TN-08_ReviewAdjointMethodPlasmaPhysicsNeptune
=============================================

.. meta::
   :description: technical note

   :keywords: Review,of,adjoint,method,in,plasma,physics,for,NEPTUNE,Technical,Report,2068625-TN-08,Deliverable,5.1,Josh,Williams∗,Ubaid,Ali,Qadri†,Sue,Thorne‡,July,2023,(Revised,September,2023),1,Introduction,Tokamak,reactors,aim,to,use,confined,plasma,to,create,energy,by,nuclear,fusion,[57].,Due,to,the,complex,combination,of,interactions,at,multiple,length-scales,,flow,instabilities,can,arise,that,cause,reduced,per-,formance,and,damage,to,the,reactor’s,walls,[41,,48].,The,interplay,between,multiple,interacting,design,and,operating,parameters,creates,a,difficult,optimisation,problem,to,maximise,power,output,of,future,tokamak,reactors.,These,parameters,are,also,likely,to,have,some,degree,of,uncertainty,,which,will,need,to,be,quantified,by,future,users,of,software,in,the,NEPTUNE,codebase.,Additionally,,the,complex,plasma,flow,is,prone,to,instabilities,that,limit,the,confinement,achieved,by,tokamaks.,Identifying,the,onset,of,instabilities,and,developing,passive/active,control,strategies,is,therefore,crucial,for,the,success,of,nuclear,fusion,reactors.,The,adjoint,method,has,been,used,extensively,in,stability,analysis,,uncertainty,quantifi-,cation,,optimisation,and,control,problems,[37,,43,,64].,Sensitivity,information,obtained,from,the,adjoint,method,is,also,useful,for,adaptive,mesh,refinement,[4].,This,report,aims,to,review,the,adjoint,method,to,determine,how,it,may,be,used,in,optimising,plasma,flows,within,the,NEPTUNE,project.,This,report,is,structured,as,follows.,In,Section,2,,we,present,the,background,and,theory,of,the,adjoint,method.,We,then,compare,different,methods,of,computing,the,adjoint,variables,,which,can,be,done,using,a,continuous,or,discrete,adjoint,method,(Section,2.1).,We,discuss,some,recent,advancements,in,algorithms,used,to,solve,the,adjoint,equations,in,Section,2.2.,In,Section,3,,we,review,a,variety,of,applications,of,the,adjoint,method,in,the,literature,that,are,relevant,to,NEPTUNE.,This,includes,(i),sensitivity,and,stability,analysis,,(ii),parameter,optimisation,and,sensitivity,,(iii),uncertainty,quantification,,(iv),adaptive,mesh,refinement,and,(v),shape,optimisation.,Finally,,in,Section,4,,we,draw,our,conclusions.,2,Adjoint,method,The,adjoint,method,computes,the,sensitivity,of,a,cost,function,(based,on,a,quantity,of,interest),relative,to,a,change,in,any,relevant,parameters,(operating,conditions,,design,parameters,,modelling,parameters),[42].,This,information,can,be,used,to,optimise,parameters,at,a,significantly,lower,cost,than,iteratively,performing,simulations,in,a,non-intrusive,way.,As,explained,by,Giles,and,Pierce,[29],,the,theory,behind,the,adjoint,method,is,easiest,understood,in,the,context,of,a,linear,problem.,In,the,series,of,problems,discussed,in,this,report,,we,are,aiming,to,find,the,change,in,some,cost,function,(or,quantity,of,interest),,J,,,relative,to,a,small,perturbation,in,the,operating,∗Josh,Williams,is,with,the,Hartree,Centre,,STFC,Royal,Observatory,Edinburgh,,Blackford,Hill,,Edinburgh,,EH9,3HJ,,UK.,†Ubaid,Ali,Qadri,is,with,the,Hartree,Centre,,STFC,Daresbury,Laboratory,,Daresbury,Science,and,Innovation,Campus,,Warrington,,Cheshire,WA4,4AD.,Email,contact:,ubaid.qadri@stfc.ac.uk,‡Sue,Thorne,is,with,the,Hartree,Centre,,STFC,Rutherford,Appleton,Laboratory,,Harwell,Campus,,Didcot,,OX11,0QX,,UK.,Email,contact:,sue.thorne@stfc.ac.uk,1,parameters,p,(dJ/dp),which,will,change,the,flow,field.,The,state,vector,containing,different,flow,field,variables,is,q,,the,perturbed,state,vector,is,q,,,and,the,perturbed,cost,function,is,′,′,gT,q,=,∂J,∂q,′,q,,,(1),where,g,is,a,vector,describing,the,cost,function,gradient.,Therefore,,we,are,aiming,to,find,gT,q,′,,,where,the,perturbed,state,vector,is,described,by,the,linear,problem,where,L,is,a,linear,operator,,and,f,is,a,vector,acting,as,a,source,term.,The,dual,of,this,problem,being,′,Lq,=,f,=,′,∂q,∂t,,,(2),where,q+,is,the,state,vector,containing,the,adjoint,solutions.,The,equivalence,between,(2),and,(3),is,LT,q+,=,g,,(cid:0)q+(cid:1)T,f,=,(cid:0)q+(cid:1)T,′,Lq,=,(cid:16),LT,q+(cid:17)T,′,q,=,gT,q,′,,,which,is,simply,(cid:0)q+(cid:1)T,f,=,gT,q,′,.,The,sensitivity,of,J,to,a,set,of,parameters,can,be,found,by,the,chain,rule,[43],dJ,dp,=,∂J,∂p,+,∂J,∂q,dq,dp,.,(3),(4),(5),(6),The,issue,is,that,computing,dq/dp,is,very,expensive,when,the,dimension,of,p,is,large.,Hence,,the,challenge,is,then,how,to,compute,this,without,directly,solving,dq/dp,for,all,values,of,p.,We,know,that,the,state,vector,must,satisfy,some,governing,equations,that,vary,in,space,based,on,the,flow,parameters,(R(q,,p),=,0),,where,R,is,the,residual,of,the,governing,equations.,The,term,dq/dp,should,also,satisfy,dR,dp,=,∂R,∂p,+,∂R,∂q,dq,dp,=,0.,In,the,explanation,of,Giles,and,Pierce,[29],,(7),is,related,to,the,linear,variables,in,(4),by,′,q,=,L,=,gT,=,,,dq,dp,∂R,∂q,∂J,∂q,,,,,which,can,be,combined,with,(1),and,(4),to,give,f,=,−,∂R,∂p,,,dJ,dp,=,=,∂J,∂p,∂J,∂p,′,+,gT,q,+,(cid:0)q+(cid:1)T,f,.,For,multiple,design/operating,parameters,(p),,we,note,that,each,design,variable,has,a,different,f,but,the,same,g.,Therefore,,instead,of,solving,multiple,equations,of,the,form,Lq,=,f,and,using,(12),,the,computational,cost,can,be,significantly,reduced,by,solving,the,single,equation,LT,q+,=,g,to,compute,q+,and,then,use,(13),,which,is,known,as,the,adjoint,method.,′,2,(7),(8),(9),(10),(11),(12),(13),Figure,1:,Gradient,error,,ε,,compared,to,step-size,,α,,for,discrete,adjoint,with,varying,floating,point,precision,,and,continuous,adjoint,(64-bit,precision),[77].,2.1,Continuous,or,discrete,adjoint,method,The,gradients,can,be,obtained,by,(i),the,continuous,method,,where,the,user,derives,the,derivatives,of,the,governing,equations,analytically,prior,to,discretisation,or,(ii),the,discrete,method,,where,the,derivatives,are,calculated,after,the,governing,equations,are,discretised,[54],,which,is,simpler,to,implement,[43].,In,the,latter,,the,exact,gradients,can,be,computed,to,machine,precision,[43].,Since,the,discrete,adjoint,method,computes,derivatives,from,the,discretised,governing,equations,,we,can,ensure,no,inconsistencies,are,introduced,to,the,simulations,during,discretisation.,This,property,is,extremely,attractive,for,chaotic,turbulent,flows,[9],(such,as,might,be,found,in,tokamak,reactors),,since,minor,inaccuracies,can,propa-,gate/amplify,to,large,errors,[38,,77].,In,aerodynamic,optimisation,,it,has,been,shown,that,the,continuous,adjoint,method,approaches,the,exact,gradients,when,the,mesh,is,sufficiently,resolved,(as,discretisation,error,decreases),[52,,53].,However,,for,aeroacoustic,control,of,a,compressible,flow,mixing,layer,,it,was,shown,by,Vishnampet,et,al.,that,the,continuous,adjoint,gradient,error,saturated,well,above,the,error,of,numerical,precision,(continuous,adjoint,error,was,O(10−8),compared,to,O(10−12),for,discrete,adjoint,,both,with,64,bit,precision,,Figure,1),[77].,This,may,be,attributed,to,the,complexity,and,chaotic,nature,of,the,compressible,flow,which,amplifies,errors,,or,the,low,amplitude,of,the,pressure,fluctuations,respon-,sible,for,aeroacoustic,noise,creating,gradients,that,are,highly,susceptible,to,discretisation,errors,[77].,Importantly,,the,authors,mention,that,even,errors,introduced,by,the,numerical,imprecision,in,the,dis-,crete,adjoint,will,eventually,accumulate,to,corrupt,the,solution,for,longer,simulations,due,to,the,chaotic,nature,of,turbulence,[77].,In,stellarator,parameter,optimisation,,the,error,from,the,continuous,adjoint,was,found,to,be,around,one,order,of,magnitude,larger,than,the,discrete,adjoint,due,to,discretisation,error,[60].,However,,the,authors,state,that,the,continuous,adjoint,error,is,still,very,small,(≈,0.1%),and,acceptable,for,their,application.,This,suggests,that,discretisation,errors,in,the,continuous,adjoint,are,broadly,negligible,except,in,extremely,chaotic,flows.,Comparing,the,discrete,and,continuous,adjoint,methods,,Chandler,[11],showed,qualitatively,the,same,global,modes,in,sensitivity,analysis,of,non-isothermal,low,density,jets,,see,Figure,2.,However,,Chandler’s,results,showed,some,numerical,artefacts,for,the,discrete,adjoint,method,that,were,not,present,in,the,continuous,adjoint,formulation.,Chandler,also,reported,that,the,continuous,adjoint,formulation,converged,faster,than,the,discrete,adjoint,but,with,increased,debugging,issues.,2.2,Adjoint,algorithm,developments,Although,the,adjoint,equation,is,significantly,less,expensive,than,computing,sensitivities,with,finite,differences,,in-practice,it,can,be,computationally,expensive,to,solve.,Here,we,review,recent,algorithmic,3,(a),(b),Figure,2:,Real,part,of,most,unstable,global,mode,from,a,(a),continuous,adjoint,and,(b),discrete,adjoint,,in,simulations,performed,by,Chandler,[11].,4,developments,improving,the,ease,of,implementing,and,solving,adjoint,problems.,2.2.1,Parallel-in-time,The,iterative,nature,of,forward-backward,computations,in,the,adjoint,method,can,create,a,compu-,tationally,expensive,process.,Recently,,a,parallel-in-time,algorithm,for,linear,[23],and,non-linear,[24],initial-value,problems,was,extended,to,improve,performance,of,adjoint,optimisation,[14,,71].,The,orig-,inal,algorithm,decomposes,a,linear,initial-value,problem,into,a,homogeneous,and,inhomogeneous,part.,The,homogeneous,equation,can,be,integrated,with,significantly,greater,speed,and,efficiency.,The,inho-,mogeneous,part,(usually,dependent,on,a,source,term),is,decomposed,in,time,and,distributed,to,each,processor,[23].,The,parallel-in-time,adjoint,method,was,tested,on,a,forced,advection-diffusion,equation,(linear),and,harmonically-forced,Burgers,equation,(non-linear),[71].,This,method,was,recently,extended,to,solve,the,adjoint,equation,in,problems,governed,by,the,incompressible,Navier-Stokes,equations,[14].,However,,it,is,not,yet,clear,if,this,approach,is,suitable,for,multiscale,,coupled,problems,such,as,plasma,transport,in,tokamaks.,Furthermore,,the,parallel-in-time,approach,would,require,significant,modification,of,the,codes,developed,throughout,the,NEPTUNE,project,thus,far.,Another,ExCALIBUR-funded,project,is,currently,working,on,advancing,parallel-in-time,algorithms,and,determining,optimal,problem-specific,parallel-in-time,approaches,for,different,applications,such,as,weather,and,fusion,modelling,[22],,which,may,be,useful,in,the,future.,2.2.2,Checkpointing,unsteady,adjoint,problems,One,major,challenge,in,using,adjoint,optimisation,of,time-dependent,problems,is,that,the,adjoint,equation,requires,knowledge,of,a,state,variable,computed,from,the,primal,solution,,u,,and,its,adjoint,,u∗,,at,each,time,step,[56].,For,long,time,series,,this,is,not,feasible,as,the,full,time-series,of,the,primal,solution,must,either,be,(1),stored,in,memory,and,accessed,at,each,step,of,the,adjoint,calculation,,which,may,not,be,possible,for,simulations,with,a,long,time,series,or,many,degrees,of,freedom,[30],,or,(2),written,to,the,disk,and,read,at,each,step,,which,is,prohibitively,slow.,To,counteract,this,,checkpointing,algorithms,can,be,used,such,as,revolve,[30],,which,saves,checkpoints,at,a,pre-defined,time,interval,(based,on,available,memory,and,number,of,timesteps),,and,the,intermediate,timesteps,are,recomputed.,For,simulations,where,the,number,of,timesteps,is,not,known,a,priori,,an,adaptive,checkpointing,scheme,must,be,used,(referred,to,as,‘online’,checkpointing),[33,,74].,A,checkpointing,algorithm,has,been,implemented,in,the,spectral,element,code,Nek5000,[66].,The,authors,also,improved,the,standard,revolve,algorithm,to,include,disk,I/O,checkpoint,(two-level,checkpointing),,which,significantly,reduced,the,number,of,recomputed,forward,steps,in,simulations,with,a,large,number,of,timesteps,[66].,The,lower,number,of,recomputation,is,only,beneficial,if,the,I/O,time,is,not,prohibitively,large,[66].,The,I/O,time,can,be,reduced,by,using,efficient,file,formats,such,as,HDF5,[79].,A,two-level,checkpointing,algorithm,based,on,revolve,has,been,implemented,in,PETSc:,the,implementation,was,described,in,[45,,80].,Recently,,an,alternative,method,to,checkpointing,was,proposed,that,uses,the,incremental,singular,value,decomposition,(iSVD),to,store,a,reduced-order,model,of,the,primal,solution,[39,,76].,In,this,method,,snapshots,of,the,flow,are,added,to,the,SVD,at,each,time,step.,Since,the,rank,of,the,SVD,is,reduced,,only,storing,the,main,modes,of,the,flow,,the,memory,consumption,is,significantly,reduced,and,this,gives,improved,performance,(CPU,time),compared,to,checkpointing,with,the,same,memory,consumption.,In,contrast,to,the,checkpointing,method,where,time-instants,of,an,exact,primal,field,are,stored,in,memory,and,the,remaining,time-series,is,re-computed,(sacrificing,CPU,time),,using,iSVD,does,not,require,expen-,sive,re-computations,since,an,approximation,of,the,primal’s,full,time-series,can,be,stored.,This,method,obviously,introduces,some,error,due,to,the,reduced,amount,of,information,contained,in,the,primal,solu-,tion,but,it,has,been,shown,that,this,has,very,little,influence,on,the,objective,function,[76].,For,example,,when,performing,a,shape,optimisation,study,,Vezyris,et,al.,found,the,iSVD,approach,to,be,within,0.01%,of,the,objective,function,obtained,with,checkpointing,but,with,a,29%,lower,CPU,time,and,0.5%,of,the,memory,overhead,[76].,Reduced-order,modelling,can,be,leveraged,using,the,recently,developed,libROM,library,from,Lawrence,Livermore,National,Lab,[12],which,has,an,iSVD,implementation.,However,,there,is,little,work,published,using,libROM,and,,therefore,,its,performance,level,is,unknown.,5,3,Applications,of,adjoint,method,relevant,to,plasma,modelling,The,adjoint,method,has,been,applied,to,many,problems,in,fluid,mechanics.,Many,of,the,studies,discussed,crossover,multiple,applications,because,topics,such,as,parameter,optimisation,often,make,use,of,the,adjoint,method’s,ability,to,interpret,sensitivity,and,stability,properties,for,understanding,optimisation,results.,The,adjoint,sensitivity,field,is,also,a,highly,useful,error,indicator,for,adaptive,mesh,refinement.,In,this,section,,we,will,discuss,various,applications,of,the,adjoint,method,in,the,literature.,3.1,Sensitivity,and,stability,analysis,A,suitable,toolkit,to,extract,information,on,the,response,of,a,base,flow,to,a,perturbation,is,stability,,receptivity,and,sensitivity,analysis,[68].,The,stability,can,be,computed,by,obtaining,the,eigenvalues,of,the,direct,problem,,which,is,done,by,first,expressing,the,perturbed,state-vector,using,a,Fourier,decomposition,′,q,=,eλtˆq,(14),where,the,notation,ˆ·,represents,a,complex,vector,field,that,describes,the,spatial,structure,of,the,pertur-,bation,(eigenvectors,of,L),,and,λ,=,σ,+,iω,is,a,complex,scalar,describing,the,temporal,evolution,of,the,perturbation,(eigenvalues,of,L),,with,a,growth,rate,,ω,,and,a,frequency,,ω.,By,substituting,(14),into,(2),,we,arrive,at,the,eigenvalue,problem,Lˆq,=,λˆq,,(15),which,represents,the,‘direct,global,modes’.,If,σ,>,0,(or,Re(λ),>,0),for,any,single,mode,,the,flow,is,asymptotically,unstable,[70].,These,instabilities,can,be,absolute,(growing,from,a,fixed,point,in,space),or,convective,(growing,in,magnitude,but,propagating,in,space),[34].,The,growth,of,convective,instabilities,was,the,subject,of,several,works,by,Blackburn,et,al.,[5–7].,In,stability,problems,,J,is,based,on,the,amplification,of,energy,(energy,at,time,T,compared,to,energy,in,the,base,flow),[67].,From,the,adjoint,problem,,we,obtain,the,receptivity,,which,represents,the,flow’s,response,to,(external),open-loop,forcing,[43].,Similar,to,the,direct,method,(15),,the,adjoint,global,modes,can,be,expressed,as,[47],(16),L+ˆq+,=,λ+ˆq+,,where,L+,is,the,adjoint,of,L.,When,both,the,direct,and,adjoint,global,modes,are,stable,but,short-term,instabilities,are,present,,this,suggests,that,the,direct,and,adjoint,operators,are,non-normal,[70].,The,(‘structural’),sensitivity,is,then,computed,as,the,product,of,the,stability,and,receptivity,,and,can,be,used,to,obtain,information,on,instabilities,such,as,the,location,of,a,‘wavemaker’,[28].,The,wavemaker,identifies,the,flow,region,where,a,change,in,problem’s,structure,would,produce,the,largest,eigenvalue,drift.,This,structural,sensitivity,is,computed,simply,as,S,=,ˆqˆq+,[62].,Qadri,used,the,Frobenius,norm,of,the,tensor,to,identify,the,wavemaker,region,but,stated,that,each,component,of,the,tensor,should,be,visualised,to,better,understand,the,flow’s,feedback,mechanism,to,perturbations,[62].,Giannetti,and,Luchini,[28],studied,the,wavemaker,region,in,a,low,Reynolds,number,flow,over,a,cylinder,near,the,critical,Reynolds,number,(Recrit,≈,47),,where,the,steady,structure,of,the,flow,is,broken.,They,compared,the,structural,sensitivity,field,with,previously,reported,experimental,data,[73],that,showed,the,placement,of,a,small,control,cylinder,a,few,diameters,downstream,of,the,wake,can,completely,suppress,the,onset,of,unsteadiness.,The,exact,recommended,location,of,the,control,cylinder,was,identified,as,being,within,the,separation,bubble,from,the,structural,sensitivity,,providing,a,theoretical,explanation,for,the,instability,suppression,observed,experimentally,that,was,not,possible,from,analysing,the,adjoint,or,direct,modes,alone,[28].,This,approach,has,been,used,in,many,studies,since,the,original,development,[28],,including,an,extension,to,3D,flow,over,a,cylinder,with,a,periodic,base,flow,[27],and,other,geometries,such,as,flow,over,an,open,cavity,[13].,Citro,et,al.,[13],considered,stability,analysis,of,flow,over,an,open,cavity.,They,evaluated,adjoint,eigenmodes,to,show,the,region,where,an,instability,is,most,receptive,to,momentum,forcing,(Figure,3,right).,They,then,computed,the,product,of,the,direct,and,adjoint,fields,to,find,the,structural,sensitivity,and,the,wavemaker,that,followed,a,streamline,around,the,cavity,(Figure,3,bottom).,Clearly,,the,sensitivity,field,is,strongest,where,the,adjoint,and,direct,modes,overlap.,Murali,[49],studied,hydrodynamic,and,magnetohydrodynamic,(MHD),flow,in,cooling,blanket,ducts,with,repeated,wedge-shaped,protrusions,,relevant,for,fusion,reactors.,They,computed,the,structural,sensitivity,and,growth,rate,of,perturbations,in,the,flow,[51],,this,was,then,used,to,later,study,the,generation,of,quasi-2D,vortices,to,enhance,heat,transfer,within,the,duct,at,various,Reynolds,numbers,[49,,50].,The,use,of,the,6,adjoint,method,to,evaluate,receptivity,and,(when,combined,with,the,direct,modes),structural,sensitivity,is,relevant,to,the,study,of,fusion,reactors,as,it,can,be,used,to,evaluate,areas,for,passive,or,active,controls,to,prevent,instabilities,that,would,hinder,the,confinement,[72].,Several,works,have,been,done,recently,using,the,adjoint,method,for,sensitivity,analysis,of,stellarator-,based,problems,[2,,60].,Paul,et,al.,used,the,adjoint,method,to,evaluate,the,sensitivity,of,ion,particle,flux,and,bootstrap,current,to,changes,in,the,Fourier,amplitudes,of,the,magnetic,field,strength,[60].,The,same,group,also,studied,shape,gradients,of,MHD,equilibria,within,a,stellarator,[2],,and,the,sensitivity,of,magnetic,island,size,to,magnetic,field,variations,[26].,Similar,methods,have,recently,been,applied,,where,Nies,et,al.,[55],derived,a,shape-gradient-like,sensitivity,function,to,quantify,quasi-symmetry,on,a,flux,surface,,with,the,aim,of,minimising,quasi-symmetry-breaking,modes,in,the,flow.,This,is,similar,to,the,study,of,Murali,on,quasi-2D,vortices,breaking,into,full,3D,flows,in,cooling,blanket,ducts,of,fusion,reactors,[49,,50].,3.2,Parameter,optimisation,and,sensitivity,When,experimental,data,is,available,,the,adjoint,method,has,been,used,to,tune,empirically-determined,parameters,in,plasma,edge-physics,models,[3].,The,authors,used,the,adjoint,method,to,attempt,to,optimise,coefficients,of,a,simple,transport,model,with,neutral,density,,ion,density,and,temperature,profiles,obtained,from,high,fidelity,simulations,(performed,with,B2-EIRENE,code,[69]).,This,did,not,appear,to,work,well,at,matching,profiles,of,plasma,in,the,edge,region,of,a,JET,configuration.,Particularly,,the,temperature,profiles,obtained,from,the,simplified,model,with,optimised,parameters,significantly,over-,estimated,the,high-fidelity,simulations.,This,was,likely,due,to,the,simplified,form,of,the,transport,model,used,[3].,Adjoint,methods,were,used,to,control,the,electron,profile,in,ITER,with,electron,cyclotron,current,drive,(ECCD),[32].,The,authors,varied,the,driving,current,beam,injection,angles,to,evaluate,the,current,density,sensitivity,and,determine,the,optimal,beam,position.,This,approach,can,be,used,to,stabilise,MHD,wave,modes.,In,tokamak,and,stellarator,reactors,,MHD,instabilities,can,arise,that,hinder,fusion,performance,by,re-,ducing,the,plasma,confinement,[72].,Paul,et,al.,[61],developed,an,adjoint,method,to,minimise,MHD,instabilities,in,stellarator,equilibria,,demonstrating,it,on,three,optimisation,problems:,(i),a,target,ro-,tational,transform,profile,that,would,enable,low,shear,flow,,(ii),creation,of,a,magnetic,well,,(iii),and,quasi-symmetry,on,a,magnetic,axis.,This,was,then,extended,by,Gaur,et,al.,[25],,who,focused,on,the,‘balloon,mode’,MHD,instability,in,a,fixed,geometry.,Specifically,,they,optimised,the,field,line,geometry,to,find,MHD,equilibria,that,were,stable,against,this,mode,by,minimising,the,maximum,eigenvalue.,Their,final,results,gave,a,maximum,eigenvalue,with,a,stable,growth,rate.,Blommaert,et,al.,[8],performed,sensitivity,analysis,of,uncertain,parameters,in,a,plasma,edge,model,in,a,tokamak,configuration.,Their,model,had,34,parameters,such,as,radial,transport,coefficients,,boundary,conditions,,rate,coefficients,(such,as,charge,exchange,rate),,and,parameters,controlling,the,magnetic,equilibrium,profile.,They,considered,two,cost,functions:,one,related,to,the,heat,load,and,the,other,related,to,the,temperature,on,the,outer,strike,point,of,the,divertor.,Their,sensitivity,analysis,showed,strong,dependence,of,both,objectives,on,the,boundary,conditions,and,equilibrium,profile,parameters.,Using,the,adjoint,method,,this,only,required,2,simulations,instead,of,34,(one,for,each,cost,function).,Their,sensitivities,were,also,validated,against,finite,difference,calculations.,In,reactive,flows,,the,adjoint,method,was,used,for,optimisation,of,spark,location,for,laminar,flame,ignition,by,Qadri,et,al.,[64].,As,is,common,in,gradient-based,methods,,they,showed,that,the,optimal,solution,is,dependent,on,the,initial,condition,[64].,Kord,and,Capecelatro,used,the,adjoint,method,to,determine,a,momentum,source,term,that,could,control,a,mixing,layer,,limiting,the,Kelvin-Helmholtz,instability,that,was,observed,in,the,baseline,solution,to,a,small,spatial,region,[36].,In,a,previous,study,,the,same,authors,used,the,adjoint,method,to,control,a,Rayleigh-Taylor,instability,[37].,Also,in,reactive,flows,,Qadri,et,al.,[63],studied,low,density,jets,and,their,global,modes.,They,used,an,adjoint,analysis,to,identify,regions,receptive,to,external,forcing,that,may,control,the,growth,rate/frequency,of,the,unstable,mode.,[44],studied,control,strategies,for,base,flow,perturbations,using,the,adjoint,method,in,a,Mao,et,al.,stenotic,flow,and,Batchelor,flow.,They,used,a,boundary,condition,that,varied,in,time,and,space,,given,by,uΓc,=,c(x)f,(t,,ω),,where,Γc,is,the,control,boundary,,c,is,the,spatial,change,in,the,velocity,amplitude,of,the,control,boundary,(to,be,optimised),,f,is,the,temporal,change,in,velocity,with,a,circular,frequency,ω.,The,cost,function,(6),was,based,on,the,kinetic,energy,of,the,flowfield,,which,was,minimised,with,7,Figure,3:,Contours,of,stability,(direct,mode,,left),,receptivity,(adjoint,mode,,right),and,structural,sensitivity,(bottom,row),in,flow,over,an,open,cavity,at,a,Reynolds,number,near,the,transition,from,2D,to,3D,flow.,Panels,show,(a,d),streamwise,direction,,(b,e),wall-normal,direction,,(c,f),spanwise,direction.,Figures,taken,from,Citro,et,al.,[13].,8,respect,to,the,amplitude,of,c.,In,tokamaks,,the,flow,may,depend,on,wall,temperature,,which,could,be,optimised/controlled,in,a,similar,way.,Related,to,parameter,optimisation,,the,adjoint,method,is,also,widely,used,in,numerical,weather,prediction,to,perform,four-dimensional,data,assimilation,[40].,This,was,discussed,in,a,previous,NEPTUNE,report,on,model,order,reduction,[21].,In,the,case,of,numerical,weather,prediction,,flow,data,of,varying,type,and,location,can,be,combined,with,a,parameterised,mathematical,model,to,determine,unknown,parameters,which,describe,the,flow.,The,unknown,parameters,are,determined,through,a,gradient-based,optimisation,process,where,the,cost,function,is,based,on,the,mean-square,error,of,the,model,and,data,[31].,The,assimilated,parameters,can,then,be,used,to,forecast,and,predict,future,weather,(or,more,generally,,future,flow,states).,Given,the,large,number,of,diagnostic,devices,in,tokamaks,which,provide,measurements,,the,use,of,adjoint,methods,for,data,assimilation,is,highly,relevant,to,NEPTUNE.,3.3,Uncertainty,quantification,Uncertainties,can,enter,complex,simulations,such,as,plasma,models,through,a,variety,of,sources,including,initial,conditions,,boundary,conditions,,geometrical,parameters,,modelling,errors,[20],,numerical,errors,,etc,[78].,The,latter,point,on,numerical,errors,is,also,relevant,for,adaptive,mesh,refinement,(Section,3.4).,Wang,[78],studied,flow,past,a,cylinder,,where,the,cylinder,is,subject,to,small,but,finite,,random,and,unknown,oscillations,in,its,angular,velocity,(around,a,mean,of,0),,given,as,ω(t),=,10,(cid:88),i=1,ξ2i−1,cos(2πifV,t),+,ξ2i,sin(2πifV,t),,(17),where,ω,is,the,angular,velocity,,ξ0...ξ20,are,random,variables,from,a,distribution,N,(0,,0.01),,fV,is,the,vortex,shedding,frequency.,The,cost,function,gradient,w.r.t,the,random,variables,is,∇ξJ,.,Wang,found,the,sensitivity,of,the,drag,coefficient,,CD,,using,the,adjoint,method,,and,used,this,in,combination,with,Monte,Carlo,sampling,to,determine,the,probability,of,the,drag,coefficient,passing,a,certain,critical,threshold,,CD,crit,,due,to,the,random,perturbations,in,(17),[78].,Wang’s,study,aimed,to,use,the,sensitivity,information,to,improve,the,efficiency,and,accuracy,of,un-,certainty,quantification,done,with,a,brute,force,Monte,Carlo,estimator.,Instead,of,simulating,random,instances,of,ξ,and,the,resultant,cost,function,,J,,,Wang,simply,evaluated,the,cost,function,once,for,one,set,of,ξ,and,used,the,following,linear,approximation,to,estimate,the,probability,of,P,(CD,>,CD,crit):,J,≈,(cid:101)J,=,J0,+,∇ξJ,·,ξ,,(18),where,the,approximation,(cid:101)J,can,be,computed,with,minimal,cost,(particularly,compared,to,evaluating,J,by,brute,force,methods).,This,is,extremely,fast,since,the,forward,and,adjoint,problems,should,only,be,solved,once,,but,the,confidence,interval,of,P,(CD,>,CD,crit),was,similar,to,a,brute,force,Monte,Carlo.,The,confidence,interval,on,P,(CD,>,CD,crit),was,reduced,one,order,of,magnitude,using,an,importance,sampling,method,,which,used,(cid:101)J,to,lower,the,variance,of,the,Monte,Carlo,solution,in,the,heavy-tailed,region,[78].,For,plasma,problems,,Carli,et,al.,[10],used,a,Bayesian,approach,to,quantify,uncertainty,in,turbulence,modelling,parameters,in,a,plasma-edge,code,(SOLPS-ITER),,where,the,Bayesian,posterior,distribution,was,found,through,gradient-descent,with,adjoint,mode,algorithmic,differentiation.,3.4,Adaptive,mesh,refinement,Several,studies,have,researched,using,adjoint,sensitivity,as,an,error,estimator,for,adaptive,mesh,refinement,[39,,58,,59,,75].,Classical,adaptive,mesh,refinement,has,used,heuristic,error,estimators,such,as,velocity,gradients,(for,wall-bounded,turbulence),[39].,This,has,been,shown,to,have,poor,performance,since,all,flow,features,are,treated,equally,and,the,user,must,iteratively,determine,controls,for,the,adaptation,process,[39].,In,contrast,,the,adjoint,method,can,be,used,as,a,‘goal-orientated’,error,estimator,that,can,identify,regions,where,a,cost,function,is,sensitive,to,perturbations,and,can,refine,there,to,avoid,introducing,numerical,artefacts,into,a,sensitive,region,[4,,65].,This,‘dual-weighted,residual’,method,uses,the,residuals,of,the,primal,solution,,weighted,by,the,adjoint,sensitivities,to,estimate,local,error,and,identify,regions,that,would,benefit,from,mesh,refinement,[65].,Initial,tests,of,steady,flow,past,a,cylinder,showed,that,9,one,could,obtain,one,order,of,magnitude,improvement,of,drag,coefficient,error,(with,the,same,number,number,of,cells),by,using,an,error,estimator,based,on,adjoint,sensitivities,[65].,Adjoint,methods,were,used,to,estimate,local,error,in,adaptive,mesh-refinement,in,spectral,element,simulations,of,a,turbulent,flow,over,a,periodic,hill,with,the,incompressible,Navier-Stokes,equations,[58].,Their,adjoint,problem,followed,the,cost,function,J,=,(cid:90),Ω,u,·,jΩ,udV,+,(cid:90),Ω,p,·,jΩ,pdV,+,(cid:90),ΓD,(cid:18),1,Re,(cid:19),∇u,·,n,−,pn,·,jΓD,ds,+,(cid:90),Γo,u,·,jΓods,,(19),where,u,is,the,fluid,velocity,,p,is,the,fluid,pressure,,Re,is,the,Reynolds,number,,n,is,a,unit,vector,normal,to,the,boundary,,Ω,is,the,internal,domain,,Γo,is,the,outlet,boundary,and,ΓD,is,a,Dirichlet,boundary,(e.g.,walls,with,no-slip,condition).,j,is,a,problem-specific,set,of,coefficients.,For,example,,in,a,lid-driven,cavity,problem,,Offermans,et,al.,used,ulid,=,−jΓD,,,and,all,other,j,=,(0,,0),(for,a,2D,case),[59].,Offermans,et,al.,then,computed,the,error,on,the,spectral,element,solution,of,the,cost,function,(19),when,a,polynomial,of,order,N,is,used,[59],with,the,following,approximation,,considering,the,domain,is,split,into,E,subdomains,,Ωe,,with,boundary,Γe:,δJ,=,J,−,JN,≈,δJ,≈,E,(cid:88),e=1,E,(cid:88),e=1,(cid:104)(cid:10)R1(q),,u+,−,u+,N,(cid:11),+,(cid:10)R2(q),,u+,−,u+,N,(cid:11),Γe,Ωe,+,(cid:10)R3(q),,p+,−,p+,N,(cid:11),(cid:105),Ωe,(20),(cid:0)||R1||,||u+,−,u+,N,||(cid:1),+,(cid:0)||R2||,||u+,−,u+,N,||(cid:1),+,(cid:0)||R3||,||p+,−,p+,N,||(cid:1),.,Ωe,Γe,Ωe,(21),N,and,p+,−,p+,where,⟨·⟩,is,the,inner,product,,||,·,||,is,the,L2-norm,,and,each,R,is,a,residual,from,either,the,cell,solution,(R1,is,from,the,momentum,equation,and,R3,is,from,the,continuity,equation),or,the,face,residual,between,two,elements,(R2).,The,terms,u+,−,u+,N,are,based,on,the,interpolation,error,and,are,approximated,with,a,relationship,based,on,the,cell,size,(volume,and,edge,length),and,polynomial,order,[59].,As,we,can,see,from,(21),,the,error,is,based,on,the,residual,of,the,primal,solutions,,which,are,weighted,by,the,adjoint,(also,known,as,‘dual’),solution,,giving,the,name,‘dual-weighted,residual’,[4].,Their,results,showed,that,the,adjoint,method,focused,the,mesh,refinement,to,areas,where,the,objective,function,sensitivity,was,high,and,left,the,mesh,unchanged,in,low-sensitivity,regions,of,the,domain.,This,was,beneficial,compared,to,the,alternative,‘spectral,error,indicator’,error,measure,that,refined,the,mesh,based,on,the,truncation,error,and,quadrature,error,,which,are,approximated,based,on,the,local,decay,rate,of,the,spectral,coefficients,[46].,This,computes,the,influence,of,the,polynomial,coefficients,,where,a,large,influence,of,high-order,terms,corresponds,to,a,high,polynomial,truncation,error,and,low,grid,resolution.,The,spectral,error,indicator,was,shown,to,be,much,simpler,to,compute,but,provided,a,more,uniform,mesh,[58].,The,computation,of,the,adjoint,error,estimator,was,complex,but,provided,a,more,focused,and,localized,refinement.,Using,the,adjoint,error,estimator,for,grid,refinement,was,shown,to,converge,to,the,correct,solution,with,fewer,refinement,iterations,[59],but,took,twice,the,time,for,one,refinement,round,due,to,increased,computational,complexity,for,a,steady,simulation.,For,an,unsteady,simulation,,the,adjoint,method,was,shown,to,be,only,10%,more,time-consuming,than,the,spectral,error,indicator,due,to,the,low,number,of,timesteps,used,for,mesh,refinement,relative,to,the,total,simulation,duration,(3,iterations,of,mesh,refinement,took,21,flow-through,times,compared,to,140,flow-through,times,on,the,final,mesh,to,get,turbulent,statistics),[58].,The,adjoint,error,estimator,was,shown,to,improve,the,accuracy,of,a,quantity,of,interest,after,fewer,rounds,of,mesh,refinement,,but,the,increased,computational,cost,of,refinement,iterations,suggests,that,adjoint-based,mesh,refinement,is,not,yet,superior,to,other,error,estimators,(with,regards,to,cost-accuracy,trade,off).,Similar,to,as,discussed,in,Section,2.2.2,,computing,the,adjoint,for,a,long,time,series,on,a,fine,mesh,invokes,computational,challenges,surrounding,how,to,manage,storing,the,full,primal,solution.,For,their,adaptive,mesh,refinement,study,on,unsteady,turbulent,flows,,Offermans,et,al.,used,checkpointing,with,the,revolve,library,[58].,An,alternative,approach,,also,discussed,in,Section,2.2.2,,is,storing,a,reduced-,order,model,of,the,primal,solution,and,using,this,when,computing,the,adjoint,equations,,as,was,done,by,Li,et,al.,[39].,This,removes,the,bottleneck,of,time-,and,memory-consuming,checkpointing,schemes,(at,the,cost,of,lower,accuracy).,However,,this,reduced-order,model,approach,makes,the,most,sense,in,an,adaptive,mesh,refinement,problem,,since,the,point,of,adaptive,mesh,refinement,is,to,minimise,the,computational,cost,required,to,obtain,the,desired,solution.,Offermans,et,al.,[58],also,mentioned,issues,in,stability,when,solving,the,adjoint,equations,due,to,highly,turbulent,flow,,which,they,speculated,may,10,be,resolved,by,using,a,reduced-order,model,to,store,the,primal,,since,the,small,and,chaotic,motion,is,generally,not,included,in,reduced-order,models.,Therefore,,the,benefit,of,using,a,reduced-order,model,for,storing,the,primal,flow,field,in,highly,turbulent,flows,may,be,two-fold:,(1),reduced,storage,,memory,and,re-computation,requirements,,(2),improved,stability,when,back-integrating,the,adjoint,problem.,3.5,Shape,optimisation,Early,adoption,of,the,adjoint,methods,in,fluid,flows,were,mainly,focused,on,shape,optimisation,of,airfoils,[35].,This,application,was,later,extended,to,fusion,by,optimising,tokamak,divertors,by,Dekeyser,[15].,Their,cost,function,aimed,to,generate,a,uniform,heat,flux,profile,by,minimising,the,least,squares,difference,between,the,computed,flux,and,a,flux,below,the,critical,value,of,the,material,[17,,18].,Different,approaches,for,treating,the,moving,shape,include:,(1),a,density-based,method,,where,the,entire,shape,design,space,is,meshed,and,cells,beyond,a,boundary,are,given,a,high,density/low,permeability,,(2),a,body,fitted,mesh,where,the,mesh,must,be,morphed,to,new,target,shapes,,or,(3),a,level-set,method,which,requires,special,numerical,treatment,at,the,wall,[1].,Dekeyser,et,al.,[17,,18],use,a,body-fitted,mesh,that,is,orthogonal,to,the,field,surfaces,which,is,common,in,plasmas.,The,divertor,region,was,then,parameterised,by,a,function,ϕ,and,the,new,geometry,in,the,divertor,region,was,generated,by,shrinking/stretching,cells,here.,They,also,use,a,high-order,(nine-point),discretisation,stencil,to,calculate,the,convective,fluxes,in,the,plasma,edge,transport,model,[16].,The,nine-point,stencil,includes,contributions,from,diagonally,adjacent,cells,,compared,to,only,contributions,from,cells,which,are,adjacent,in,the,horizontal,and,vertical,directions,[15].,This,five-point,stencil,gave,some,inaccuracies,in,shape,sensitivities,compared,to,finite,differences,,whereas,the,nine-point,stencil,agreed,perfectly,[16].,Later,,a,1D,version,of,the,finite,volume,code,used,in,Dekeyser’s,previous,work,for,edge,simulations,was,coupled,to,a,Monte,Carlo,code,for,a,kinetic,simulation,of,neutrals,[19].,Even,in,a,1D,analysis,,there,was,a,severe,issue,computing,the,cost,function,gradients,due,to,Monte,Carlo,statistical,error.,The,continuous,adjoint,required,around,a,factor,of,100,more,particles,to,obtain,similar,cost,function,accuracy,as,the,discrete,adjoint,,although,their,discrete,adjoint,code,was,10,times,more,expensive,(when,using,the,same,number,of,particles).,This,shows,the,use,of,adjoint,methods,for,shape,optimisation,and,other,applications,mentioned,in,Section,3,becomes,problematic,when,coupled,to,a,kinetic,simulation.,This,may,lead,to,the,requirement,of,an,unfeasibly,large,number,of,particles,for,accurate,adjoint,analyses,in,NEPTUNE.,4,Conclusions,We,have,reviewed,the,adjoint,method,,its,developments,and,various,applications,within,the,context,of,plasma,flows.,The,adjoint,method,is,an,effective,tool,for,optimisation,and,sensitivity,analysis,in,flow,configurations,with,a,large,number,of,operating,parameters,or,modelling,parameters,in,tokamak,simulations.,A,key,issue,when,using,the,adjoint,method,to,study,unsteady,flows,is,that,the,primal,solution,must,be,known,at,each,timestep,,which,presents,some,computational,issues,that,have,been,addressed,through,checkpointing,,reduced-order,modelling,and,parallel-in-time,algorithms.,Of,these,,checkpointing,appears,the,least,intrusive,and,is,already,implemented,in,libraries,such,as,PETSc,,meaning,it,is,a,viable,fit,for,NEPTUNE,developments,in,the,Nektar++,framework.,Applications,of,the,adjoint,method,mainly,consist,of,analysing,flow,sensitivity,to,perturbations.,Such,information,about,sensitivity,is,often,used,to,optimise,design,parameters,,shape,or,control,strategies.,Sensitivity,information,can,also,be,used,to,understand,and,quantify,error,in,the,presence,of,unknown,parameters,,or,as,an,error,estimator,in,adaptive,mesh,refinement,algorithms.,References,1.,Alexandersen,,J.,&,Andreasen,,C.,S.,A,review,of,topology,optimisation,for,fluid-based,problems.,Fluids,5,,29,(2020).,2.,Antonsen,,T.,,Paul,,E.,J.,&,Landreman,,M.,Adjoint,approach,to,calculating,shape,gradients,for,three-dimensional,magnetic,confinement,equilibria.,Journal,of,Plasma,Physics,85,,905850207,(2019).,11,3.,Baelmans,,M.,,Blommaert,,M.,,De,Schutter,,J.,,Dekeyser,,W.,&,Reiter,,D.,Efficient,parameter,estimation,in,2D,transport,models,based,on,an,adjoint,formalism.,Plasma,Physics,and,Controlled,Fusion,56,,114009,(2014).,4.,Bangerth,,W.,&,Rannacher,,R.,Adaptive,finite,element,methods,for,differential,equations,(Springer,Science,&,Business,Media,,2003).,5.,Blackburn,,H.,M.,,Barkley,,D.,&,Sherwin,,S.,J.,Convective,instability,and,transient,growth,in,flow,over,a,backward-facing,step.,Journal,of,Fluid,Mechanics,603,,271–304,(2008).,6.,Blackburn,,H.,M.,&,Sherwin,,S.,J.,Instability,modes,and,transition,of,pulsatile,stenotic,flow:,pulse-,period,dependence.,Journal,of,Fluid,Mechanics,573,,57–88,(2007).,7.,Blackburn,,H.,M.,,Sherwin,,S.,J.,&,Barkley,,D.,Convective,instability,and,transient,growth,in,steady,and,pulsatile,stenotic,flows.,Journal,of,Fluid,Mechanics,607,,267–277,(2008).,8.,Blommaert,,M.,,Reiter,,D.,&,Baelmans,,M.,An,efficient,methodology,to,analyze,plasma,edge,model,parameter,sensitivities.,Nuclear,Materials,and,Energy,12,,1049–1054,(2017).,9.,Capecelatro,,J.,,Bodony,,D.,J.,&,Freund,,J.,B.,Adjoint-based,sensitivity,and,ignition,threshold,mapping,in,a,turbulent,mixing,layer.,Combustion,Theory,and,Modelling,23,,147–179,(2019).,10.,Carli,,S.,et,al.,Bayesian,maximum,a,posteriori-estimation,of,κ,turbulence,model,parameters,us-,ing,algorithmic,differentiation,in,SOLPS-ITER.,Contributions,to,Plasma,Physics,62,,e202100184,(2022).,11.,Chandler,,G.,J.,Sensitivity,analysis,of,low-density,jets,and,flames,PhD,thesis,(University,of,Cam-,bridge,,2011).,12.,Choi,,Y.,,Arrighi,,W.,J.,,Copeland,,D.,M.,,Anderson,,R.,W.,&,Oxberry,,G.,M.,libROM,version,1.0.0.,Oct.,2019.,https://github.com/LLNL/libROM.,13.,Citro,,V.,,Giannetti,,F.,,Brandt,,L.,&,Luchini,,P.,Linear,three-dimensional,global,and,asymptotic,stability,analysis,of,incompressible,open,cavity,flow.,Journal,of,Fluid,Mechanics,768,,113–140,(2015).,14.,Costanzo,,S.,,Sayadi,,T.,,de,Pando,,M.,F.,,Schmid,,P.,&,Frey,,P.,Parallel-in-time,adjoint-based,optimization–application,to,unsteady,incompressible,flows.,Journal,of,Computational,Physics,471,,111664,(2022).,15.,Dekeyser,,W.,Optimal,Plasma,Edge,Configurations,for,Next-Step,Fusion,Reactors,(2014).,16.,Dekeyser,,W.,,Reiter,,D.,&,Baelmans,,M.,A,one,shot,method,for,divertor,target,shape,optimization.,PAMM,14,,1017–1022,(2014).,17.,Dekeyser,,W.,,Reiter,,D.,&,Baelmans,,M.,Automated,divertor,target,design,by,adjoint,shape,sen-,sitivity,analysis,and,a,one-shot,method.,Journal,of,Computational,Physics,278,,117–132,(2014).,18.,Dekeyser,,W.,,Reiter,,D.,&,Baelmans,,M.,Divertor,target,shape,optimization,in,realistic,edge,plasma,geometry.,Nuclear,fusion,54,,073022,(2014).,19.,Dekeyser,,W.,et,al.,Divertor,design,through,adjoint,approaches,and,efficient,code,simulation,strate-,gies.,Contributions,to,Plasma,Physics,58,,643–651,(2018).,20.,Dow,,E.,&,Wang,,Q.,Quantification,of,Structural,Uncertainties,in,the,k-w,Turbulence,Model,in,52nd,AIAA/ASME/ASCE/AHS/ASC,Structures,,structural,dynamics,and,materials,conference,19th,AIAA/ASME/AHS,adaptive,structures,conference,13t,(2011),,1762.,21.,Ed,Threlfall,,W.,A.,Select,techniques,for,MOR,(Model,Order,Reduction),tech.,rep.,CD/EXCALIBUR-,FMS/0031-M2.5.1,(UK,Atomic,Energy,Authority,,Mar.,15,,2021).,https://github.com/ExCALIBUR-,NEPTUNE/Documents/blob/main/reports/ukaea_reports/CD-EXCALIBUR-FMS0031-M2.5.1.pdf.,22.,Exposing,parallelism:,Parallel,in,Time,Excalibur,projects.,https://excalibur.ac.uk/projects/,exposing-parallelism-parallel-in-time.,23.,Gander,,M.,J.,&,G¨uttel,,S.,PARAEXP:,A,parallel,integrator,for,linear,initial-value,problems.,SIAM,Journal,on,Scientific,Computing,35,,C123–C142,(2013).,24.,Gander,,M.,J.,,G¨uttel,,S.,&,Petcu,,M.,A,nonlinear,ParaExp,algorithm,in,Domain,Decomposition,Methods,in,Science,and,Engineering,XXIV,24,(2018),,261–270.,12,25.,Gaur,,R.,et,al.,An,adjoint-based,method,for,optimizing,MHD,equilibria,against,the,infinite-n,,ideal,ballooning,mode.,arXiv,preprint,arXiv:2302.07673,(2023).,26.,Geraldini,,A.,,Landreman,,M.,&,Paul,,E.,An,adjoint,method,for,determining,the,sensitivity,of,island,size,to,magnetic,field,variations.,Journal,of,Plasma,Physics,87,,905870302,(2021).,27.,Giannetti,,F.,,Camarri,,S.,&,Luchini,,P.,Structural,sensitivity,of,the,secondary,instability,in,the,wake,of,a,circular,cylinder.,Journal,of,Fluid,Mechanics,651,,319–337,(2010).,28.,Giannetti,,F.,&,Luchini,,P.,Structural,sensitivity,of,the,first,instability,of,the,cylinder,wake.,Journal,of,Fluid,Mechanics,581,,167–197,(2007).,29.,Giles,,M.,B.,&,Pierce,,N.,A.,An,introduction,to,the,adjoint,approach,to,design.,Flow,,turbulence,and,combustion,65,,393–415,(2000).,30.,Griewank,,A.,&,Walther,,A.,Algorithm,799:,revolve:,an,implementation,of,checkpointing,for,the,reverse,or,adjoint,mode,of,computational,differentiation.,ACM,Transactions,on,Mathematical,Soft-,ware,(TOMS),26,,19–45,(2000).,31.,Griffith,,A.,K.,&,Nichols,,N.,K.,Adjoint,methods,in,data,assimilation,for,estimating,model,error.,Flow,,turbulence,and,combustion,65,,469–488,(2000).,32.,Hamamatsu,,K.,&,Fukuyama,,A.,Controllability,of,driven,current,profile,in,ECCD,on,ITER.,Fusion,engineering,and,design,53,,53–58,(2001).,33.,Heuveline,,V.,&,Walther,,A.,Online,checkpointing,for,parallel,adjoint,computation,in,PDEs:,Appli-,cation,to,goal-oriented,adaptivity,and,flow,control,in,Euro-par,(2006),,689–699.,34.,Huerre,,P.,&,Monkewitz,,P.,A.,Absolute,and,convective,instabilities,in,free,shear,layers.,Journal,of,Fluid,Mechanics,159,,151–168,(1985).,35.,Jameson,,A.,Aerodynamic,design,via,control,theory.,Journal,of,scientific,computing,3,,233–260,(1988).,36.,Kord,,A.,&,Capecelatro,,J.,A,discrete-adjoint,framework,for,optimizing,high-fidelity,simulations,of,turbulent,reacting,flows.,Proceedings,of,the,Combustion,Institute,(2022).,37.,Kord,,A.,&,Capecelatro,,J.,Optimal,perturbations,for,controlling,the,growth,of,a,Rayleigh–Taylor,instability.,Journal,of,Fluid,Mechanics,876,,150–185,(2019).,38.,Lea,,D.,J.,,Allen,,M.,R.,&,Haine,,T.,W.,Sensitivity,analysis,of,the,climate,of,a,chaotic,system.,Tellus,A:,Dynamic,Meteorology,and,Oceanography,52,,523–532,(2000).,39.,Li,,X.,,Hulshoff,,S.,&,Hickel,,S.,Towards,adjoint-based,mesh,refinement,for,Large,Eddy,Simulation,using,reduced-order,primal,solutions:,Preliminary,1D,Burgers,study.,Computer,Methods,in,Applied,Mechanics,and,Engineering,379,,113733,(2021).,40.,Lorenc,,A.,C.,Analysis,methods,for,numerical,weather,prediction.,Quarterly,Journal,of,the,Royal,Meteorological,Society,112,,1177–1194,(1986).,41.,Luce,,T.,Realizing,steady-state,tokamak,operation,for,fusion,energy.,Physics,of,Plasmas,18,,030501,(2011).,42.,Luchini,,P.,&,Bottaro,,A.,Adjoint,equations,in,stability,analysis.,Annual,Review,of,fluid,mechanics,46,,493–517,(2014).,43.,Magri,,L.,Adjoint,methods,as,design,tools,in,thermoacoustics.,Applied,mechanics,reviews,71,(2019).,44.,Mao,,X.,,Blackburn,,H.,&,Sherwin,,S.,Optimal,suppression,of,flow,perturbations,using,boundary,control.,Computers,&,Fluids,121,,133–144,(2015).,45.,Marin,,O.,,Constantinescu,,E.,&,Smith,,B.,A,scalable,matrix-free,spectral,element,approach,for,unsteady,PDE,constrained,optimization,using,PETSc/TAO.,Journal,of,Computational,Science,47,,101207,(2020).,46.,Mavriplis,,C.,Adaptive,mesh,strategies,for,the,spectral,element,method.,Computer,methods,in,applied,mechanics,and,engineering,116,,77–86,(1994).,47.,Meliga,,P.,,Chomaz,,J.-M.,&,Sipp,,D.,Global,mode,interaction,and,pattern,selection,in,the,wake,of,a,disk:,a,weakly,nonlinear,expansion.,Journal,of,Fluid,Mechanics,633,,159–189,(2009).,48.,Mordijck,,S.,Taking,Control,of,Fusion,Reactor,Instabilities.,Physics,15,,154,(2022).,13,49.,Murali,,S.,Stability,and,Heat,Transfer,Characteristics,of,Hydrodynamic,Channel,Flows,and,Quasi-,two-dimensional,Magnetohydrodynamic,Duct,Flows,with,Repeated,Wedge-shaped,Protrusions,PhD,thesis,(Monash,University,,2022).,50.,Murali,,S.,,Hussam,,W.,K.,&,Sheard,,G.,J.,Heat,transfer,enhancement,in,quasi-two-dimensional,magnetohydrodynamic,duct,flows,using,repeated,flow-facing,wedge-shaped,protrusions.,Interna-,tional,Journal,of,Heat,and,Mass,Transfer,171,,121066,(2021).,51.,Murali,,S.,,Ng,,Z.,Y.,&,Sheard,,G.,J.,Stability,of,flow,in,a,channel,with,repeated,flow-facing,wedge-,shaped,protrusions.,Journal,of,Fluid,Mechanics,941,,A59,(2022).,52.,Nadarajah,,S.,&,Jameson,,A.,A,comparison,of,the,continuous,and,discrete,adjoint,approach,to,automatic,aerodynamic,optimization,in,38th,Aerospace,sciences,meeting,and,exhibit,(2000),,667.,53.,Nadarajah,,S.,&,Jameson,,A.,Studies,of,the,continuous,and,discrete,adjoint,approaches,to,viscous,automatic,aerodynamic,shape,optimization,in,15th,AIAA,computational,fluid,dynamics,conference,(2001),,2530.,54.,Newman,III,,J.,C.,,Taylor,III,,A.,C.,,Barnwell,,R.,W.,,Newman,,P.,A.,&,Hou,,G.,J.-W.,Overview,of,sensitivity,analysis,and,shape,optimization,for,complex,aerodynamic,configurations.,Journal,of,Aircraft,36,,87–96,(1999).,55.,Nies,,R.,,Paul,,E.,J.,,Hudson,,S.,R.,&,Bhattacharjee,,A.,Adjoint,methods,for,quasi-symmetry,of,vacuum,fields,on,a,surface.,Journal,of,Plasma,Physics,88,,905880106,(2022).,56.,Nobis,,H.,,Schlatter,,P.,,Wadbro,,E.,,Berggren,,M.,&,Henningson,,D.,S.,Topology,optimization,of,unsteady,flows,using,the,spectral,element,method.,Computers,&,Fluids,239,,105387,(2022).,57.,Nordlund,,K.,H.,et,al.,European,research,roadmap,to,the,realisation,of,fusion,energy,(2018).,58.,Offermans,,N.,,Massaro,,D.,,Peplinski,,A.,&,Schlatter,,P.,Error-driven,adaptive,mesh,refinement,for,unsteady,turbulent,flows,in,spectral-element,simulations.,Computers,&,Fluids,251,,105736,(2023).,59.,Offermans,,N.,,Peplinski,,A.,,Marin,,O.,&,Schlatter,,P.,Adaptive,mesh,refinement,for,steady,flows,in,Nek5000.,Computers,&,Fluids,197,,104352,(2020).,60.,Paul,,E.,J.,,Abel,,I.,G.,,Landreman,,M.,&,Dorland,,W.,An,adjoint,method,for,neoclassical,stellarator,optimization.,Journal,of,Plasma,Physics,85,,795850501,(2019).,61.,Paul,,E.,J.,,Landreman,,M.,&,Antonsen,,T.,Gradient-based,optimization,of,3D,MHD,equilibria.,Journal,of,Plasma,Physics,87,,905870214,(2021).,62.,Qadri,,U.,A.,Global,stability,and,control,of,swirling,jets,and,flames,PhD,thesis,(University,of,Cambridge,,2014).,63.,Qadri,,U.,A.,,Chandler,,G.,J.,&,Juniper,,M.,P.,Passive,control,of,global,instability,in,low-density,jets.,European,Journal,of,Mechanics-B/Fluids,72,,311–319,(2018).,64.,Qadri,,U.,A.,,Magri,,L.,,Ihme,,M.,&,Schmid,,P.,J.,Using,adjoint-based,optimization,to,enhance,ignition,in,non-premixed,jets.,Proceedings,of,the,Royal,Society,A,477,,20200472,(2021).,65.,Rannacher,,R.,Adaptive,finite,element,discretization,of,flow,problems,for,goal-oriented,model,re-,duction.,Computational,Fluid,Dynamics,2008,,31–45,(2008).,66.,Schanen,,M.,,Marin,,O.,,Zhang,,H.,&,Anitescu,,M.,Asynchronous,two-level,checkpointing,scheme,for,large-scale,adjoints,in,the,spectral-element,solver,Nek5000.,Procedia,Computer,Science,80,,1147–,1158,(2016).,67.,Schmid,,P.,J.,Nonmodal,stability,theory.,Annu.,Rev.,Fluid,Mech.,39,,129–162,(2007).,68.,Schmid,,P.,J.,&,Brandt,,L.,Analysis,of,Fluid,Systems:,Stability,,Receptivity,,Sensitivity.,Applied,Mechanics,Review,66,,024803,(2014).,69.,Schneider,,R.,et,al.,Plasma,edge,physics,with,B2-eirene.,Contributions,to,Plasma,Physics,46,,3–191,(2006).,70.,Sipp,,D.,,Marquet,,O.,,Meliga,,P.,&,Barbagallo,,A.,Dynamics,and,control,of,global,instabilities,in,open-flows:,a,linearized,approach.,Applied,Mechanics,Reviews,63,(2010).,71.,Skene,,C.,S.,,Eggl,,M.,F.,&,Schmid,,P.,J.,A,parallel-in-time,approach,for,accelerating,direct-adjoint,studies.,Journal,of,Computational,Physics,429,,110033,(2021).,14,72.,Strait,,E.,J.,Magnetic,control,of,magnetohydrodynamic,instabilities,in,tokamaks.,Physics,of,Plasmas,22,,021803,(2015).,73.,Strykowski,,P.,J.,&,Sreenivasan,,K.,R.,On,the,formation,and,suppression,of,vortex,‘shedding’at,low,Reynolds,numbers.,Journal,of,Fluid,Mechanics,218,,71–107,(1990).,74.,Stumm,,P.,&,Walther,,A.,New,algorithms,for,optimal,online,checkpointing.,SIAM,Journal,on,Scientific,Computing,32,,836–854,(2010).,75.,Venditti,,D.,A.,&,Darmofal,,D.,L.,Adjoint,error,estimation,and,grid,adaptation,for,functional,outputs:,Application,to,quasi-one-dimensional,flow.,Journal,of,Computational,Physics,164,,204–,227,(2000).,76.,Vezyris,,C.,,Papoutsis-Kiachagias,,E.,&,Giannakoglou,,K.,On,the,incremental,singular,value,decom-,position,method,to,support,unsteady,adjoint-based,optimization.,International,Journal,for,Numer-,ical,Methods,in,Fluids,91,,315–331,(2019).,77.,Vishnampet,,R.,,Bodony,,D.,J.,&,Freund,,J.,B.,A,practical,discrete-adjoint,method,for,high-fidelity,compressible,turbulence,simulations.,Journal,of,Computational,Physics,285,,173–192,(2015).,78.,Wang,,Q.,Uncertainty,quantification,for,unsteady,fluid,flow,using,adjoint-based,approaches,PhD,thesis,(Stanford,University,,2009).,79.,Zhang,,H.,&,Constantinescu,,E.,Revolve-based,adjoint,checkpointing,for,multistage,time,integration,in,International,Conference,on,Computational,Science,(2021),,451–464.,80.,Zhang,,H.,,Constantinescu,,E.,M.,&,Smith,,B.,F.,PETSc,TSAdjoint:,a,discrete,adjoint,ODE,solver,for,first-order,and,second-order,sensitivity,analysis.,SIAM,Journal,on,Scientific,Computing,44,,C1–,C24,(2022).,15

:pdfembed:`src:_static/TN-08_ReviewAdjointMethodPlasmaPhysicsNeptune.pdf, height:1600, width:1100, align:middle`