AD-based perturbation methods for uncertainties and errors. M. Martinelli*, A. Dervieux, L. Hascoët, V. Pascual and A. Belme

Transcription

1 Int. J. Engineering Systems Modelling and Simulation, Vol., Nos. 1/, AD-based perturbation metods for uncertainties and errors M. Martinelli, A. Dervieux, L. Hascoët, V. Pascual and A. Belme Institut National de Recerce en Informatique et en Automatique, INRIA Sopia Antipolis Méditerranée, route des Lucioles, BP 93, 0690 Sopia Antipolis Cedex, France Corresponding autor Abstract: Te progress of automatic differentiation (AD) and its impact on perturbation metods is te object of tis paper. AD studies sow an important activity for developing metods addressing te management of modern CFD kernels, taking into account te language evolution and intensive parallel computing. Te evaluation of a posteriori error analysis and of resulting correctors will be addressed. Recent works in te AD-based construction of second-derivatives for building reduced-order models based on a Taylor formula will be presented on te test case of a steady compressible flow around an aircraft. Keywords: automatic differentiation; AD; second-order derivatives; uncertainty; error correction; design; CFD; systems modelling; simulation. Reference to tis paper sould be made as follows: Martinelli, M., Dervieux, A., Hascoët, L., Pascual, V. and Belme, A. (010) AD-based perturbation metods for uncertainties and errors, Int. J. Engineering Systems Modelling and Simulation, Vol., Nos. 1/, pp Biograpical notes: Massimiliano Martinelli received is PD in Applied Matematics at Scuola Normale Superiore of Pisa (007) and is MSc in Applied Matematics at University of Bologna (003). After two years at INRIA Sopia Antipolis, e is now a Contract Professor at Università Politecnica delle Marce and as a researc grant at te Department of Matematics at University of Pavia. Its researc interests are CFD, automatic differentiation and ig-performance computing. Alain Dervieux received is PD at University of Paris VI and is a Researc Scientist at INRIA. He as created and directed a CFD team tere during 15 years. His scientific contributions concern approximations metods on unstructured meses for CFD applications, multigrid metods, sape optimal design, mes adaption metods and teory. Laurent Hascoët, Ingenieur from Ecole Polytecnique, Palaiseau, France, received is PD from University of Nice in He is a Permanent Researcer at INRIA since He as also been working in a small company, CONNEXITE, from 1991 to 1993, wen it merged into SIMULOG. CONNEXITE was building tools for restructuring and parallelising FORTRAN programs. Since 1998, e as been working on automatic differentiation, and leading a researc team named TROPICS. Valérie Pascual as a student at Ecole Normale Supérieure de Jeunes Filles in Paris, se received er PD in Computer Science from Paris XI University in Se joined INRIA in 1984 as a Permanent Researcer. Se developed programming environments and structured editors for Lisp and Java languages and worked on interactive structured editing tools using program transformations. In 1999 se joined te new TROPICS team. Her current area of researc includes analysis and transformation of programs applied to automatic differentiation. Se participates in te development of te TAPENADE automatic differentiation tool. Anca Belme received er Master degree in Applied Matematics at University of Montpellier II, France in 008. Se started er PD tesis te same year at INRIA Sopia-Antipolis, France, under te direction of M. Alain Dervieux. Her subject is adjoint metods for numerical error correction of unsteady flows. Copyrigt 010 Inderscience Enterprises Ltd.

2 66 M. Martinelli et al. 1 Introduction Wile ig fidelity models are mainly used for deterministic design, wic assumes a perfect knowledge of te environmental and operational parameters, uncertainty can arise in many aspects of te entire design-production-operational process: from te assumptions done in te matematical model describing te underlying pysical process to te manufacturing tolerances and to te operational parameters and conditions tat could be affected by unpredictable factors (e.g., atmosperic conditions). Exact and approximate tecniques for propagating tese uncertainties require additional computational effort but are progressively well-establised. Te proposed study takes place in NODESIM-CFD FP6 project (Martinelli and Hascoët, 008; NODESIM-CFD, 008). Te automatic differentiation (AD) tool TAPENADE (Hascoët and Pascual, 004) is discussed in Section. It as been developed for a large range of applications were te code-to-code direct and reverse differentiation is needed. Direct and reverse ADs are used for addressing numerical error reduction since tey elp building correctors (Section 3). In Section 4, uncertainty propagation is addressed by a perturbation tecnique using te first terms of Taylor series of te ig-fidelity model (metod of moments). Previous investigation of tese metods can be found in Gate and Giles (006, 007). We present as example te response surface of a wing. AD improvements Our AD tool TAPENADE as been extended to deal wit Fortran95 and wit ANSI C (Pascual and Hascoët, 005; Pascual and Hascoët, 008). Figure 1 sows te arcitecture of TAPENADE. It is implemented mostly in Java (115,000 lines) except for te separate front-ends wic can be written in teir own languages. Front- and back-ends communicate wit te kernel via an intermediate abstract language ( IL ) tat makes te union of te constructs of individual imperative languages. Notice also te clear separation between te general-purpose program analysis and te differentiation engine itself. Figure 1 Overall arcitecture of TAPENADE pointers and allocation. Anoter development concerned declarations. Te differentiated program respects te order of declarations, uses te include files and keeps te comments from te original program. Generated codes are more readable and often smaller. We also investigated extensions to TAPENADE to successive differentiations, in particular to efficiently andle tangent differentiation of te stack primitives present in te reverse differentiated codes. We implemented user directives for te reverse differentiation of a frequent class of parallel loops (directive II-LOOP) and for optimal ceckpointing in reverse differentiation (Naumann et al., 008; Hascoët et al., 008; Tber et al., 007). TAPENADE lets te user specify finely wic procedure calls must be ceckpointed or not wit te directive NOCHECKPOINT. 3 Numerical errors reduction 3.1 Error estimates and correctors Let us recall first ow linearised direct or adjoint states can be useful for improving numerical accuracy issues. Numerical error involves te deviation between te solution W = W( x, y, z ) of matematical model, i.e., of te non-linear PDE symbolised by: Ψ ( W ) = 0, (1) and te output data produced by te computations, i.e., te more or less perfect numerical solution of te discrete system: N Ψ ( W ) = 0 R. () Te discrete unknown degrees of freedom: N i W R, W =[( W ) ]. W is te N-dimensional array of Te output data produced by te computation do not involve a function W but, instead te array W wic needs to be transformed via an interpolation: let V L ( ) a space of rater smoot function. In practice, V C 0 ( ). Let R be a linear interpolation operator transforming an array of N degrees of freedom into a continuous function: R : R N V v R v. (3) Let: W( xyz,, ) = ( RW )( xyz,, ). Tanks to te language-independent internal representation of programs, tis still makes a single and only tool and every development benefits to differentiation of eac input language. One of tese developments concerned te pointer analysis. Te reverse mode now accepts most uses of Similarly, we need an operator from continuous functions to arrays. Let T be an operator transforming a continuous function into an array of N degrees of freedom: N T : V R v T v. (4) It is useful to take te adjoint of R :

3 AD-based perturbation metods for uncertainties and errors 67 T = R. (5) Te deviation between te PDE solution and te numerical one can be defined as W W. It consists mainly of approximation errors, of algoritmic errors arising typically because iterative algoritms are not iterated infinitely, and of round-off errors due to te fact tat te programme is run in floating point aritmetic. We discuss ere mainly of approximation errors, altoug te oter ones may be also addressed in part by te metod studied ere. Anoter way to post-process a computation is to use it for evaluating a scalar functional: Let j a smoot linear functional applying W into te scalar number: ju ( ) = ( gw, ) L ( ) were g is a given g = Tg L ( ) function. Tis allows to define: g = R g = R T g (6) Te continuous adjoint writes: Ψ ( ) p = g. Te discrete adjoint equation is ten defined by: T Ψ p =. Tg (7) And we can ten consider: p = R p. A fundamental assumption of te present analysis is tat tis discrete adjoint is a good enoug approximation of continuous adjoint p for allowing to replace p by p te calculations wic follow. In order to evaluate te approximation error, two kinds of estimates can be applied: A posteriori estimate: Ψ( W) Ψ( W ) = Ψ( W ) (8) were Ψ( W ) is te continuous residual applied to discrete solution. Ten: Ψ W W Ψ( W ). (9) A priori estimate: Ψ ( TW) Ψ ( W ) = Ψ ( TW ) (10) were Ψ ( TW ) is te discrete residual applied to discretised continuous solution. Ten: Ψ Ψ TW W ( TW). (11) We observe tat tese estimates involve unavailable continuous functions. In te a posteriori estimate, te solution of te continuous linearised system can be approximated tanks to te discrete Jacobian. For te a priori estimate, we can also solve tis issue in some particular case (see Loseille, 008), by replacing Ψ ( TW ) by an expression TΘW ( ) depending only of W. Corresponding to tese estimates, we ave te following field correctors : Ψ = Ψ δ W R T ( W ) (1) Ψ = W Θ δ R R T W ( ) (13) and te following direct-linearised goal-oriented correctors : 1, Ψ δ j = g R T Ψ( ) W (14) L ( ), Ψ δ j = g R T Θ ( ) W (15) L ( ) Also follows te adjoint-based goal-oriented correctors : δ j = ( p, T Ψ( W )) L (16) 1 ( ) ( ) δ j = ( p, T Θ ( W )) L (17) We recognize ere te superconvergent corrector of Pierce and Giles (1998). We observe tat, tanks to te coice of T as te adjoint operator of R, te linearised-based and adjoint-based formulation are perfectly equivalent. At te contrary, te effort to compute tem is very different, particularly in te case of unsteady PDE, since te adjoint system as to be solved reverse in time wile using te state solution at all time levels (Hascoët and Dauvergne, 008). Tis remark leads to te following recommendations: use te direct linearised formulation in any case, you only need a corrector for te field as well as a corrector for one or several output functionals te adjoint formulation is compulsory wen you wis to derive a goal-oriented optimal mes. Te second recommendation is motivated by te fact tat an optimal mes will be derived from minimisation of te error term in wic we need to put in evidence te dependance of error wit respect to mes. Since te adjoint is an approximation of a continuous function, it does not muc depend of mes. At te contrary, te continuous residual T Ψ( W ) or te truncation error T Θ ( W ) are

4 68 M. Martinelli et al. proportional to a power of te mes size. In Loseille (008), te truncation error is expressed in terms of second derivatives of solution field and allows te derivation of an optimal mes. 3. An example To end tis discussion, we give a numerical example of corrector evaluation built on a finite-element approximation. We can write Euler equations under te form: 1 5 W V = H ( ), φ V, F( W) φd φfˆ ( W). ndγ = 0 (18) wit F( W). n = F( W). n Fˆ ( W). n. A Gauss quadrature is applied for te evaluation of te rigt and side. We ave applied tis to a steady subsonic flow and give some preliminary results. Figure compares te entropy generation in te flow computed directly and te same flow corrected by formula (1). Entropy level is one order of magnitude smaller. Figure Entropy spurious generation for, (a) direct computation of a steady flow (b) for a corrected one (see online version for colours) were F ˆ ( W ) accounts for te different boundary conditions. Let us introduce a discretisation of te previous EDP. Let τ a tetraedrisation of wit N vertices. It will rely on a discrete space of functions: 1 5 φ τ φ V = { H ( ), T, T P } te canonical basis of wic is denoted: V = span[ N ], N ( x ) = δ i, j, vertices of τ, i i j ij 1 and on te interpolation operator: : V V, φ( x ) = φ( x ), i, vertex of τ. i i Comparing wit te previous abstract teory, we get: (a) 5N R f f = i f i i 5N : V R, φ φ = [ φ( i)]. R : V, R [ ] N, T T x Te discretisation is set into te discrete space, but also it differs from te continuous statement in two features, a discrete flux F instead of F : F : V V and an extra term of artificial diffusion D : W V, φ V,( Ψ( W), φ) V V = 0, wit ( Ψ ( W ), φ ) = φ F ( W ). n dγ + V V φ D ( W ) d. k Te discrete fluxes are cosen as follows: F( W) = F( W) = F( W). F ( W) = F ( W) = F( W). (19) (0) After some calculations and simplifications, te main error term appears as follows: Ψ ( δw, φ) = φ( F( W) F( W)) d (1) + φ ( Fˆout ( ) Fˆout W ( W)). nd Γ (b) 4 Uncertainty propagation In optimisation problems, uncertainty propagation analysis may concern te study of te cost functional j : γ j( γ): = J( γ, W) R () were all varying parameters are represented by te uncertain (i.e., not-deterministic) control variables; γ R n, and were te state variables W = W( γ ) R N are solution of te (non-linear) state equation Ψ( γ, W ) = 0. (3)

5 AD-based perturbation metods for uncertainties and errors 69 It is important to note tat te state equation (3) contains te governing PDE of te matematical model of te pysical system of interest (for example, te stationary part of te Euler or Navier-Stokes equations) and it can be viewed as an equality constraint for te functional (). Te basic probabilistic approaces for analysing te propagation of uncertainties are Monte-Carlo metods. A full non-linear Monte-Carlo metod gives us complete and exact information about uncertainty propagation in te form of its PDF, but wit a proibitively expensive cost in terms of CPU time. In NODESIM-CFD, several oter probabilistic approaces for analysing te propagation of uncertainties are considered suc as Latine Hypercubes and Polynomial Caos. We contribute on perturbative metods based on te Taylor expansion (Martinelli, 007). 4.1 Perturbation metods To reduce te computational cost, we may tink to use only some (derivate) quantities caracterising te distribution of te input variables instead of an entire sample drawn from a population wit a given PDF. Terefore, te idea beind te metod of moments is based on te Taylor series expansion of te original non-linear functional () around te mean value of te input control ( μγ = E[ γ ]), and ten computing some statistical moments of te output (usually mean and variance). In tis way, we are assuming tat te input control γ can be decomposed as sum of a fully deterministic quantity μ γ wit a stocastic perturbation δγ u wit te property E[ δγ u ] = 0. Wit tese definitions, te Taylor series expansion of te functional j( γ ) around te mean value μ γ is: j( γ) = j( μγ + δγu) = j( μγ) + Gδγ δγuh δγu + O( δγu ) were = j G γ u μ γ u (4) is te gradient of te functional respect j to te uncertain variables and H = is te Hessian γ u μ γ matrix, bot evaluated at te mean of te input variables μ γ. By considering various orders of te Taylor expansion (4) and taking te first and te second statistical moment, we can approximate te mean μ j and te variance σ j of te functional j( γ ) in terms of its derivatives evaluated at μ γ and in terms of statistical moments of te control γ. First order moment metods: ( E u ) O( δγ ) μj = j( μγ ) + O [ δγ ] 3 σ = j E ( Gδγu) + E[ u] (5) Second order moment metods: 1 μ ( ) j = j μγ + E δγuhδγu + O σ j 3 ( E[ δγ ]) ( δγ ) ( )( ) u δγu δγu δγu δγ δγ ( δγ δγ ) u u u u = E G + E G H 1 1 E H + E 4 4 H + O 4 ( δγ ) u u Wit tis metod, it is clear tat we are using only some partial information about te input uncertainties, in fact, we are using only some statistical moments of te control variable instead of full information available wit its PDF, and we will not ave anymore te PDF of te propagated uncertainty, but only its approximate mean and variance. Anoter important point is tat te metod of moments is applicable only for small uncertainties, due to te local nature of Taylor expansion approximation. Two tings sould be noted ere: te first one is tat for te metod of moments we need te derivatives of te functional respect to te control variables affected by uncertainties: in particular, we need te gradient for te first order metod, and gradient and Hessian for te second order metod. Due to te fact tat j( γ) = j( γ, W), were W = W( γ ) is te solution of te state equation (3), we ave for te derivative: j J J = + γ γ γ u u u Since we know te solution W ( γ ) by its numerical values as result of a program (implementing an appropriate metod, e.g., fixed point metod), it is interesting to use of AD tools (like TAPENADE) in order to obtain te needed derivatives (Martinelli et al., 007). Te same remarks apply to te computation of te Hessian matrix. In particular, we note tat te derivatives are computed at te mean value of te control μ γ, so tey are fully deterministic and can be picked out from te expectations in te equations (5) or (6). In oter words, we can write ( ) ( ) ( ) i k δγ u i k δγu δγu i k ik ik, ik, () i ( k) δγ u δγu ik δγu δγu i k ik ik, ik, () ( ) () ( )( ) i k l δγ u δγu δγu l ik δγu δγu δγu ikl,, ( ) ( ) ( ( ) i k u u l m ik lm u u u δγu iklm,,, E = G = GG E = GG C E = H = H E = GG C E = G H = GH E E = δγ Hδγ = H H E δγ δγ δγ ) ( ) (6)

6 70 M. Martinelli et al. were H j Gi = γ () i u μ γ j ik = () i ( k) γu γu μ γ () i ( k) () i ( k) ik = u u = u u are te elements of te gradient, are te elements of te Hessian matrix and C E[ δγ δγ ] cov( γ γ ) are te elements of te covariance matrix. Every expectation term E [ ] in te equation (6) is defined by te statistical model of te uncertainties and could be computed in a pre-processing pase. For example, for te important case were te uncertainties are random and normally distributed, we ave: E () i ( k) () l δγ u δγu δγu = 0 () i ( k) () l ( m) δγ u δγu δγu δγu = ik lm + il km + im kl E C C C C C C and if tese (normal) uncertainties are independent, ten old te relation Cik = σi δ ij were σ ( i ) ( i = ) i E δγu δγ u and te equations (6) become E = ( G ) = G δγu iσi i u δγu iiσi i δγ u δγu δγu u δγu ii kk ik σiσk ik, E = δγ H = H E = ( G )( H ) = 0 E = ( δγ H ) = ( H H + H ) (7) Since we ave te term E = ( δγ u Hδγ u ) 4, te error is still of te order of E = δγ 4 u. Computing te oter terms of same order require te knowledge of order of derivatives iger tan te second. From te previous discussion, it is clear tat in order to apply te metod of moments we need to solve only one (expensive) non-linear system wit derivatives (at te mean μ γ ) and ten apply te (inexpensive) equations (5) or (6) were, for te fully non-linear Monte-Carlo approac of te previous section, we need to solve N 1 non-linear systems (3). 4. First and second-order derivatives of a functional We are interested by obtaining te first and second derivatives of a functional j depending of γ R N, and expressed in terms of a state W R N as follows: { ψγ ( ) = Ψ( γ, W ( γ)) = 0 (8) j( γ) = J( γ, W( γ)) Our problem can be viewed from two different points of view: te first one is consider te solution algoritm for state equation as part of j itself, i.e., considering j as a function of te control variables γ only. Te second one is consider te system made by two different routines: one of tem is te routine tat solves te non-linear system Ψ( γ, W ( γ )) = 0 (and contains te evaluation te residual Ψ( γ, W )), and te oter is te routine J( γ, W) tat computes te value of te functional from te state variables W and (eventually) te control variables γ. Te first approac leads to a straigtforward algoritm for first order derivatives, in fact, we just need to differentiate te entire routine j wit tangent or reverse mode. In tis context, te routine j contains te iterative solver metod for te state equation, and te differentiated routines will also contain tis loop in differentiated form. If we need n iter loop iterations in order to obtain te non-linear solution, and we assume for eac iteration a unitary cost, we can analyse te cost for te gradient of te functional. Using tangent mode, te cost for te entire gradient will be nn ( iterα T ) were n is te number of components of te gradient and 1< α T < 4 is te overead associated wit te differentiated code respect to te original one. For tis strategy, te memory requirements will be of te same order of te undifferentiated code. Wit reverse mode (Hascoët et al., 005) we are able to obtain te entire gradient wit a single evaluation of te differentiated routine, but te total cost (in terms of CPU time and memory) will depends on te strategy used by te AD tool to solve te problem of inverse order differentiation for te original routine. For te case of a store-all (SA) strategy, te CPU cost will be ( niterα R ) wit 1 < α R <, i.e., α R times te undifferentiated code, but te required memory will be n times greater. For a recompute-all (RA) strategy, te CPU cost will be ( niterα R ), i.e., ( niterα R ) te non-linear solution, but te memory will be te same of te undifferentiated routine. For real large programs, neiter SA nor RA strategy can work, so we need a special storage/recomputation trade-off in order to be efficient using ceckpoints. Obviously, wit ceckpointing te CPU cost will be greater tan te cost of SA strategy and can be sown tat te cost for te differentiated code will be of te order of s n iter (were s is te number of snapsots available). It is clear tat for gradient computation wit n 1, te reverse mode is faster tan tangent mode, but for a program containing an iterative algoritm, te reverse mode is not always applicable. Te problem relies on te fact tat te reverse mode computation is performed in te opposite way of te original code ( backward sweep ) after a forward sweep needed to store te variable needed in te successive pases. For te previous arguments, we prefer differentiate not te entire program (solution of te state equation + functional evaluation), but te two main component in a separate way, using te fact tat at te solution, te residuals will be zero (i.e., we do not differentiate te routine containing te main loop, but only te quantities involved after te last iteration). For tis second approac,

7 AD-based perturbation metods for uncertainties and errors 71 we ave to analyse te influence of state equation and te functional evaluation in more details. Tis is te purpose of te next sections First derivative Using te cain rule, te gradient of te functional j( γ) = J( γ, W( γ )) is given by: dj J J dw = + dγ γ dγ were te derivatives of te state variables Wc () are obtained solving te linear system dψ Ψ Ψ dw = + =0. dγ γ dγ Terefore, two strategies can be applied. Direct differentiation It consists in computing te Gateaux-derivatives wit respect to eac component direction (( e = (0, 0,1,0,,0) T i, were 1 is at te i -esim component): dj dj J J dw = ei = + dγ dγ γ dγ wit: i i i Ψ dw dγ Ψ = ei i γ (9) (30) Tis as to be applied to eac component of γ, i.e., n times and te cost is n linearised N-dimensional systems to solve. If we coose to solve te single system (30) wit an iterative matrix-free metod, and te solution is obtained after n iter step, te total cost will be of te order of α T n iter, T, i.e., n iter,t evaluation of te matrix-by-vector Ψ operation, x were eac evaluation costs α T times te evaluation of te state residual Ψ( γ, W ) (and te cost of te state residual is taken as reference equal to 1). Terefore, te cost of te full gradient will be n n iter,. Inverse differentiation (reverse mode) Te complete gradient is given by te equation dj j Ψ = Π0 dγ γ γ were Π 0 is te solution of te linear system Ψ j Π= α T T (31) (3) Tis computation needs only one extra linearised N-dimensional system, te adjoint system (some metods for calculation of te adjoint solutions are described in). If we coose to solve te adjoint system (3) wit an iterative matrix-free metod, we can apply te same estimate done as in te case of te tangent mode differentiation, but tis time te overead associated wit te evaluation of te matrixby-vector operation Ψ x respect to te state residual evaluation will be α R and usually αr > αt, and te number of iteration n iter,r for te convergence of te solution could be different from n iter,t of te previous case (but te asymptotical rate of convergence will be te same Ψ of te original linear system =, x b see Pierce and Giles (1998) for more details. Terefore, te cost for te gradient will be α R n iter, R, and te reverse mode differentiation for te gradient computation is ceaper tan te tangent mode if n Second derivative For second derivatives we ave different possibilities. Direct-direct option Tis metod was initially investigated along wit various oter algoritms, but te publication does not go into te implementation details for a generic fluid dynamic code. Here we present te matematical background beind te idea and te efficient AD implementation of Gate and Giles but wit a different analysis of te computational cost. Starting from te derivative (9), we perform anoter differentiation respect to te variable γ k obtaining = Dik, J + i k i k d j j d W dγdγ W dγdγ were J J dw Dik, J = ei ek + ei γ γ γ dγk J dw J dw dw + ek + γ dγi dγi dγk Differentiating te equation (30) we get D were Ψ dw ik, + = 0 Wd id k (33) Ψ γ γ (34) dw Dik, Ψ Ψ Ψ = ei ek + e i γ γ W γ dγk Ψ dw Ψ dw dw + ek + γ dγi dγi dγk

8 7 M. Martinelli et al. Substituting te second derivatives of te state respect to te dw control variables in equation (33) from equation dγidγk (34) we get J Ψ ik, ik, d j = D J D Ψ dγidγ k (35) = D J D Ψ ik, 0 ik, were 0 is te solution of te adjoint system (3) evaluated at te point ( γ, W ( γ )) solution of te state dw equation Ψ( γ, W ) = 0. Te n derivatives sould be dγi computed (and stored) using tangent mode differentiation of te non-linear solver algoritm, and eac derivatives costs n iter α T. If we need te full Hessian matrix, we ave to evaluate te quantity (35) nn ( + 1) times, i.e., we ave to evaluate te terms Dik, Ψ and Dik, J for i = 1,, n and j = i,, n due to te symmetry of te Hessian, and eac evaluation of ( Dik, Ψ costs α T (te evaluation of Dik, J ) is negligible respect to ( Dik, Ψ )). Terefore, te full Hessian costs nαt[ niter, T + ( n + 1) αt ]. Wit similar arguments, if we want only te diagonal part of te Hessian, te cost is nα [ n + α ]. T iter, T T Inverse-direct Tis consists in te direct derivation in any direction ei i, = 1, n of te (non-scalar) function: j J ( γ, W( γ)) = ( γ, W( γ)) γ γ Ψ ( γ, W ( γ)) γ Ψ Ψ θi = ei γ Ψ J λi = ei γ J Ψ + θi 0 γ Ψ 0 θi ei were all te functions in te equations (36) (37) are evaluated at te final state (in order to verify Ψ( γ, W ( γ )) = 0). Inverse-inverse If we are interested in a (scalar!) functional depending on te gradient, ten it can be interesting to apply a second inverse differentiation. We do not focus on tis direction at te moment. 5 Numerical experiments Te interest of tis approac is briefly illustrated by te building of a response surface for te wing sape of a business aircraft (courtesy of Piaggio Aero Ind.), for a transonic regime (see te sape and te mes in Figure 3). Te nominal operational conditions are defined by te free-stream Mac number M = 0.83 and te incidence α =. We suppose tat only tese two quantities are subject to random fluctuations. For simplicity, we assume tat teir PDF are Gaussian wit given mean and variance. Te mean values correspond to te nominal values. Te section of te initial wing sape corresponds to te NACA 001 airfoil. Figure 3 Wing sape and mes in te symmetry were W ( γ ) and ( γ, W ( γ )) are solutions of te above two state systems. Wit some algebra we obtain j j J = γ e = e i i γi γ γ γ j Ψ + θi 0 ei γ γ γ Ψ Ψ 0 θi λi γ γ Te derivation needs te solution of te adjoint systems Ψ j 0 = (36) and n perturbed N-dimensional linear systems (for te full Hessian): For te present work, due to te fact tat we consider only two uncertain variables, we used a ToT approac for te Hessian evaluation. Te accuracy of te second-order response surface obtained wit te differentiated software is not different from te one obtained wit oter works, suc

9 AD-based perturbation metods for uncertainties and errors 73 as tose of Gate and Giles (007) (wo, by te way, also used TAPENADE, but on anoter CFD software). We illustrate tis accuracy in Figure 4. Te direct evaluation required 1 1 non-linear simulations. Te second-order approximation required only one non-linear state equation Ψ = 0 plus four linear systems using ToT. Relative error is less tan % wile using only first derivatives produce errors of 16%. Let us mention tat tis metod compares also well wit Kriging metods as was demonstrated in te comparison paper of Martinelli and Duvigneau (008). Figure 4 Drag coefficient vs. Mac number and angle of attack (first-order spatial accuracy) for te transonic wing, (a) non-linear simulations (b) percentage relative difference between te non-linear simulations and te second order Taylor approximation (see online version for colours) (b) Note: For te top plot we ave solved 1 1 non-linear systems Ψ = 0. 6 Concluding remarks (a) AD metods and tools take place in a process wic tends to make numerical simulation more secure by contributing to build te derived software necessary for addressing uncertainty and error. Tanks to AD, te derivation of tis software is performed wit more and more safety. Tis paper as displayed two main examples of tis process. First, uncertain data are modelled by random variables and teir impact on te simulation process is evaluated by a metod of moments. Te complexity of te proposed metod is analysed for large number of uncertain variables. Te two metods studied are already competitive for a small number of variables and even more for a large one. Acknowledgments Tese studies are supported in part by NODESIM-CFD, an FP-6 European project. References Gate, D. and Giles, M.B. (007) Efficient Hessian calculation using automatic differentiation, Proc. of 5t Applied Aerodynamics Conference, AIAA Miami, Florida, Vol Gate, D. and Giles, M.B. (006) Inexpensive Monte Carlo uncertainty analysis, Recent Trends in Aerospace Design and Optimization Book, pp.03 10, available at ttp:// Hascoët, L. and Dauvergne, B. (008) Adjoints of large simulation codes troug automatic differentiation, European Journal of Computational Mecanics, Vol. 17, pp Hascoët, L. and Pascual, V. (004) TAPENADE.1 user s guide, INRIA Tecnical Report, available at ttp:// Hascoët, L., Naumann, U. and Pascual, V. (005) To be recorded analysis in reverse-mode automatic differentiation, Future Generation Comp. Syst., Vol. 1, No. 8, pp Hascoët, L., Utke, J. and Naumann, U. (008) Ceaper adjoints by reversing address computations, Scientific Programming, Vol. 16, No. 1. Loseille, A. (008) Adaptation de maillage anisotrope 3D multiecelles et ciblée à une fonctionnelle pour la Mécanique des Fluides, application à la prédiction du bang sonique, in Frenc, PD dissertation at te University P.M. Curie, available at ttp://tel.arcives-ouvertes.fr/tel /fr/. Martinelli, M. (007), Sensitivity evaluation in aerodynamics optimal design, PD dissertation, Scuola Normale di Pisa, Italy. Martinelli, M. and Duvigneau, R. (008) Comparison of secondorder derivatives and metamodel-based Monte Carlo approaces to estimate statistics for robust design of a transonic wing, Proceedings of te 10t AIAA Non-Deterministic Approaces Conference, AIAA , Scaumburg, IL, USA. Martinelli, M. and Hascoët, L. (008) Tangent-on-tangent vs. tangent-on-reverse for second differentiation of constrained functionals, Proceedings of te AD008 Conference, Bonn, Germany, to appear in Lecture Notes in Computational Science and Engineering, Springer. Martinelli, M., Dervieux, A. and Hascoët, L. (007) Strategies for computing second-order derivatives in CFD design problems, Proc. of West-East Hig Speed Flow Field Conference, 19 November, Moscou, Russia. Naumann, U., Hascoët, L., Hill, C., Hovland, P.D., Rieme, J. and Utke, J. (008) A Framework for Proving Correctness of Adjoint Message-Passing Programs, PVM/MPI Dublin, Ireland, pp NODESIM-CFD (008) Non-deterministic Simulation for CFDbased Design Tecnologies, FP6 Program, available at ttp://

10 74 M. Martinelli et al. Pascual, V. and Hascoët, L. (005) Extension of TAPENADE towards Fortran 95, Automatic Differentiation: Applications, Teory, and Tools, Lecture Notes in Computational Science and Engineering, Selected papers from AD004, Cicago, Springer. Pascual, V. and Hascoët, L. (008) TAPENADE for C, Advances in Automatic Differentiation, Springer. Pierce, N.A. and Giles, M.B. (1998) Adjoint recovery of superconvergent functionals from approximate solutions of partial differential equations, Oxford University Computing Laboratory Report, AMS (MOS): 65G99,76N15. Tber, M-H., Hascoët, L., Vidard, A. and Dauvergne, B. (007) Building te tangent and adjoint codes of te ocean general circulation model OPA wit te automatic differentiation tool TAPENADE, Researc Report, INRIA, No. 637, available at ttp://al.inria.fr/inria , also in Te Computing Researc Repository (CoRR) abs/