inputs,m,and,α.,The,blue,points,are,design,input,locations,generated,from,the,Latin,hypercube,design,and,the,red,points,are,testing,input,locations;,(Right):,cumulative,explained,variance,given,by,the,POD.,GP,surrogates,are,then,constructed,independently,for,each,of,the,25,dimensions,of,the,reduced,order,data.,GP,surrogates,are,trained,with,the,Mat´ern-2.5,kernel,using,the,RobustGaSP,package,in,R.,3.2,Experimental,Results,We,test,the,constructed,non-intrusive,ROM,at,four,testing,input,positions,(m,,α),=,(6,,2),,(10,,2),,(1,,2),and,(10,,0),(see,the,left,plot,of,Figure,2).,The,FE,solutions,(from,the,Firedrake),and,the,predicted,solutions,from,the,built,ROM,are,compared,in,Figure,3.,The,normalised,(to,the,range,of,FE,solutions),errors,between,the,FE,solutions,and,the,predicted,solutions,from,the,built,ROM,are,shown,in,Figure,4.,The,coverage,of,the,ROM,(i.e.,,the,instances,that,the,FE,solutions,fall,within,the,predictive,bounds,of,GP-based,ROM),are,also,given,in,Figure,5.,It,can,been,seen,from,these,results,that,the,constructed,ROM,with,GP,could,predict,well,the,FE,solutions,of,the,anisotropic,problem,at,input,locations,that,are,not,realised.,Among,the,four,testing,positions,,the,final,case,with,m,=,10,and,α,=,0,presents,the,largest,normalised,errors,up,to,13%.,This,is,not,a,surprising,result,because,m,has,no,effect,on,the,FE,solution,of,the,problem,when,α,=,0.,However,,this,information,is,not,fully,captured,in,the,training,data,and,thus,not,gained,by,the,non-intrusive,ROM,with,GP,,which,is,pure,data-driven,method,that,only,understands,the,functional,relation,between,m,,α,and,the,solution,field,from,the,training,set.,As,a,result,,we,could,observe,5,blurred,oscillation,periods,in,the,predicted,solutions,from,ROM,in,Figure,3.,However,,the,predictive,interval,(whose,upper,and,lower,bounds,are,given,at,two,standard,deviations,2(cid:98)σ,above,and,below,the,predictive,mean,(cid:98)µ),of,the,4,Figure,3:,Comparisons,of,FE,solutions,to,the,predicted,solutions,given,by,the,constructed,GP-based,ROM.,The,first,row,gives,the,FE,solutions.,The,second,row,gives,the,predicted,solutions,from,the,GP-,based,ROM.,The,columns,from,left,to,right,correspond,to,testing,input,positions,(m,,α),=,(6,,2),,(10,,2),,(1,,2),and,(10,,0),respectively.,Figure,4:,The,normalised,errors,between,FE,solutions,and,the,predicted,solutions,from,the,ROM,with,GP,surrogate.,The,plots,from,left,to,right,correspond,to,testing,input,positions,(m,,α),=,(6,,2),,(10,,2),,(1,,2),and,(10,,0),respectively.,Figure,5:,The,coverage,of,constructed,ROM,with,GP,,giving,the,instances,that,FE,solutions,fall,within,the,predictive,bounds,provided,by,the,ROM,with,GP.,1,indicates,that,the,FE,solution,is,covered,by,the,predictive,interval,(whose,upper,and,lower,bounds,are,given,at,two,standard,deviations,2(cid:98)σ,above,and,below,the,predictive,mean,(cid:98)µ),and,0,indicates,otherwise.,The,plots,from,left,to,right,correspond,to,testing,input,positions,(m,,α),=,(6,,2),,(10,,2),,(1,,2),and,(10,,0),respectively.,5,GP-based,ROM,covers,the,FE,solutions,sufficiently,in,this,case,,demonstrating,that,one,can,benefit,from,the,predictive,uncertainty,embedded,in,the,non-intrusive,ROM,coupled,with,GP,emulation.,4,Future,Directions,We,demonstrate,in,this,report,that,a,non-intrusive,ROM,with,GP,surrogate,could,be,used,to,replace,computationally,expensive,computer,solvers,for,problems,with,high-dimensional,output,,in,one,of,the,building,blocks,of,nuclear,fusion,modelling.,However,,the,predictive,performance,of,GP-based,ROM,relies,on,the,information,contained,in,the,training,data,,i.e.,,the,quality,of,computer,experimental,design.,We,use,Latin,hypercube,sampling,(LHS),in,this,experiment,,but,it,is,worth,exploring,the,benefits,provided,by,the,sequential,designs,[1],,especially,in,cases,where,FE,solutions,are,changing,rapidly,in,small,regions,of,the,input,space.,The,rapid,changes,of,FE,solution,field,also,indicates,that,the,GP,surrogate,should,incorporate,the,non-stationary,features,,giving,rise,to,the,more,advanced,Gaussian,process,models,with,deep,hierarchies,[4].,Furthermore,,dimension,reduction,techniques,such,as,POD,lose,information,when,the,original,data,are,projected,onto,a,lower,dimensional,space,,and,thus,some,extreme,but,important,events,could,be,masked,in,the,low,dimensional,data,,a,scenario,called,masking,effect.,As,a,result,,if,the,GP,surrogate,is,built,on,the,low,dimensional,data,one,may,not,be,able,to,recover,these,outlying,events,using,the,constructed,non-intrusive,ROM.,Although,the,non-intrusive,ROM,requires,no,domain,knowledge,and,access,to,the,source,code,of,original,problems,,it,ignores,the,physics,implied,by,the,underlying,problem,and,thus,may,be,inaccurate,comparing,to,the,its,intrusive,counter-party.,Therefore,,it,would,be,worth,exploring,the,trade-off,between,the,speed,and,accuracy,of,intrusive,and,non-intrusive,MOR,,especially,in,context,of,UQ.,It,would,also,be,interesting,to,find,a,middle,ground,where,one,could,exploit,the,benefits,(e.g.,,accuracy,,speed,and,uncertainty),of,both,intrusive,and,non-intrusive,ROM,,producing,a,physics-informed,non-intrusive,ROM.,Some,relevant,literature,on,physics-informed,machine,learning,(say,using,a,boundary,condition,or,other,approaches),include,[22,,10,,23].,Layer,1,Layer,2,x1,x2,f1,f2,w1,w,2,f3,y,Figure,6:,An,illustrative,example,of,a,system,of,three,computer,models,f1,,f2,and,f3.,Note,this,is,only,for,illustration.,Linked,GP,in,[14],can,work,on,any,feed-forward,computer,systems.,Since,fusion,models,are,often,multi-disciplinary,and,multi-physics,,the,recent,advances,on,linked,Gaussian,process,surrogates,[14],could,be,considered,and,explored,as,a,potential,candidate,to,construct,non-intrusive,ROM,for,fusion,systems,by,linking,non-intrusive,ROM,of,individual,sub-models.,For,example,,to,construct,the,ROM,of,the,two-layered,system,in,Figure,6,,one,could,first,build,GP-based,non-intrusive,ROM,(as,demonstrated,above),for,all,individual,sub-models,(f1,,f2,and,f3),and,then,construct,the,non-intrusive,ROM,of,the,whole,system,by,linking,the,non-intrusive,ROM,of,f1,and,f2,to,that,of,f3,through,the,reduced,space,w1,and,w2,analytically.,One,key,benefit,of,this,approach,for,system-wise,reduced,order,modelling,is,that,one,only,needs,to,do,dimensionality,reduction,to,the,outputs,of,sub-models.,Whereas,,to,build,intrusive,ROM,,one,has,to,make,extra,challenging,efforts,to,reformulate,the,original,high-fidelity,model,(e.g.,,f3),under,both,reduced,input,(e.g.,,w1,and,w2),and,output,(e.g.,,y).,References,[1],J.,Beck,and,S.,Guillas,,Sequential,design,with,mutual,information,for,computer,experiments,(MICE):,emulation,of,a,tsunami,model,,SIAM/ASA,Journal,on,Uncertainty,Quantification,,4,(2016),,pp.,739–766.,[2],K.-L.,Chang,,S.,Guillas,,et,al.,,Computer,model,calibration,with,large,non-stationary,spatial,outputs:,application,to,the,calibration,of,a,climate,model,,Journal,of,the,Royal,Statistical,Society,Series,C,,68,(2019),,pp.,51–78.,6,[3],K.,R.,Dalbey,,Efficient,and,Robust,Gradient,Enhanced,Kriging,Emulators,,Tech.,Rep.,SAND2013–,7022,,Sandia,National,Laboratories:,Albuquerque,,NM,,USA,,2013.,[4],A.,Damianou,and,N.,D.,Lawrence,,Deep,Gaussian,processes,,in,Artificial,intelligence,and,statis-,tics,,PMLR,,2013,,pp.,207–215.,[5],F.,Deluzet,and,J.,Narski,,A,two,field,iterated,asymptotic-preserving,method,for,highly,anisotropic,elliptic,equations,,Multiscale,Modeling,&,Simulation,,17,(2019),,pp.,434–459.,[6],M.,Gu,and,J.,O.,Berger,,Parallel,partial,Gaussian,process,emulation,for,computer,models,with,massive,output,,The,Annals,of,Applied,Statistics,,10,(2016),,pp.,1317–1347.,[7],M.,Gu,,J.,Palomo,,and,J.,O.,Berger,,RobustGaSP:,robust,Gaussian,stochastic,process,emulation,i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