CD-EXCALIBUR-FMS0040-M5.1_ManagementExternalResearchSupportsUqProcurement
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   :description: technical note

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ExCALIBUR,Management,of,external,research.,Supports,UQ,Procurement,M5.1,Abstract,The,report,describes,work,for,ExCALIBUR,project,NEPTUNE,at,Milestone,5.1.,This,report,collates,technical,material,used,to,inform,the,preparation,of,the,call,for,procurement,of,support,for,Uncertainty,Quantification,(UQ).,The,work,performed,primarily,consisted,of,collating,material,produced,internally,by,UKAEA,and,by,external,grantees,at,UCL,,plus,hands-on,exploration,of,sam-,pling,techniques,,the,construction,of,surrogates,and,the,study,of,dimensionally,reduced,models.,In,addition,,information,from,Exeter,contractors,gathered,at,NEPTUNE,workshops,is,included.,Besides,paying,for,contractors,,UKAEA,is,also,sponsoring,a,PhD,in,UQ,which,is,described,in,order,to,avoid,duplication.,The,report,has,been,slightly,delayed,so,that,it,could,be,updated,with,material,from,a,second,NEPTUNE,UQ,workshop,on,30,June,,in,order,to,facilitate,assessment,of,bids,for,grants,for,the,NEPTUNE,2021,call.,UKAEA,REFERENCE,AND,APPROVAL,SHEET,Client,Reference:,UKAEA,Reference:,CD/EXCALIBUR-FMS/0040,Issue:,Date:,1.00,July,5,,2021,Project,Name:,ExCALIBUR,Fusion,Modelling,System,Prepared,By:,Name,and,Department,Ed,Threlfall,Joseph,Parker,Will,Saunders,Wayne,Arter,Signature,N/A,N/A,N/A,N/A,Date,July,5,,2021,July,5,,2021,July,5,,2021,July,5,,2021,BD,Reviewed,By:,Rob,Akers,July,5,,2021,Advanced,Dept.,Manager,Computing,Approved,By:,Rob,Akers,July,5,,2021,Advanced,Dept.,Manager,Computing,2,1,Introduction,This,report,collates,technical,material,used,to,inform,the,preparation,of,the,call,for,procurement,of,support,for,Uncertainty,Quantification,(UQ).,The,work,performed,primarily,consisted,of,collating,material,produced,internally,[1,,2,,3,,4],and,by,external,grantees,at,UCL,[5,,6],,plus,hands-on,exploration,of,sampling,techniques,,the,construction,of,surrogates,and,the,study,of,dimensionally,reduced,models.,The,production,of,this,call,reflects,the,importance,attributed,to,UQ,in,project,NEPTUNE,,so,that,the,software,produced,should,be,capable,of,producing,‘actionable’,results,,that,could,form,a,significant,input,to,multi-million,pound,procurements.,It,seems,worthwhile,to,start,by,recapitulating,the,project,NEPTUNE,requirements,for,UQ,,insofar,as,they,are,currently,understood.,In,particular,,to,note,a,clear,distinction,between,the,experiments,that,NEPTUNE,code,needs,to,model,,and,the,outputs,of,the,simulations,themselves.,Relevant,experiments,are,almost,entirely,medium-sized,and,large,tokamaks,,where,it,seems,that,even,the,edge,plasma,has,a,temperature,of,around,10,eV,(105,K).,The,presence,of,magnetic,fields,of,order,1,−,10,T,,and,electric,fields,of,around,104,Vm−1,implies,a,hostile,environment,,subject,further,to,large,electromagnetic,transients,not,just,at,start-up,and,ramp-down,of,a,discharge,,but,also,caused,by,plasma,instabilities,such,as,ELMs,and,sawteeth.,Most,diagnostics,struggle,to,achieve,absolute,accuracies,of,10,%,,although,they,usually,detect,the,direction,of,smaller,changes,reliably.,Many,are,“one-of-a-kind”,so,require,careful,interpretation,,and,all,signals,may,have,been,subject,to,filtering,both,at,low,and,high,frequencies,,to,remove,‘spikes’,and,‘drift’.,In,comparison,,signals,from,simulation,are,very,‘clean’,,but,the,simulation,may,lack,crucial,realism,,in,that,either,important,physical,processes,are,not,treated,or,if,included,,that,the,spatio-temporal,discretisation,may,not,be,sufficiently,fine,to,represent,them,accurately.,Turning,to,the,expected,usage,of,the,NEPTUNE,software,,this,covers,both,physics,and,engi-,neering.,Both,physicist,and,engineers,are,potentially,interested,in,everything,computed,by,the,software,,which,is,of,course,likely,to,be,an,overwhelming,amount,of,detail.,A,physicist,is,more,likely,to,begin,by,formulating,simpler,models,and,use,the,software,to,determine,their,appropriate-,ness,and,accuracy,,an,engineer,probably,more,likely,to,use,the,output,of,simulation,as,a,basis,for,producing,simplified,models.,An,experimental,physicist,may,be,satisfied,with,qualitative,accu-,racy,,provided,it,is,obtained,rapidly,enough,to,help,formulate,the,next,experimental,‘shot’,(eg.,by,indicating,that,more,or,less,gas,input,would,be,helpful),,whereas,a,more,theoretical,counterpart,may,want,to,quantify,the,difference,between,model,predictions,and,experiment,sufficiently,well,to,help,produce,a,more,refined,model.,An,engineer,seeking,to,design,a,reactor,may,be,interested,only,in,a,relatively,small,number,of,quantities,of,interest,(QoI),but,which,need,to,be,determined,more,precisely,,such,as,local,maxima,in,time-averaged,heat,flux,,whereas,other,engineers,may,see,edge,code,as,forming,a,part,of,a,digital,twin,,to,be,used,to,help,control,operation,of,an,actual,It,is,notable,that,tokamak,device,where,accurate,time-dependent,modelling,will,also,be,critical.,most,of,the,preceding,requirements,involve,extrapolation,,from,model,to,new,features,of,an,exist-,ing,experiment,or,,in,the,design,case,,to,more,extreme,values,of,geometrical,size,,magnetic,field,etc.,representing,a,fusion,reactor,relative,to,existing,experiments,such,as,JET.,For,the,above,it,is,therefore,as,Smith,[7],describes,(see,also,the,NEPTUNE,report,[4,,§,1]),,im-,portant,that,UQ,involves,more,than,just,clever,sampling,to,identify,parameter,sensitivities,,and,the,standard,definitions,of,ways,of,quantifying,these,such,as,Sobol,indices,[1].,In,particular,,the,production,of,surrogate,models,is,within,scope,of,UQ.,It,is,worth,saying,that,in,one,sense,even,3,the,most,complex,partial,differential,equation,(PDE),models,of,plasma,are,simply,surrogates,for,experimental,‘reality’.,The,most,detailed,and,least,controversial,model,of,tokamak,plasma,con-,sists,in,treating,it,as,of,order,1020,or,more,particles,evolving,under,the,influence,of,external,and,self-generated,electromagnetic,fields,,but,this,is,of,course,not,tractable,even,at,the,Exascale,[8],,without,very,selective,sampling,of,the,particles.,Currently,,particle,models,are,seen,as,at,best,struggling,to,capture,edge,plasma,behaviour,,with,no,certain,likelihood,of,significant,improvement,,and,in,any,event,there,is,a,separate,calls,that,could,cover,research,into,particle,sampling.,The,PDE,models,attempt,to,capture,the,key,physics,of,the,plasma,particles,,viz.,their,appearance,,mo-,tion,,interactions,with,other,particles,and,eventual,loss,,as,terms,representing,respectively,source,,advection,,diffusion,and,sinks,in,continuum,models.,Unfortunately,,long,mean,free,paths,(mfps),mean,that,at,least,in,some,regions,it,is,necessary,explicitly,to,represent,phase-velocity,depen-,dence,of,plasma,density,,and,the,resulting,basic,6-D,dependence,of,fields,(5-D,if,gyro-averaged,models,are,used),stretches,current,and,foreseen,computing,facilities.,In,any,event,,above,applica-,tions,eg.,preliminary,design,studies,and,operational,control,,require,models,capable,of,evaluation,within,seconds,or,less,,ie.,surrogates,for,the,PDEs,,expected,to,have,lower,spatial,dimensionality,(2-D,or,1-D),or,even,as,systems,of,ordinary,differential,equations,(ODEs).,Unfortunately,for,application,to,NEPTUNE,,most,surrogates,have,been,designed,for,use,as,in-,terpolants.,Nonetheless,,interpolation,is,likely,to,be,important,at,least,within,the,software,,eg.,to,connect,different,models,,and,the,book,by,de,Boor,[9],has,interesting,material,relating,to,the,use,of,splines,in,this,context.,In,particular,,de,Boor,lists,theorems,relating,to,the,optimality,of,spline,poly-,nomial,interpolation,,so,that,there,might,be,wonder,that,any,other,form,of,interpolation,is,required.,In,fact,,spline,optimality,is,found,only,in,the,energy,or,least-squares,norm,,so,there,is,anyway,the,possibility,of,fitting,the,data,with,polynomial,functions,that,have,under-,and,over-shooting,oscil-,lations,between,sample,points,(called,nodes,in,spline,jargon).,Moreover,,de,Boor,illustrates,the,potential,for,dramatic,failings,using,what,he,refers,to,as,the,Titanium,heat,data,set,[9,,Fig.,XIII.1].,The,data,is,a,set,of,some,50,apparently,regularly,spaced,samples,of,a,function,which,is,practically,flat,except,for,one,distinct,‘hump’,,rising,to,four,times,the,value,of,the,‘plateau’,over,c.,15,%,of,the,range.,Since,the,interpolation,is,by,a,piecewise,(5th,order,polynomial),the,undershoot,by,greater,than,the,maximum,‘hump’,value,is,in,many,respects,more,impressive,than,that,exhibited,in,Boyd’s,book,[10,,Fig.,4.3],as,part,of,his,demonstration,of,the,Runge,phenomenon,,ie.,that,polynomial,interpolation,at,evenly,spaced,points,may,diverge.,In,both,books,,the,problem,is,identified,as,due,to,poor,choice,of,sample,points.,de,Boor,[9,,Fig.,XIV.5],shows,improved,fitting,if,the,number,of,samples,in,the,plateau,region,of,the,Ti,heat,data,is,reduced,,whereas,Boyd,points,out,that,optimal,N,th,order,polynomial,interpolation,is,achieved,using,sample,points,that,are,the,roots,of,the,Cheby-,chev,polynomial,of,degree,(N,+,1),,that,cluster,as,close,together,as,1/N,2,near,the,end-points,of,the,sample,region.,In,any,event,,for,many,interpolation,purposes,,it,seems,that,polynomial,spline,fitting,with,an,order,of,three,or,so,,is,adequate,provided,the,nodes,are,chosen,suitably.,The,cost,scales,as,the,operation,count,of,a,banded-diagonal,matrix,solver,,so,O(N,),where,N,is,the,number,of,points,to,be,interpolated,per,dimension.,There,are,however,strong,proponents,for,the,use,of,ra-,tional,polynomial,interpolation,[11],and,in,particular,radial,basis,function,[12].,Wavelet,bases,[13],also,have,their,adherents,,and,special,conditions,such,as,periodicity,or,infinite,domain,mean,that,respectively,Fourier,series,and,the,Hermite,basis,are,to,be,preferred.,The,Hermite,basis,is,em-,ployed,in,the,Polynomial,Chaos,Expansion,(PCE),,which,is,advocated,by,P.,Coveney,et,al,,see,the,NEPTUNE,report,[5].,Accurate,spline,fitting,requires,a,sufficient,amount,of,data,to,enable,best,location,of,the,nodes,,4,which,,since,they,generally,aim,to,use,as,few,nodes,as,possible,,is,commonly,unavailable,in,the,output,of,highly,expensive,numerical,calculations.,This,lack,of,data,means,also,that,inter-,polation,using,Neural,Networks,(NNs),is,mostly,unsuitable,,and,motivates,the,use,of,Gaussian,Processes,(GPs),for,interpolation,despite,their,expense,,that,O(N,3),of,inverting,a,full,matrix,[2].,They,come,with,the,benefit,of,accompanying,error,bounds,,and,seem,to,be,relatively,stable,when,used,for,extrapolation,[14].,They,may,be,very,effective,in,combination,with,a,special,basis,pro-,duced,by,Principal,Component,Analysis,(PCA),,aka,Proper,Orthogonal,Decomposition,or,POD,,see,the,NEPTUNE,report,[6].,Moreover,,S.,Guillas,[15],describes,a,number,of,techniques,which,outperform,PCA,,in,particular,favouring,a,reproducing,kernel,approach,to,estimating,derivatives,needed,to,optimise,the,basis.,This,is,coupled,with,a,sequential,approach,to,selecting,sample,points.,Indeed,,choosing,sample,points,appropriately,is,still,a,major,issue,for,GPs,,which,P.,Chal-,lenor,(private,communication),has,explored,extensively,and,favours,‘Leave,One,Out’,(LOO).,Nonetheless,,given,the,general,need,for,extrapolation,,it,is,desirable,to,incorporate,as,much,phys-,ical,knowledge,as,possible,into,the,surrogate,model.,The,hopes,are,that,the,parameters,of,the,surrogate,are,either,invariant,under,the,extrapolation,,or,that,they,have,well-determined,scaling,properties,,and,that,all,important,physical,processes,have,been,identified.,The,parameters,of,the,surrogate,may,then,be,fitted,using,existing,data,from,experiment,and/or,simulation,as,described,above.,The,surrogate,may,need,to,be,space,and/or,time,dependent,as,required,by,the,user,and,application.,Of,course,,such,surrogates,can,be,very,useful,for,interpolation,also.,Worthy,of,partic-,ular,mention,is,Model,Order,Reduction,[3],,which,for,linear,and,some,nonlinear,problems,,offers,reliable,estimates,of,error,in,interpolating,Quantities,of,Interest,(QoI),,hence,likely,to,be,particularly,useful,for,the,design,engineer.,The,body,of,the,current,document,consists,of,hands-on,exploration,of,sampling,techniques,in,Section,2,,the,construction,of,surrogates,by,Gaussian,Processes,in,Section,3,and,of,surrogates,consisting,of,dimensionally,reduced,models,in,Section,4.,The,implications,for,the,call,are,sum-,marised,in,Section,5.,5,2,BOUT++,BOUT++,[16,,17],is,a,framework,for,writing,fluid,and,plasma,simulations,in,curvilinear,geometry.,It,is,designed,primarily,to,target,reduced,plasma,fluid,models,,but,it,can,evolve,any,number,of,partial,differential,equations,,with,equations,appearing,in,a,readable,form,via,its,domain,specific,language.,BOUT++,is,being,used,in,Project,NEPTUNE,as,the,framework,in,which,to,build,the,fluid,referent,model,proxyapp.,A,team,of,BOUT++,developers,attended,the,three,VECMAtk,hackathons,in,Spring,2021.,The,goal,of,the,hackathon,work,was,to,create,the,software,framework,to,enable,BOUT++,to,use,VECMAtk,,and,to,develop,workflows,for,uncertainty,quantification,and,the,creation,of,surrogate,models.,In,this,report,we,focus,on,the,development,of,the,software,infrastructure,and,the,workflows,for,un-,certainty,quantification.,The,code,produced,at,the,hackathons,is,available,in,the,repository,[18].,2.1,Software,Infrastructure,The,VECMA,toolkit,wraps,around,user’s,software,,providing,a,library,of,python,functions,for,non-,intrusive,UQ.,From,VECMAtk’s,perspective,,BOUT++,is,a,black,box,code,to,which,it,must,provide,input,parameters,and,from,which,it,must,read,output,quantities,of,interest,(QoIs).,Therefore,the,workflow,is,written,in,Python,,and,is,driven,by,calls,to,VECMAtk,functions.,VECMAtk,selects,the,parameter,values,to,be,sampled,,collects,the,QoI,values,,and,calculates,and,plots,statistical,quantities,such,as,Sobol,indices,and,confidence,intervals.,The,only,necessary,interaction,between,VECMA,and,BOUT++,is,for,VECMA,to,communicate,input,parameters,to,BOUT++,and,to,read,back,output,values.,This,is,achieved,by,providing,an,encoder,,a,routine,that,is,capable,of,writing,BOUT++,input,files,,and,a,decoder,,a,routine,to,parse,BOUT++,output,files.,Both,are,actually,classes,providing,additional,functions,,such,as,routines,to,check,that,BOUT++,simulations,have,finished.,Examples,of,decoders,and,encoders,used,for,BOUT++,are,given,in,Appendices,A,and,B,respectively.,This,framework,structure,makes,it,very,easy,to,reuse,the,same,workflow,with,different,codes.,BOUT++,could,be,replaced,by,any,other,code,that,produces,the,same,QoIs,from,the,same,pa-,rameters,simply,by,replacing,the,BOUT++,encoder/decoder,with,corresponding,functions,for,that,other,code.,This,pattern,also,makes,it,easy,to,use,synthetic,data,instead,of,an,actual,code,execution.,Instead,of,calling,the,encoder/decoder,,one,simply,calls,a,function,to,return,the,QoIs,as,a,function,of,parameters.,This,is,particularly,useful,in,developing,the,workflow,,such,as,testing,the,sampling,routines,or,plotting,functions,,as,executing,BOUT++,is,by,far,the,most,significant,runtime,cost,in,the,workflow.,The,encoders/decoders,also,provide,the,method,for,changing,the,QoIs,for,different,physical,cases.,For,example,in,our,work,with,temperature,T,(x,,y,,z,,t),as,the,QoI,,each,of,the,different,cases,–,fixed,point,over,time,T,(0,,0,,0,,t),,spatial,range,at,the,final,time,T,(x,,0,,0,,tend),,or,logarithm,of,temperature,log[T,(x,,0,,0,,tend)],–,were,each,achieved,by,implementing,a,separate,decoder.1,1A,similar,approach,allowed,us,to,work,around,a,bug,in,the,chaospy,sampling,routine,(a,library,dependence,of,VECMAtk).,The,chaospy,log-uniform,probability,distribution,was,broken,,but,the,uniform,distribution,was,functional.,6,Figure,1:,Mean,and,confidence,intervals,(left),and,Sobol,indices,(right),plotted,against,x,array,index,for,quantity,of,interest,T,.,2.2,Uncertainty,Quantification,The,first,workflow,considered,introduced,UQ,into,BOUT++,simulations.,Instead,of,performing,a,single,simulation,to,obtain,a,single,result,,the,VECMAtk,UQ,framework,performs,an,ensemble,of,simulations,,varying,chosen,input,parameters,in,a,specified,range.,From,this,,VECMAtk,fits,a,surrogate,model:,it,finds,the,QoIs,as,a,function,of,the,input,parameters.,With,this,model,it,calculates,statistics,,such,as,the,mean,value,of,a,QoI,and,corresponding,confidence,intervals,,and,the,Sobol,indices,,which,give,a,way,of,quantifying,how,much,uncertainty,in,the,QoIs,is,due,the,variation,of,each,parameter.,One-dimensional,heat,conduction,was,used,as,an,initial,test,problem,,∂T,∂t,=,χ,∂2T,∂x2,,,(1),where,T,is,the,temperature,on,the,spatial,domain,x,∈,[0,,1],with,boundary,conditions,T,(0,,t),=,T,(1,,t),=,0,and,initial,condition,T,(x,,0),=,T0,exp[−(x,−,0.5)2/0.04].,Here,,the,thermal,conductivity,χ,and,the,initial,amplitude,T0,are,uncertain,parameters.,2.2.1,Polynomial,chaos,expansion,As,a,first,example,,we,implemented,a,workflow,for,1D,heat,conduction,(1),varying,χ,∈,(0.2,,4),and,T0,∈,(0.5,,1.5),using,polynomial,chaos,sampling.,The,resulting,statistics,and,Sobol,indices,for,T,at,the,final,time,T,=,10,is,shown,in,Figure,1.,The,left,figure,shows,the,mean,final,temperature,and,its,confidence,intervals,.,The,right,figure,shows,the,first,Sobol,indices,,which,describe,how,much,of,the,uncertainty,is,due,to,each,parameter.,For,example,,at,the,centre,of,the,domain,,about,Therefore,,in,order,to,obtain,a,log-uniform,distribution,,we,provided,an,encoder,that,wrote,input,parameter,p,to,the,BOUT++,input,file,as,variable=10**p,instead,of,variable=p.,7,65%,of,the,uncertainty,is,due,to,varying,χ,(independently,of,T0),,about,30%,is,due,to,varying,T0,(independently,of,χ),,and,the,remaining,5%,is,due,to,varying,both,in,a,coupled,fashion.,The,statistics,are,qualitatively,converged,using,only,a,3rd-order,polynomial,chaos,expansion,which,requires,(3,+,1)2,=,16,code,evaluations,in,two,dimensions.,However,the,confidence,intervals,encompass,negative,values,of,temperature,near,the,boundary,,even,though,the,temperature,is,non-negative,and,all,simulation,data,provided,is,also,non-negative.,One,approach,fixing,this,is,to,use,log(T,),as,the,QoI.,The,logarithm,is,a,bijective,map,between,the,positive-valued,temperature,T,∈,(0,,∞),and,the,real,value,log(T,),∈,(−∞,,∞).,Therefore,this,allows,the,same,analysis,routines,to,be,performed,on,log(T,),without,concern,about,sign.,The,resulting,confidence,intervals,now,only,allow,positive,temperatures.,Moreover,the,Sobol,indices,are,qualitatively,the,same,,indicating,that,the,transformed,model,ascribes,uncertainty,to,the,various,parameters,in,a,similar,fashion.,However,,considering,higher-order,polynomial,chaos,it,becomes,apparent,that,the,negative,values,are,an,artefact,of,the,data,analysis,routine.,Near,T,=,0,,the,temperature,is,a,steeply-varying,function,of,the,input,parameters,,and,low-order,polynomial,fitting,is,simply,inaccurate,,predicting,negative,values.,Increasing,the,order,of,the,polynomial,fit,(by,increasing,the,order,of,PCE,and,providing,more,simulation,data),leads,to,this,problem’s,no,longer,being,observed.,2.2.2,Adaptive,stochastic,collocation,Using,higher-order,polynomial,chaos,is,reasonable,when,varying,a,small,number,of,parameters,,but,varying,more,parameters,is,beset,by,the,“curse,of,dimensionality”:,the,number,of,sam-,ples,needed,for,nth-order,PC,with,d,parameters,is,(n,+,1)d.,Therefore,,we,also,implemented,EasyVVUQ’s,sampling,with,adaptive,stochastic,collocation,(ASC),,which,is,based,on,the,algo-,rithm,by,Gerstner,and,Griebel,[19],In,ASC,,the,grid,of,sampling,points,for,each,parameter,is,adaptively,refined,,beginning,from,a,grid,with,only,one,sampling,point,for,each,parameter.,Estimates,are,made,for,the,reduction,in,the,error,that,would,result,from,adding,one,more,sampling,point,in,each,parameter;,the,refinement,is,made,to,the,parameter,which,would,reduce,the,error,the,most.,As,a,result,,computing,resources,are,directed,where,they,are,most,effective,,yielding,results,that,are,comparable,with,high-order,PCE,but,which,require,many,fewer,code,evaluations.,Implementing,a,new,sampling,algorithm,(from,a,VECMA,user’s,perspective),is,a,simple,matter,of,changing,the,class,on,which,sampling,routines,are,called.,However,,adaptive,stochastic,colloca-,tion,allows,additional,functionality,which,we,may,incorporate,in,our,workflow.,In,particular,,ASC,uses,a,Clenshaw–Curtis,collocation,grid,which,has,a,nested,quadrature,grid,(unlike,in,Gaussian,quadrature,,the,grid,for,(n,+,1)th,order,accuracy,contains,all,the,points,for,nth,order,accuracy).,This,means,we,may,increase,the,order,of,accuracy,by,adding,one,new,code,evaluation,,rather,than,(n,+,1),as,would,be,required,in,PC.,This,allows,us,to,use,an,iterative,approach,,continuing,to,refine,the,order,in,each,parameters,dimension,until,some,convergence,criteria,is,met.,In,our,workflow,,we,used,relative,changes,in,the,Sobol,indices,from,one,iteration,to,the,next,,but,this,criterion,is,not,ideal,as,it,is,noisy,and,non-monotonic.,Finally,,as,a,proof,of,concept,we,implemented,uncertainty,quantification,for,a,two-dimensional,8,simulation,of,plasma,filament,propagation,,∂n,∂t,∂Ω,∂t,=,−{φ,,n},+,2,=,−{φ,,Ω},+,2,∇2φ,=,Ω,,ρs,Rc,ρs,Rc,∂n,∂z,∂n,∂z,+,Dn∇2n,,+,DΩ,n,∇2Ω,,(2a),(2b),(2c),in,a,two-dimensional,box,(x,,z),with,Dirichlet,boundary,conditions,in,x,and,periodic,boundary,conditions,in,z.,In,(2),,n,is,plasma,density,,Ω,is,vorticity,,φ,is,electrostatic,potential,,t,is,time,,ρs,is,the,Bohm,gyroradius,,Rc,is,the,radius,of,curvatuve,,Dn,and,DΩ,are,dissipation,parameters,,and,{A,,B},=,(∂A/∂x)(∂B/∂z),−,(∂A/∂z)(∂B/∂x),is,a,Poisson,bracket.,We,use,the,two-dimensional,Laplacian,∇2,≡,∂2/∂x2,+,∂2/∂z2.,Between,each,timestep,,the,vorticity,equation,(2c),must,be,solved,for,φ,so,its,value,can,be,used,in,(2a),and,(2b),to,advance,n,and,Ω.,We,use,φ,at,the,current,timestep,as,an,initial,guess,for,the,iterative,method,(and,φ,=,0,as,the,guess,for,the,first,time,step).,Four,input,parameters,were,varied,,the,constant,coefficients,ρs,,Rc,,Dn,and,DΩ.,While,still,rea-,sonably,small,,individual,simulations,require,around,10,minutes,on,16,cores,on,the,University,of,York’s,cluster.,Third,order,PCE,required,1296,simulations,,while,comparable,results,using,ASC,only,required,256,simulations.,9,3,Nektar++,Nektar++,[20,,21],is,a,framework,for,solving,partial,differential,equations,(PDEs),using,the,spectral,/,hp-element,method.,The,software,contains,several,different,solvers,targeting,canonical,problems,in,fluid,mechanics.,It,is,possible,to,extend,the,framework,by,adding,new,domain-specific,solvers;,this,currently,involves,low-level,coding,as,there,is,no,DSL.,Nektar++,is,being,used,in,Project,NEPTUNE,as,the,framework,for,the,higher-order,solution,of,fluid,equations,and,currently,is,being,extended,to,handle,plasma,fluids.,Data,is,extracted,from,Nektar++,simulations,by,means,of,filters,and,written,to,text-based,output,files.,A,small,team,from,UKAEA,attended,the,three,VECMAtk,hackathons,in,Spring,2021.,The,goal,of,the,hackathon,work,was,to,create,the,software,framework,to,enable,Nektar++,to,use,VECMAtk,,and,to,develop,workflows,for,uncertainty,quantification,and,the,creation,of,surrogate,models.,3.1,Software,Infrastructure,The,VECMA,toolkit,wraps,around,an,individual,Nektar++,solver,to,provide,non-intrusive,UQ,(only,non-coupled,workflows,are,considered,here).,The,toolkit,interacts,with,Nektar++,via,the,inputs,and,outputs,of,the,latter,,through,respectively,an,encoder,and,a,decoder,;,these,objects,can,be,interchanged,to,enable,different,workflows,(for,example,,the,relevant,quantities,of,interest,(QoIs),are,parsed,in,the,decoder).,3.2,Uncertainty,Quantification,The,initial,test,problem,used,the,incompressible,Navier-Stokes,solver,of,Nektar++,to,simulate,classical,convection:,a,two-dimensional,fluid-filled,slot,with,vertical,walls,maintained,at,different,temperatures;,starting,with,an,initial,uniform,temperature,gradient,from,the,hot,side,to,the,cold,side,,the,system,is,time-evolved,until,a,steady,state,is,reached.,Warmer,fluid,is,subject,to,an,upward,buoyancy,force,,leading,to,a,steady,convective,cell,solution,with,fluid,travelling,up,the,hot,side,and,down,the,cold,one.,In,dimensionless,form,,the,equations,for,fluid,velocity,u,,temperature,T,and,pressure,p,are,1,P,r,(cid:18),∂u,∂t,(cid:18),∂T,∂t,(cid:19),(cid:19),+,u,·,∇u,+,u,·,∇T,=,−∇p,+,Ra,T,ˆy,+,∇2u,=,∇2T,∇,·,u,=,0.,(3),(4),(5),Here,,the,Rayleigh,number,Ra,and,the,Prandtl,number,P,r,are,the,governing,parameters,and,are,taken,to,be,subject,to,uncertainty.,10,Figure,2:,Steady-state,colour,map,of,dimensionless,temperature,for,Ra,=,104,,P,r,=,1.0.,The,left,side,is,maintained,at,a,temperature,one,unit,higher,than,the,right,and,a,convective,cell,carries,fluid,in,a,clockwise,circulation.,The,Nektar++,simulations,were,designed,so,as,to,run,quickly,on,a,single,desktop,computer.,Com-,putational,parameters,governing,accuracy,were,chosen,to,give,plausible,results,without,detailed,analysis,of,convergence.,3.2.1,Polynomial,chaos,expansion,A,square,slot,was,used,as,a,first,example;,the,results,follow,closely,the,original,investigation,in,[22].,An,example,temperature,profile,over,the,full,domain,can,be,see,in,Figure,2.,The,QoI,was,the,temperature,profile,T,(x),evaluated,half-way,up,the,slot,,parsed,via,the,decoder,SlicedHisDecoder,from,data,extracted,using,the,Nektar++,HistoryPoints,filter,to,record,field,values,at,chosen,sample,points.,Note,that,the,sample,points,cannot,lie,on,the,vertical,boundaries,,as,the,Dirichlet,boundary,condition,enforces,constant,temperature,values,there,,which,causes,the,analysis,to,fail,(a,singular,matrix,error,is,reported,by,VECMA).,P,r,was,taken,to,be,in,the,range,1,−,10,(values,typical,for,the,gases,and,liquids,commonly,investigated,in,the,literature),,with,a,uniform,distribution.,Ra,was,taken,to,be,in,the,range,1.0,×,104,−,3.2,×,104,(values,consistent,with,a,laminar,convective,steady,state),,with,a,log-uniform,distribution.,The,polynomial,chaos,expansion,(PCE),produces,a,fit,to,the,sampled,QoI,data,using,a,sum,over,orthogonal,polynomials,in,the,input,variables,,where,the,orthogonality,is,defined,using,the,input,probability,distributions,as,the,measure,and,the,fit,is,performed,by,either,spectral,projection,or,linear,regression.,The,polynomial,chaos,analysis,of,EasyVVUQ,provides,the,mean,,variance,,and,standard,deviation,of,the,outputs,and,also,the,minimum,,maximum,,median,and,the,1st,,10th,,90th,,and,99th,percentiles,(a,useful,supported,stats(),routine,lists,the,available,statistics).,Some,of,these,quantities,are,shown,in,Figure,3,,which,was,generated,using,a,fifth-order,PCE.,Also,shown,are,the,first,Sobol,indices,,which,show,correctly,that,the,uncertainty,in,the,output,is,dominated,by,the,sensitivity,to,Ra,in,this,case;,in,the,regime,in,question,,the,effect,of,varying,P,r,is,small.,11,Figure,3:,Mean,and,confidence,intervals,(left),and,first,Sobol,indices,(right),as,a,function,of,po-,sition,for,the,central,temperature,profile,T,(x),fitted,by,a,polynomial,chaos,expansion.,Note,that,the,uncertainty,tends,to,zero,in,regions,where,the,temperature,is,either,fixed,(end,points),or,is,constrained,by,symmetry,(centre).,3.2.2,Stochastic,collocation,Stochastic,collocation,(SC),creates,a,representation,of,the,QoIs,that,is,exact,at,the,sample,points;,the,solution,is,a,linear,combination,of,sample-point,values,multiplying,Lagrange,polynomials,at,those,points.,The,stochastic,collocation,analysis,of,EasyVVUQ,provides,the,mean,,variance,and,standard,deviation,of,the,outputs.,The,set,of,available,statistics,is,smaller,than,for,the,PCE,,but,the,method,is,more,efficient,(as,explained,earlier,in,this,report).,The,decoder,used,in,the,preceding,section,is,switchable,between,PCE,and,SC.,3.3,Surrogate,models,Note,that,,although,it,concerns,surrogate,models,,the,work,in,this,subsection,used,EasyVVUQ,and,not,EasySurrogate.,3.3.1,Polynomial,chaos,expansion,and,stochastic,collocation,Either,of,the,orthogonal,polynomial,fit,constructed,in,the,polynomial,chaos,expansion,or,the,La-,grange,polynomial,fit,constructed,in,the,stochastic,collocation,method,can,serve,as,a,computationally-,inexpensive,surrogate,for,the,full,model.,As,an,example,,PCE,and,SC,surrogates,for,the,Nusselt,number,N,u,as,a,function,of,Ra,and,P,r,were,constructed.,N,u,is,the,dimensionless,heat,flux,across,the,cavity,,normalized,to,the,Ra,=,0,no-convection,solution,,calculated,as,N,u,=,(cid:82),H,0,∂xT,(x0,,y),+,∂xT,(x1,,y)dy,(cid:82),H,0,∂xT,(x0,,y),+,∂xT,(x1,,y)dy|Ra=0,.,(6),12,Figure,4:,Polynomial,chaos,and,stochastic,collocation,surrogates,for,Nusselt,number,as,a,function,of,Rayleigh,number,,for,Prandtl,number,1.0.,where,∂x,denotes,partial,derivative,with,respect,to,coordinate,x.,The,Nusselt,number,was,extracted,using,a,customized,filter,in,Nektar++,and,the,output,file,was,parsed,by,a,decoder,(HeatFluxDecoder),using,a,simple,text,processing,routine.,An,illustration,of,the,SC,and,PCE,fits,(both,using,fifth-order,polynomial,expansions),for,the,simple,case,of,Nusselt,number,as,a,function,of,the,Rayleigh,number,(P,r,=,1),is,shown,in,Figure,4.,The,Nusselt,number,is,seen,to,follow,a,typical,power-law,curve.,3.3.2,Gaussian,process,surrogate,A,different,set-up,,though,again,using,the,Nektar++,incompressible,Navier-Stokes,solver,,was,used,as,an,initial,exploration,of,the,Gaussian,process,surrogate,capability,in,EasyVVUQ;,this,was,a,slot,10,units,high,and,1,wide,,operating,at,larger,Rayleigh,numbers,where,the,flow,exhibits,a,boundary-layer,instability,with,waves,moving,up,the,hot,wall,and,down,the,cold,one,(illustrated,in,Figure,5).,This,gave,interesting,time,series,for,the,maximum,temperature,near,the,affected,parts,of,the,walls,(the,lower,part,of,the,cold,wall,was,used),and,the,position,of,the,hottest,point.,These,QoIs,are,of,interest,in,fusion,applications,as,they,characterize,the,hot,spots,on,the,wall,of,a,reactor,containing,hot,plasma.,The,time,series,were,extracted,over,a,chosen,time,window,using,a,customized,filter,in,Nektar++;,the,simulation,runs,were,performed,once,and,the,decoder,(HotSpotDecoder),was,given,the,ability,to,read,stored,simulation,data,in,order,to,avoid,repeating,calculations.,In,view,of,the,weak,dependence,on,the,parameter,P,r,,only,the,parameter,Ra,was,varied,in,this,activity,(values,in,the,range,105,−,107,were,used).,A,kernel,with,zero,mean,and,a,Mat,´ern,covariance,with,parameters,ν,=,3/2,and,scale,107,were,used.,This,investigation,was,an,attempt,at,finding,a,surrogate,for,computations,that,are,more,demanding,than,the,laminar,convection,ones,in,the,preceding,section,and,also,concerned,a,system,that,exhibits,deterministic,chaos.,The,time,series,data,used,to,train,the,Gaussian,process,is,shown,in,Figure,5.,Note,that,for,Ra,sufficiently,large,for,the,instability,to,form,,temperature,maxima,enter,from,the,top,of,the,monitored,section,and,proceed,downwards,until,usurped,by,the,next,wave.,For,smaller,values,of,Ra,,the,system,is,quiescent,and,the,time,series,eventually,become,constant.,13,Figure,5:,Boundary-layer,instability,at,the,cold,wall,for,Ra,=,2.3,×,108,(top).,The,maximum,temperature,on,this,part,of,the,wall,and,also,the,position,of,the,hottest,point,are,shown,as,time,series,(respectively,left,and,right,below).,14,Figure,6:,Output,of,the,Gaussian,process,surrogates,for,the,maximum,temperature,and,the,posi-,tion,of,the,hottest,point,evaluated,at,positions,away,from,the,sample,parameter,values.,Some,sample,evaluations,of,the,Gaussian,process,surrogate,away,from,the,sample,points,can,be,seen,in,Figure,6.,It,can,be,seen,that,the,surrogate,correctly,reproduces,the,more,active,nature,of,the,time,series,for,larger,values,of,Ra.,15,4,Dimensionally,reduced,models,4.1,Boltzmann,Equation,One,suggestive,example,of,the,production,of,a,surrogate,is,the,reduction,of,the,Boltzmann,equa-,tion,to,the,equations,of,fluid,dynamics.,The,model,is,a,combination,of,Lie,derivative,and,solenoidal,constraints,∂f,∂t,+,ux,∂f,∂x,∂ux,∂x,+,uy,+,∂f,∂y,∂uy,∂y,=,S,=,0,(7),(8),where,f,(x,,y,,t),is,the,phase,space,density,as,a,function,of,time,t,,ux,is,the,flow,in,position,space,x,,and,uy,is,the,flow,in,velocity,space,y.,Collision,and,other,source,terms,are,represented,as,S(x,,y,,t).,For,simplicity,,it,is,assumed,that,position,and,velocity,space,are,each,1-D.,In,more,conventional,notation,where,v,is,the,velocity,space,coordinate,and,a,the,acceleration,experienced,by,a,particle,,then,y,↔,v,,ux,=,v,and,uy,=,a.,Integrating,Equation,(7),over,velocity,space,,ie.,forming,(cid:82),·dy,leads,to,(cid:90),∂uy,∂y,(cid:90),∂ux,∂x,uxf,dy,−,uyf,dy,−,∂n,∂t,∂,∂x,f,dy,=,f,dy,+,∂,∂y,Sdy,(9),+,(cid:90),(cid:90),(cid:90),upon,using,and,introducing,ux,∂f,∂x,=,∂(uxf,),∂x,−,f,∂ux,∂x,(cid:90),n,=,f,dy,Using,Equation,(8),so,that,two,terms,cancel,,Equation,(9),becomes,(cid:90),∂n,∂t,+,∂,∂x,uxf,dy,+,[uyf,]±yb,=,(cid:90),Sdy,(10),(11),(12),If,f,and/or,uy,vanish(es),at,the,boundaries,of,the,velocity,domain,y,=,±yb,,the,third,term,is,zero.,Suppose,there,is,a,mean,flow,,ie.,that,ux,may,be,written,such,that,and,thus,ux,=,U,(x),+,˜u(x,,y),(cid:90),˜uf,dy,=,0,(cid:90),uxf,dy,=,nU,then,substituting,Equation,(13),in,the,second,term,of,Equation,(12),results,in,∂n,∂t,+,∂nU,∂x,=,S0,16,(13),(14),(15),(16),which,is,recognised,as,the,fluid,dynamical,equation,of,mass,conservation,with,source,S0(x),=,(cid:82),Sdy,−,[uyf,]±yb.,To,obtain,the,momentum,conservation,equation,from,Equation,(7),,form,(cid:82),·uxdy,and,rearrange,f,-derivative,terms,,leading,to,(cid:90),∂nU,∂t,+,∂,∂x,u2,xf,dy,−,(cid:90),∂u2,x,∂x,f,dy,+,(cid:90),∂,∂y,uxuyf,dy,−,(cid:90),∂(uxuy),∂y,(cid:90),f,dy,=,Suxdy,(17),Using,∂(uxuy),∂y,=,ux,∂uy,∂y,+,∂ux,∂y,uy,=,−ux,∂ux,∂x,+,∂ux,∂y,uy,(18),(the,second,manipulation,using,Equation,(8)),,Equation,(17),gives,(cid:90),∂nU,∂t,+,∂,∂x,u2,xf,dy,−,(cid:90),ux,∂ux,∂x,f,dy,+,[uxuyf,]±yb,−,(cid:90),uy,∂ux,∂y,(cid:90),f,dy,=,Suxdy,(19),Writing,ux,=,U,(x,,t),+,˜u,,the,second,term,expands,to,(cid:90),∂,∂x,(U,2,f,dy),+,2,(cid:90),∂,∂x,(U,˜uf,dy),+,(cid:90),∂,∂x,˜u2f,dy,which,since,(cid:82),˜uf,dy,=,0,,reduces,to,∂(nU,2),∂x,+,∂,∂x,(cid:90),˜u2f,dy,The,third,term,in,Equation,(19),is,zero,provided,Indeed,if,ux,=,y,,then,upon,introducing,∂ux,∂x,=,0,(cid:90),p,=,˜u2f,dy,and,(cid:90),G,=,uyf,dy,=,(cid:90),af,dy,(20),(21),(22),(23),(24),there,results,the,fluid,dynamical,equation,of,momentum,conservation,with,momentum,source,S1(x),=,(cid:82),Suxdy,−,[uxuyf,]±yb,,namely:,∂nU,∂t,+,∂nU,2,∂x,=,−,∂p,∂x,+,G,+,S1,It,may,be,helpful,to,note,that,the,third,term,in,Equation,(19),expands,to,−,(nU,∂U,∂x,+,U,(cid:90),∂,˜u,∂x,(cid:90),f,dy,+,˜u,∂,˜u,∂x,f,dy),where,Equation,(14),has,been,used,to,zero,one,term.,17,(25),(26),4.2,Lie,Derivative,Surrogate,In,the,absence,of,sources,Equation,(7),may,be,written,for,a,flow,u,in,any,number,of,dimensions,as,+,u.∇f,=,0,(27),To,proceed,to,derive,a,widely,used,class,of,surrogate,for,models,involving,Lie,derivatives,,it,is,helpful,to,run,over,a,small,amount,of,mathematics.,The,streamlines,of,a,vector,field,such,as,u,by,definition,satisfy,=,u(x),(28),∂f,∂t,dx,ds,where,x(s),is,the,streamline,parameterised,by,s,,and,where,often,s,is,chosen,to,be,arc-length,along,the,curve.,Hence,substituting,Equation,(28),in,u.∇f,gives,the,Lie,derivative,as,df,/ds,which,illustrates,its,coordinate,independence,,since,s,is,an,almost,arbitrary,label.,Further,,substituting,Equation,(28),in,Equation,(27),gives,+,=,0,∂f,∂t,∂f,∂s,where,now,f,(s,,t).,Equation,(29),is,well-known,to,have,the,exact,solution,f,=,F,(t,−,s),for,arbitrary,scalar,functions,of,a,single,variable,F,.,In,realistic,applications,,S,(cid:54)=,0,,but,the,property,that,f,is,a,function,only,of,arclength,(and,time),survives,,so,that,the,physics,of,sources,and,sinks,may,be,dealt,with,by,a,surrogate,which,has,only,one,space,dimension,supposing,the,streamlines,are,known.,Moreover,,in,the,plasma,context,,the,main,flow,may,often,be,directed,along,lines,of,magnetic,field,B,,so,that,exactly,the,same,statements,could,be,made,replacing,streamline,with,fieldline.,(29),Now,,Equation,(25),can,be,expressed,in,Lie,derivative,form,by,eliminating,∂n,and,division,by,n,,giving,∂t,using,Equation,(16),,∂U,∂t,+,U,∂U,∂x,=,−,(cid:18),∂p,∂x,1,n,+,G,+,S1,−,U,S0,(cid:19),(30),Thus,provided,u,is,expressed,in,Cartesians,,each,component,separately,obeys,an,equation,of,the,form,Equation,(27),and,similar,remarks,apply.,There,is,also,[23],a,(pseudo-)vector,Lie,formulation,for,B,and,the,vorticity,∇,×,u,which,can,lead,to,similar,simplifications,and,potential,surrogates.,4.3,Surrogates,for,advection,Equations(7),and,(8),combine,to,give,an,equation,identical,to,that,for,mass,conservation,in,2-D,,namely,∂f,∂t,+,∂(uxf,),∂x,+,∂(uyf,),∂y,=,S,(31),where,now,f,(x,,y,,t),is,the,fluid,density,as,a,function,of,time,t,,and,ux,and,uy,are,components,of,the,flow,u.,Source,terms,may,still,be,represented,as,S(x,,y,,t),,collision,terms,may,be,represented,as,a,further,contribution,to,S(x,,y,,t),,of,form,Sd,=,∂,∂x,(cid:18),κxx,∂f,∂x,+,κxy,(cid:18),(cid:19),∂f,∂y,+,∂,∂y,κyx,∂f,∂x,+,κyy,(cid:19),∂f,∂y,(32),18,The,above,2-D,expressions,should,be,sufficient,for,understanding,how,to,deal,with,many,situations,involving,advection,,diffusion,and,sources,,which,are,more,generally,formulated,as,∂f,∂t,+,∇,·,(uf,−,κ∇f,),=,S,(33),for,tensor,diffusion,coefficient,κ.,Analogous,to,Section,4.1,,integrating,Equation,(31),over,coordinate,y,,ie.,forming,(cid:82),·dy,leads,to,∂,(cid:82),f,dy,∂t,+,∂,∂x,(cid:90),uxf,dy,+,[uyf,]y=±yb,=,(cid:90),Sdy,(34),There,is,then,a,simplification,to,a,1-D,surrogate,upon,supposing,that,as,in,Section,4.1,ux,may,be,written,and,introducing,as,before,ux,=,U,(x),+,˜u(x,,y),(cid:90),n,=,f,dy,to,give,without,further,approximation,,the,equation,(cf.,Equation,(16),∂n,∂t,+,∂nU,∂x,=,S0,(35),(36),(37),where,S0,includes,the,difference,in,mass,fluxes,at,the,boundaries,y,=,±yb.,However,,if,instead,of,Equation,(35),is,assumed,ux,=,V,(y),+,˜v(x,,y),(38),then,it,is,necessary,to,assume,a,weak,dependence,of,f,on,y,to,give,an,equivalent,of,Equation,(37),that,is,necessarily,approximate,as,(cid:90),∂n,∂t,+,1,2yb,V,dy,·,∂n,∂x,=,S0,(39),The,production,of,a,so-called,0-D,surrogate,is,achieved,by,further,integrating,over,coordinate,x,,ie.,forming,(cid:82),·dx,of,Equation,(34):,d,(cid:82),(cid:82),f,dxdy,dt,(cid:20)(cid:90),+,(cid:21),uxf,dy,(cid:20)(cid:90),+,(cid:21),uyf,dx,(cid:90),(cid:90),=,Sdxdy,(40),x=±xb,y=±yb,This,is,evidently,an,ODE,for,the,total,mass,ntot(t),=,(cid:82),(cid:82),f,dxdy,in,terms,of,the,total,source,Stot(t),=,(cid:82),(cid:82),Sdxdy,provided,the,boundary,fluxes,are,known.,Note,that,diffusion,may,be,explicitly,included,in,the,above,expressions,Equations(34),and,(40),on,making,the,replacements,uxf,−→,uxf,−,κxx,uyf,−→,uyf,−,κyx,∂f,∂x,∂f,∂x,−,κxy,−,κyy,∂f,∂y,∂f,∂y,19,(41),(42),Again,analogously,to,Section,4.1,,form,(cid:82),·ψdy,for,an,arbitrary,scalar,ψ,and,rearrange,f,-derivative,terms,,leading,to,(cid:90),∂nΨ,∂t,+,∂,∂x,uxψf,dy,−,(cid:90),ux,∂ψ,∂x,f,dy,+,[ψuyf,]±yb,−,(cid:90),uy,∂ψ,∂y,(cid:90),f,dy,=,Sψdy,(43),where,now,also,is,introduced,Ψ,such,that,and,thus,ψ,=,Ψ(x),+,˜ψ(x,,y),(cid:90),ψf,dy,=,nΨ,(44),(45),Further,expanding,ux,and,ψ,in,Equation,(43),,there,results,,using,similar,manipulations,to,Sec-,tion,4.1,,but,without,assuming,ux,=,y,that,where,∂nΨ,∂t,+,∂nU,Ψ,∂x,=,−,∂pψ,∂x,+,Gψ,+,S1ψ,(cid:90),pψ,=,˜u,˜ψf,dy,,S1ψ,=,(cid:90),Sψdy,−,[ψuyf,]±yb,and,exploiting,Equation,(26),Gψ,=,nU,(cid:90),+,∂Ψ,∂x,uy,∂,˜ψ,∂y,f,dy,+,U,(cid:90),∂,˜ψ,∂x,f,dy,+,(cid:90),∂,˜ψ,∂x,˜u,f,dy,As,before,,using,the,equation,of,mass,conservation,simplifies,the,dynamics,,resulting,in,where,n,∂Ψ,∂t,−,U,(cid:90),∂,˜ψ,∂x,f,dy,=,−,∂pψ,∂x,+,Hψ,+,S1ψ,−,ΨS0,Hψ,=,(cid:90),uy,∂,˜ψ,∂y,f,dy,+,(cid:90),∂,˜ψ,∂x,˜u,f,dy,(46),(47),(48),(49),(50),As,might,be,expected,from,Section,4.1,,taking,ψ,=,ux,achieves,only,minor,further,simplification.,20,5,Summary,ExCALIBUR,project,NEPTUNE,work,on,UQ,to-date,(including,two,UQ,workshops,organised,by,each,of,the,two,UCL,grantees),has,already,helped,established,a,significant,number,of,points,rel-,evant,to,the,design,of,the,NEPTUNE,software.,The,points,may,be,dealt,with,under,the,two,main,headings,of,sampling,and,surrogate,production,,however,there,is,a,strong,suggestion,that,sam-,pling,and,the,production,of,a,surrogate,should,operate,sequentially,,meaning,that,the,surrogate,informs,the,choice,of,sample,points,,which,in,turn,modifies,the,surrogate,&c.,For,sampling,,there,are,recommendations,from,external,NEPTUNE,reports,[5],that,adaptive,stochastic,collocation,(ASC),is,to,be,preferred,,supported,by,the,study,in,Section,2.2.2,above.,The,ASC,method,is,expected,to,be,resilient,in,the,face,of,the,“curse,of,dimensionality”,,although,not,completely,resistant,[1].,It,seems,natural,to,accompany,most,sampling,by,polynomial,interpo-,lation,when,the,spectral/hp,element,approach,is,used,to,represent,spatial,dependences,in,terms,of,pth,order,polynomials.,Section,3.3.1,above,confirms,that,surrogates,should,incorporate,“appropriate”,physical,knowl-,edge,,such,as,the,need,for,say,Nusselt,numbers,not,to,be,less,than,unity.,Section,3,confirms,that,turbulence,,even,when,arising,in,the,context,of,a,2-D,fluid,model,,is,hard.,The,work,shows,that,there,remain,questions,concerning,adequate,numerical,resolution,,probably,tied,up,with,the,existence,of,relatively,long,timescale,transients.,In,such,circumstances,,the,additional,expense,of,GP,is,likely,to,be,worthwhile,,and,to,become,even,more,so,with,the,higher,dimensional,models.,Given,the,obvious,need,to,couple,models,of,different,dimensionality,,the,linked,GP,approach,to,producing,surrogates,described,external,NEPTUNE,report,[6],(see,also,ref,[24]),appears,worth,pursuing.,There,is,also,a,strong,suggestion,from,the,discussion,[1],that,active,subspace,methods,be,con-,sidered,to,help,produce,surrogates.,Clearly,UQ,is,an,area,where,there,is,scope,for,the,devel-,opment,of,significant,new,techniques,,indicating,that,it,is,worth,supporting,a,post-doc,if,only,in,order,to,keep,abreast,of,the,latest,research.,Since,,there,is,lacking,a,convincing,demonstration,at,scale,of,the,proposed,techniques,,a,new,grant,could,usefully,direct,a,post-doc,to,explore,prac-,tical,details,of,implementation.,These,could,also,include,the,use,of,PCA,aka,POD,in,application,to,time-dependent,problems,,as,described,in,the,closing,chapters,of,the,book,by,Brunton,and,Kutz,[25,,§,11,12].,Probably,there,is,no,particular,worry,regarding,PCA,itself,as,it,seems,to,be,so,widely,used.,Application,to,time-dependent,problems,is,already,being,pursued,by,a,UKAEA,spon-,sored,research,student,at,Leeds,(Alasdair,Roy).,One,technique,for,identifying,parameters,which,is,known,to,scale,because,of,its,use,in,weather,and,climate,studies,is,Data,Assimilation,(DA),,discussed,briefly,in,internal,project,NEPTUNE,report,[3,,§,3],,specifically,by,means,of,the,En-,semble,Kalman,Filter,(EnKF).,EnKF,does,however,rely,on,having,good,starting,estimates,for,the,parameters,,now,discoverable,by,an,optimisation,front-end,in,the,UKAEA,sponsored,EnKF-Python,software,[26],,and,also,at,least,crude,estimates,of,the,uncertainties,in,the,parameters.,A.,Roy,is,exploring,the,possibility,of,using,algorithms,for,SINDy,-,Sparse,Identification,of,Nonlinear,Dynam-,ics,[25,,§,7.3],-,to,help,provide,the,information,required,by,DA.,(Exa)scalability,will,need,ultimately,to,be,considered,for,use,in,NEPTUNE.,There,has,also,been,consideration,of,the,use,of,adjoints,,see,discussion,in,NEPTUNE,report,[27,,§,2.4.1],of,their,usefulness,and,economy,for,design,,and,discussion,in,NEPTUNE,report,[3,,§,3.2],of,their,use,in,calculating,functional,derivatives.,Again,,there,is,the,issue,of,scalability,to,consider.,21,However,,so,as,not,to,be,too,prescriptive,,there,are,other,aspects,of,UQ,that,would,withstand,further,examination,,such,the,utility,of,other,sparse,approximation,techniques,,eg.,Proper,gen-,eralized,decomposition,(PGD),described,in,ref,[28],as,example,of,use,of,tensor,train,ideas,and,already,mentioned,in,previous,call,documents,,along,with,the,Discrete,Empirical,Interpolation,Method,(DEIM).,In,particular,,a,method,would,be,useful,for,selecting,automatically,amongst,the,various,possible,techniques,,having,regard,to,application,and,resources,available.,Acknowledgement,The,support,of,the,UK,Meteorological,Office,and,Strategic,Priorities,Fund,is,acknowledged.,References,[1],E.,Threlfall,and,W.,Arter.,Assessment,of,which,UQ,methods,are,required,to,make,NEPTUNE,software,actionable.,Technical,Report,CD/EXCALIBUR-FMS/0035-M2.4.1,,UKAEA,,2021.,[2],W.,Arter,and,E.,Threlfall.,Selection,of,techniques,for,Uncertainty,Quantification,(in,prepara-,tion).,Technical,Report,CD/EXCALIBUR-FMS/00XX-M2.4.2,,UKAEA,,2021.,[3],E.,Threlfall,and,W.,Arter.,Select,techniques,for,MOR,(Model,Order,Reduction).,Technical,Report,CD/EXCALIBUR-FMS/0031-M2.5.1,,UKAEA,,2021.,[4],W.,Arter,,E.,Threlfall,,and,J.,Parker.,Report,on,user,layer,design,for,Uncertainty,Quantification.,Technical,Report,CD/EXCALIBUR-FMS/0024-M3.1.3,,UKAEA,,2020.,[5],M.,Vassaux,,W.,Edeling,,and,P.,V.,Coveney.,Review,of,methods,and,toolkits,for,uncertainty,quantification,of,single,and,coupled-model,applications.,Technical,Report,2047352,1-TN-01-,4,,UKAEA,Project,Neptune,,2021.,[6],D.,Ming,and,S.,Guillas.,Report,on,suitability,and,potential,of,ROM,to,fusion,models,:,A,Non-intrusive,ROM,for,Solvers,with,High-dimensional,Outputs.,Technical,Report,2047352,2-,TN-01,,UKAEA,Project,Neptune,,2021.,[7],R.C.,Smith.,Uncertainty,Quantification:,Theory,,Implementation,,and,Applications.,SIAM,,2014.,[8],W.,Arter.,Numerical,simulation,of,magnetic,fusion,plasmas.,Reports,on,Progress,in,Physics,,58:1–59,,1995.,http://dx.doi.org/10.1088/0034-4885/58/1/001.,[9],C.,de,Boor.,A,practical,guide,to,splines.,Springer,,New,York,,1978.,[10],J.P.,Boyd.,Chebyshev,and,Fourier,Spectral,Methods.,Dover,Publications,,Mineola,,NY,,2001.,http://www-personal.umich.edu/,jpboyd/BOOK,Spectral2000.html.,[11],J.,Stoer,and,R.,Bulirsch.,Introduction,to,Numerical,Analysis,,3rd,Edition.,Springer,,2002.,22,[12],B.,Fornberg,and,N.,Flyer.,A,Primer,on,Radial,Basis,Functions,with,Applications,to,the,Geo-,sciences.,SIAM,,2015.,[13],M.,Farge,and,K.,Schneider.,Wavelet,transforms,and,their,applications,to,mhd,and,plasma,turbulence:,a,review.,Journal,of,Plasma,Physics,,81(6):435810602,,2015.,[14],P.,Challenor.,Private,communication.,Technical,Report,2049XXX-TN-01,,UKAEA,Project,Neptune,,2021.,[15],X.,Liu,and,S.,Guillas.,Dimension,reduction,for,Gaussian,process,emulation:,An,applica-,tion,to,the,influence,of,bathymetry,on,tsunami,heights.,SIAM/ASA,Journal,on,Uncertainty,Quantification,,5(1):787–812,,2017.,[16],B.,Dudson,,P.,Hill,,D.,Dickinson,,J.,Parker,,A.,Allen,,G.,Breyiannia,,J.,Brown,,L.,Easy,,S.,Farley,,B.,Friedman,,E.,Grinaker,,O.,Izacard,,I.,Joseph,,M.,Kim,,M.,Leconte,,J.,Leddy,,M.,Liten,,C.,Ma,,J.,Madsen,,D.,Meyerson,,P.,Naylor,,S.,Myers,,J.,Omotani,,T.,Rhee,,J.,Sauppe,,K.,Savage,,H.,Seto,,D.,Schwrer,,B.,Shanahan,,M.,Thomas,,S.,Tiwari,,M.,Umansky,,N.,Walkden,,L.,Wang,,Z.,Wang,,P.,Xi,,T.,Xia,,X.,Xu,,H.,Zhang,,A.,Bokshi,,H.,Muhammed,,M.,Estarellas,,and,F.,Riva.,BOUT++,v4.3.2,,March,2020.,[17],B.D.,Dudson.,BOUT++,website.,https://boutproject.github.io/,,2020.,Accessed:,June,2020.,[18],P.,A.,Hill,,J.,T.,Parker,,D.,Dickinson,,and,B.,D.,Dudson.,BOUT++,VECMA,hackaton,repository.,https://github.com/boutproject/VECMA-hackathon,,2021.,Accessed:,June,2021.,[19],Thomas,Gerstner,and,Michael,Griebel.,Dimension–adaptive,tensor–product,quadrature.,Computing,,71(1):65–87,,2003.,[20],D.,Moxey,,C.D.,Cantwell,,Y.,Bao,,A.,Cassinelli,,G.,Castiglioni,,S.,Chun,,E.,Juda,,E.,Kazemi,,K.,Lackhove,,J.,Marcon,,et,al.,Nektar++:,enhancing,the,capability,and,application,of,high-fidelity,spectral/hp,element,methods.,Computer,Physics,Communications,,249:107110,,2020.,[21],D.,Moxey,et,al.,Nektar++,website.,https://www.nektar.info,,2020.,Accessed:,June,2020.,[22],J.W.,Elder.,Numerical,experiments,with,free,convection,in,a,vertical,slot.,Journal,of,Fluid,Mechanics,,24:823–843,,1965.,[23],W.,Arter.,Potential,Vorticity,Formulation,of,Compressible,Magnetohydrodynamics.,Physi-,cal,Review,Letters,,110(1):015004,,2013.,http://dx.doi.org/10.1103/PhysRevLett.110.,015004.,[24],D.,Ming,and,S.,Guillas.,Linked,Gaussian,Process,Emulation,for,Systems,of,Computer,Models,using,Mat,´ern,Kernels,and,Adaptive,Design.,arXiv,preprint,arXiv:1912.09468,,2021.,[25],S.L.,Brunton,and,J.N.,Kutz.,Data-driven,science,and,engineering:,Machine,learning,,dynam-,ical,systems,,and,control.,CUP,,2019.,23,[26],W.,Arter,,A.,Osojnik,,C.,Cartis,,G.,Madho,,C.,Jones,,and,S.,Tobias.,Data,assimilation,ap-,In,2018,IEEE,International,proach,to,analysing,systems,of,ordinary,differential,equations.,Symposium,on,Circuits,and,Systems,(ISCAS),,Florence,,Italy,,number,8351751,,27-30,May,2018.,http://dx.doi.org/10.1109/ISCAS.2018.8351751.,[27],W.,Arter,,L.,Anton,,and,D.,Samaddar.,Year,One,Summary,Report.,Technical,Report,CD/EXCALIBUR-FMS/0011-1.00-M1.2.1,,UKAEA,,2020.,[28],F.,Chinesta,,R.,Keunings,,and,A.,Leygue.,The,Proper,Generalized,Decomposition,for,Ad-,vanced,Numerical,Simulations:,A,Primer.,Springer,Briefs,in,Applied,Science,and,Technology,,2014.,A,Example,BOUT++,decoder,def,__init_subclass__,(,cls,):,super,().,__init_subclass__,(,cls,.,__name__,),Filename,or,glob,to,read,in,(,default,:,",BOUT,.,dmp,.*.,nc,"),""",self,.,target_filename,=,target_filename,or,",BOUT,.,dmp,.*.,nc,",def,__init__,(,self,,,target_filename,=,None,):,@staticmethod,def,_get_output_path,(,run_info,=,None,,,outfile,=,None,):,""",Parameters,==========,target_filename,:,str,or,None,1,class,BaseBOUTDecoder,(,BaseDecoder,,,decoder_name,=",bout,++,-,base,"):,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,raise,RuntimeError,(",Passed,’,None,’,to,’,run_info,’"),run_path,=,run_info,.,get,(",run_dir,",,,",data,"),if,not,os,.,path,.,isfile,(,settings_filename,):,""",Get,the,path,the,run,directory,,,and,optionally,the,file,outfile,""",if,run_info,is,None,:,return,os,.,path,.,join,(,run_path,,,outfile,),def,sim_complete,(,self,,,run_info,=,None,):,if,not,os,.,path,.,isdir,(,run_path,):,return,False,""",Return,True,if,the,simulation,has,finished,""",settings_filename,=,self,.,_get_output_path,(,run_info,,,",BOUT,.,settings,"),raise,RuntimeError,(,f,",Run,directory,does,not,exist,:,{,run_path,}"),24,#,Check,for,normal,,,clean,finish,settings_file,=,BoutOptionsFile,(,settings_filename,),return,",run,:,finished,",in,settings_file,""",Read,the,BOUT,++,outputs,into,an,xarray,dataframe,""",data_files,=,self,.,_get_output_path,(,run_info,,,self,.,target_filename,),return,open_boutdataset,(,data_files,,,info,=,False,),""",Serialise,the,class,to,a,dict,of,JSON,serialisable,variables,""",#,This,is,possibly,too,magic,...,get,the,variable,names,from,#,the,__init__,method,init_vars,=,inspect,.,signature,(,self,.,__init__,).,parameters,.,keys,(),#,Assuming,that,the,__init__,does,‘,self,.,var,=,var,‘,,,we,can,#,just,return,a,dict,of,everything,try,:,return,{,var,:,getattr,(,self,,,var,),for,var,in,init_vars,},#,Let,’,s,try,to,give,a,helpful,error,message,if,this,does,go,wrong,match,=,re,.,search,(,r,",no,attribute,’(.*),’",,,str,(,e,)),if,match,is,None,:,def,get_restart_dict,(,self,):,except,AttributeError,as,e,:,def,get_outputs,(,self,,,run_info,):,#,Oops,,,something,else,raise,e,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,class,SimpleBOUTDecoder,(,BaseBOUTDecoder,):,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,""",Parameters,==========,variables,:,iterable,or,None,@staticmethod,def,element_version,():,return,"0.1.0",varname,=,match,.,group,(1),raise,AttributeError,(,df,=,self,.,get_outputs,(,run_info,),),25,""",Just,collects,individual,variables,at,the,last,timestep,""",def,__init__,(,self,,,target_filename,=,None,,,variables,=,None,):,Iterable,of,variables,to,collect,from,the,output,.,If,None,,,return,everything,""",super,().,__init__,(,target_filename,=,target_filename,),#,TODO,:,check,it,’,s,an,iterable,or,otherwise,sensible,type,?,self,.,variables,=,variables,def,parse_sim_output,(,self,,,run_info,=,None,,,*,args,,,**,kwargs,):,f,",’{,varname,},’,doesn,’,t,seem,to,be,a,member,of,this,instance,.\,nDid,you,miss,’,self,.{,varname,},=,{,varname,},’,in,’,__init__,’?",88,89,90,91,92,93,94,95,96,return,{,variable,:,f,l,a,t,t,e,n,_,d,a,t,a,f,r,a,m,e,_,f,o,r,_,J,S,O,N,(,df,[,variable,][,-1,,,...]),for,variable,in,self,.,variables,},@staticmethod,def,element_version,():,return,"0.1.0",B,Example,BOUT++,encoder,""",Read,an,existing,BOUT,.,inp,file,to,use,as,a,template,.,If,no,input,is,given,,,an,empty,set,of,options,will,be,created,Example,-------,from,boutvecma,.,encoder,import,BOUTEncoder,encoder,=,BOUTEncoder,(",data,/,BOUT,.,inp,"),""",if,template_input,:,self,.,_options,=,BoutOptionsFile,(,template_input,),else,:,self,.,_options,=,BoutOptions,(),self,.,template_input,=,template_input,def,__init__,(,self,,,template_input,=,None,):,def,encode,(,self,,,params,=,None,,,target_dir,=""):,1,class,BOUTEncoder,(,BaseEncoder,,,encoder_name,=",bout,++"):,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,options,=,deepcopy,(,self,.,_options,),for,key,,,value,in,params,.,items,():,options,=,self,.,_options,options,[,key,],=,value,f,.,write,(,str,(,options,)),else,:,""",Create,a,BOUT,.,inp,file,in,target_dir,with,modified,parameters,.,Sub,-,sections,are,specified,using,colons,in,the,params,keys,Example,-------,encoder,.,encode,({",section,:,key,":42},,,target_dir,=",data,"),modifies,key,so,that,the,file,",data,/,BOUT,.,inp,",contains,[,section,],key,=,42,""",if,params,:,#,Note,:,Here,options,could,be,BoutOptions,or,BoutOptionsFile,#,so,can,’,t,just,use,options,.,write,with,open,(,os,.,path,.,join,(,target_dir,,,",BOUT,.,inp,"),,,",w,"),as,f,:,26,39,40,41,42,43,44,def,element_version,(,self,):,return,"0.1",def,get_restart_dict,(,self,):,return,{",template_input,":,self,.,template_input,},27

:pdfembed:`src:_static/CD-EXCALIBUR-FMS0040-M5.1_ManagementExternalResearchSupportsUqProcurement.pdf, height:1600, width:1100, align:middle`