Improvement of adjoint based methods for efficient computation of response surfaces

Transcription

1 Center for Turbulence Researc Annual Researc Briefs Improvement of adjoint based metods for efficient computation of response surfaces By F. Palacios, K. Duraisamy AND J. J. Alonso 1. Motivation and objectives In computational fluid dynamics, te solution of te dual or adjoint) equations in conjunction wit te primal or flow equations) offers valuable information tat can be used in design, sensitivity analysis, error estimation, mes adaptation, uncertainty quantification, etc. From te view point of implementation and application of te adjoint tecnique, one of te following two approaces may be pursued: te continuous metod and te discrete one. In te continuous approac Jameson 1988; Anderson & Venkatakrisnan 1997; Kim et al. 2002; Castro et al. 2007), te adjoint equations are derived from te continuous flow equations and ten subsequently discretized using a numerical metod of coice, wereas in te discrete approac Griewank 2000; Nadaraja & Jameson 2000; Giles et al. 2003; Mavriplis 2006; Mader et al. 2008), te adjoint equations are directly derived from te discretized governing equations. Tese approaces offer distinct advantages in different scenarios and ave bot been implemented for purposes of Verification and Validation V&V) and Uncertainty Quantification UQ) in ypersonic flow applications. In tis work, we outline preliminary work in two areas of researc tat involve te use of adjoints in uncertainty quantification: Uncertainty quantification using non-intrusive metods suc as stocastic collocation Eldred 2009) typically involves te construction of a response surface of te quantity of interest as a function of te uncertain parameters. To acieve te level of desired accuracy, a large number of Computational Fluid Dynamics CFD) simulations is typically required. To improve te efficiency of tis process, we propose a new robust grid adaptation tecnique tat is aimed at minimizing te numerical error over a small variation of te parameters around a nominal state. In tis approac, computational grids are generated wit te knowledge tat small variations of te uncertain parameters can locally cange te solution and ence, te error distribution). Tis is in contrast to classical adjoint tecniques Pierce & Giles 2002; Fidkowski & Darmofal 2009), wic seek to adapt te grid wit te aim of minimizing numerical errors for a nominal flow condition. In te discrete sensitivity framework suc as discrete adjoints, complex step, and finite differences), te grid convergence of nonlinear flow calculations does not guarantee te convergence of linear sensitivities Giles et al. 2003). Tis attribute as a special relevance in problems involving strong socks, were te gradient of an output functional of interest may exibit noise, and ence, cannot be directly utilized. To counter tis difficulty, we utilize a bi-grid filtering strategy Glowinski 1992; Zuazua 2005) tat offers te promise of converged linear sensitivities using a discrete computation tecnique. In summary, te pursuit of UQ science in ypersonic flow applications, togeter wit sensitivity computations using adjoints, as brougt to te table some interesting limitations of te state-of-te-art in adjoint metodologies. In tis report, our early work FuSim-E project, AIRBUS Spain.

2 396 F. Palacios, K. Duraisamy and J. J. Alonso in two novel metodologies is introduced and some preliminary numerical results are presented. 2. Application of adjoint tecniques to enance te computation of response surfaces In tis section, we present a brief introduction to te teoretical basis of te metods tat ave been developed to improve and accelerate te computation of response surfaces, taking advantage of te adjoint metodology to minimize te numerical error incurred wen variations of uncertain input parameters are considered New robust grid metodology A response surface represents te explicit dependence of a desired output of a simulation model as a function of all possible values of te decision/input variables. Te response surface Cung & Alonso 2001; Cantrasmi & Iaccarino 2009) is a powerful tool in UQ, altoug, in practice, a large number of samples is required. Typically tese samples constitute small variations of te parameters of interest in some local neigborood. We seek to develop a new grid adaptation strategy for creating wat we call robust computational grids. Flow computations on tese grids must be able to result in low numerical error even if we sligtly cange te nominal flow conditions suc as te Mac number, geometry, pysical constants, etc.) tat were used in generating te original adapted grid. Te objective is to perform a single grid adaptation about a nominal value of te parameters and to ten use tis grid for many additional simulations wile guaranteeing tat te numerical error is eld constant. A baseline grid tat exibits tis property under reasonable variations of te input parameters is called a robust grid. In te following discussion, te basis for error estimation in integral outputs functionals) of partial differential equations Giles & Pierce 2001; Venditti & Darmofal 2002; Park 2003; Fidkowski & Darmofal 2009) are reviewed. It is well known tat tis local error estimate could be used as an adaptation indicator in a grid adaptive strategy for producing specially tuned grids for accurately estimating some functional. Let us consider a fine grid Ω and a coarse grid Ω H. Te goal is to obtain an accurate estimate of some functional f U ) in te fine grid witout solving te flow or adjoint equations on te fine grid. An expansion of f U ) about te coarse-grid solution is f U ) = f ) + f U U H ) + H.O.T., 2.1) were U H represents te coarse-grid solution expressed on te fine-grid via some consistent interpolation operation. Te nonlinear residual operator representing te flow system of equations is denoted by R U U ) = ) Linearizing te residual about te coarse-grid solution yields R U U ) = R U UH ) + RU U ) + H.O.T., 2.3)

3 Adjoint based metods for computation of response surfaces 397 were, neglecting te ig order terms, we can write U U H ) R U = Now we define te linear adjoint residual operator R Ψ Ψ) as ) T R Ψ R U Ψ) Ψ f ) 1 R U ). 2.4), 2.5) were it is important to note tat in tis last expression we are using a prolongated solution U H to evaluate te Jacobian. By definition RΨ Ψ ) = 0, and ence, ) T R U Ψ = f. 2.6) Introducing 2.6) and 2.4) into 2.1) we finally obtain te following expression for te estimate of te numerical error: ǫ U, ) = f ) f U ) = Ψ ) T R U ). 2.7) Unfortunately, to evaluate tis last expression it is necessary to solve te adjoint problem 2.6) on te fine grid. Te next step is to look for a more appropriate way for estimating te error. Defining Ψ H as te coarse-grid adjoint solution expressed on te fine grid via some consistent projection operation, te error can be written as ǫ U, ) = Ψ H ) T R U ) + Ψ Ψ H ) T R U ), 2.8) were te difference Ψ Ψ H ) can be estimated using 2.6), and using RΨ Ψ ) = 0 we finally obtain ǫ U, ) = Ψ H )T R U UH ) + ) 1 R Ψ ΨH )) T R U R U UH ). 2.9) Once again, using 2.4) it is possible to write te error estimation in te following useful form: ǫ U, ) = Ψ H ) T R U ) + R Ψ Ψ H ) ) T U ). 2.10) Tis expression can be used to generate specially tuned grids for accurately estimating te error in calculating a functional. Note tat te first term on te rigt and side of te above equation is known via coarse-grid computations and fine-grid interpolations. Previous work Fidkowski & Darmofal 2009; Venditti & Darmofal 2002) as successfully used tis expression to elp guide te mes adaptation process. Our aim is to introduce an adaptation tecnique tat produces low numerical error for a range of variation in some flow or geometric parameters. Te first step in te new metodology is to define a robust grid as a grid in wic, for small variations of te solution, te numerical error is nearly constant. In oter words, te variation of te numerical error wit respect to variations of a parameter k geometric or flow properties of te problem), must be small or zero in te ideal scenario. Te derivative of te numerical error 2.10) owing to a variation of te parameter k can be evaluated as dǫ dk = dψ H dk ) T R U ) + Ψ H ) T dru UH ) dk

4 398 F. Palacios, K. Duraisamy and J. J. Alonso dr Ψ + Ψ H ) ) T U ) + R Ψ Ψ H ) ) T du ), 2.11) dk dk were te adjoint and te flow solutions are functions of te parameter k. Te value of U ) can be estimated using 2.4) to obtain dǫ dk = ΨH ) T dru UH ) ) + dψ H T dr Ψ Ψ H ) ) T ) 1 R U R U ) dk dk dk du + ) ) T R Ψ Ψ H ). 2.12) dk In te limit of an extremely well adapted mes for te direct and adjoint problems, we expect tat R ΨΨH ) 0 and RU UH ) 0. In tat situation, te variation of te error due to a cange in te parameter k can be computed as dǫ dk = ΨH ) T dru UH ). 2.13) dk Te key idea of tis development is tat if we adapt te computational grid for solving te direct and te adjoint problems independently, te last expression gives us a very easy way to compute te numerical error wen we cange te flow conditions to determine te complete response surface. Furtermore, supposing tat V H is te new solution on te coarse grid, and Ψ H te adjoint solution in te original problem, te correction to te functional value can be computed as ǫv H ) ǫu, ) = Ψ H )T R V V H ) RU UH )). 2.14) Using 2.10), te error of te new solution can be computed as ǫv H ) = Ψ H ) T R V V H ) + R Ψ Ψ H ) ) T U ) Ψ H ) T R V V H ), 2.15) were it is important to note tat tis error does not depend on te finest grid solution. In principle, for a reconstruction-based finite volume sceme, te numerical error is likely to cange even wen te solution is sligtly perturbed by a cange in input conditions. Fortunately, tis error can be estimated using coarse-grid calculations. On te oter and, if we ave Galerkin ortogonality, te first term in 2.10) will be zero and te adapted grid for te adjoint and flow problem) will provide excellent results even wen some flow parameter is sligtly canged. Te adaptation tecnique tat we ave developed is based in obtaining R ΨΨH ) 0 and R UUH ) 0 for te baseline flow solution. To acieve tis objective, it is natural to employ a residual-based indicator to drive te flow and adjoint adaptation process. Eiter of te following approaces can be pursued: A direct approac, wic consists of computing te value of R ΨΨH ) and RU UH ) in a very fine grid and subsequently adapting te original grid in zones were tese values are large. An indirect strategy, in wic te residual is driven to zero in te dual control volumes of te grid. Note tat te original numerical sceme is designed to ave a zero flow residual on te primal control volumes. Once we ave an estimation of te residual for te direct and adjoint problems, it can be used as an indicator for mes adaptation. Wit regard to te refinement process itself Boroucaki et al. 2005), tere is only one element subdivision tat can preserve

5 Adjoint based metods for computation of response surfaces 399 te element sape quality: omotetic subdivision, wic for a planar triangle, splits it into four by adding tree new vertices at te middle of te edges. It as to be noted tat elements tat sare tese edges must also be divided if anging nodes are to be avoided. In te residual based strategy we typically define te error as e = Ω α R 2 ρ + R2 ρ v + R2 ρe, 2.16) were Ω is te area of te control volume, and R i are te four two-dimensional) or five tree-dimensional) residuals at eac node. A widely used formulation utilizes α = 1. Te advantage of scaling te indicator wit a positive value of α is tat te adaptation stops automatically in te corresponding area after several cycles even wen discontinuities are present in te flowfield. Finally an indicator of te error is computed as e lim = e m + c lim e s, 2.17) were e lim is te error limit, e m is te mean of te error indicator, e s is te standard deviation of te error indicator, and c lim is a constant. Typically a value near unity is used for te constant. Te omotetic element refinement is used in grid elements were te error indicator is greater tan a corresponding error tolerance Sensitivity computation on small pressure sensor bi-grid filtering tecnique) It is known tat wit te discrete sensitivity approac automatic differentiation, complex step, and finite differences), te convergence of nonlinear flow equations to macine precision does not guarantee te convergence of functional sensitivities Giles et al. 2003). In particular, we ave observed tat in te presence of strong socks and for functionals tat extend across only a small number of mes cells, tere are significant differences between te gradient of a functional computed wit te continuous and te discrete adjoints or exact finite differences), in tat te latter can be expected to be noisy mainly for functionals wit a small spatial footprint). Te use of automatic differentiation Griewank 2000; Martins et al. 2006; Gauger et al. 2007; Mader et al. 2008) is a very attractive tecnique to tackle derivative computations wen dealing wit complex multi-pysics systems. However, as we will see, in te presence of strong socks te computed gradient can be noisy, and tis situation as critical relevance if we are interested in computing second derivatives using te same metodology. To obtain a useful expression to process te discrete sensitivities, te starting assumption is tat te continuous adjoint metodology will provide a gradient approximation tat will be closer to te analytical gradient in fact, in te limit of a very fine grid it must be te analytical one). Tis as been confirmed by performing numerical experiments in flows wit strong socks. Tus, an attempt will be made at building a discrete adjoint tat mimics te beavior of te continuous adjoint. Once suc a discrete adjoint is constructed, te next step will be to compare tis new discrete adjoint wit te classical one. Consider te stationary Euler equations in a domain Ω: RU) = FU) = 0, in Ω, 2.18) were R is te residual, U is te vector of conservative variables, and F is te convective flux vector. Te Euler equations ave to be completed wit te flow tangency boundary condition at all solid walls, and at te inlet and outlet boundary conditions are specified

6 400 F. Palacios, K. Duraisamy and J. J. Alonso for incoming waves, wile outgoing waves are determined by te solution inside te fluid domain. In a numerical practice, a modified version of tese equations is solved. Tis modified governing equation can be written as RU) + DU) = FU) + ǫ GU) U) = 0, in Ω, 2.19) were te te stabilizing viscosity term can be interpreted as te discretization of a second derivative Gu) u)) for some viscosity function G depending on te particular sceme under consideration, and ǫ contains powers of spatial coordinates. It is important to note tat ǫ sould approac zero as te grid spacing approaces zero. If we define a functional JU, k) tat depends on te conservative variables and a parameter k tat depends on te flow or geometry, te complete variation of te functional is δj = J J δu + δk. 2.20) k Typically, te discrete adjoint problem will be introduced troug te Lagrange multipliers to remove te explicit dependence on δu. It is important to note tat our objective is to compare two discrete adjoints, te classical one and a new discrete adjoint tat mimics te continuous adjoint beavior: In te classical discrete adjoint approac, te modified state equation 2.19) is fulfilled, and we will build te Lagrangian: LU, k, λ D ) = JU) + λ D [RU, k) + DU)]. On te oter and, using ideas inspired in te continuous adjoint problem, we also define te following Lagrangian: LU, k, λ C ) = JU) + λ C [RU, k) + C], were C is a constant wic includes ig-order terms of te discretization tat do not depend on U in te continuous adjoint procedure). Te variation of te objective function J in tese approaces is, terefore δj = J δj = J δu + J k R δk λd + D R J δu + δk λc k ) R δu λ D k ) δu λ C R k ) δk, 2.21) ) δk. 2.22) Te final expressions for te sensitivities old if λ D and λ C satisfy te following adjoint equations: J R = λd + D ) J R, = λc, 2.23) were te classical discrete adjoint can be written as λ D = J ) 1 R λ D D ) 1 R. 2.24) Te last step of our formulation involves computing te difference between te discrete adjoint, wic mimics te continuous adjoint beavior, and te classical discrete adjoint: ) 1 λ C = λ D D R + I), 2.25) were it must be possible to ceck tat D R ) 1 is noting more tan a smooter of te continuous adjoint solution. Te main issue appears wen we are looking for solutions wit strong socks. In tat

7 Adjoint based metods for computation of response surfaces 401 R ) 1. Te analogy of tis problem wit oter similar problems Glowinski situation we ypotesize tat tis operator produces a strong damping in ig frequencies of te continuous adjoint beyond a critical wave number. In oter words, te ig frequency numerics of te continuous adjoint solution are limited by te dispersion produced by D 1992; Zuazua 2005) suggests a filtering of te ig frequencies to eliminate te armful effect of te smooter described above. Tis may be done by a bi-grid algoritm. Te bi-grid strategy is a powerful numerical tool tat can be used to damp or filter te spurious numerics in control problems. To acive tis radical filtering it suffices to define a finite grid of step size 2. Following tis, te flow solution is approximated in tat grid and re-interpolated again to te fine mes. Wit tis procedure, te sort wavelengts are eliminated from te adjoint solution. Te major component of te noise was introduced at tese scales. Finally, an iterative process is proposed to compute te new discrete adjoint solution λ D n+1 = λc Filter { λ D n D ) )} 1 R, were λ D 0 = λc. 2.26) It is important to note tat tis filtering strategy will be especially useful if we are interested in computing sensitivities on small sensors or for second derivatives Continuous Euler adjoint formulation for two-dimensional ypersonic problems Te sensitivity computation in internal aerodynamics problems is formulated on a fluid domain Ω, enclosed by boundaries divided into a inlet S in, outlet S out, and solid wall boundaries S wall. Te objective is to compute te variation of J = jp, n S )ds, 2.27) S S wall wit respect to canges in te geometry and inlet flow. It is important to note tat jp, n S ) is a function tat depends on n S inward-pointing unit normal to S) and te pressure P. Hypersonic inviscid flows are caracterized by te appearance of strong sock discontinuities Σ. If te te discontinuity touces te surface S and x b = Σ S we ave tat, in order to account for discontinuities Bardos & Pironneau 2002) in te variation of J, δj = + S [ j P np + t tg S\x b j n S ) κ j + j n S n S )] δs ds j δp ds [j] x b [δσx b ) n S n Σ )δsx b )]. 2.28) P n S t Σ Ideal fluids are governed by te Euler s equations introduced in te previous section) tat [ can] develop strong sock waves. Wen tis occurs, te Rankine-Hugoniot conditions F nσ = 0 relate te flow variables on bot sides of te discontinuity. Σ In te presence of sock discontinuities te formal linearization of te state equations fails to be true Baeza et al. 2009) and te sensitivity of te model needs to take into account perturbations of te solution and perturbations of te location of te sock. In tis case, δu stands for te infinitesimal cange in te state on bot sides of te discontinuity and results from te solution of te linearized Euler equations, wereas δσ describes te infinitesimal normal deformation of te discontinuity and it results from a linearization of te Rankine-Hugoniot conditions:

8 402 F. Palacios, K. Duraisamy and J. J. Alonso ) AδU = 0, in Ω \ Σ, δ v n S = δs n v n S + tg δs) v t S, on S wall \ x b, δw) + = δw in, on S in, δw) [ + = 0, [ ] on S out, AδΣ n U + δu)] Σ + F Σ = 0 on Σ, Σ Σ 2.29) wit δw) + representing te incoming caracteristics and were F/ = A is te Jacobian matrix. In order to eliminate δp and δσ from 2.28), te adjoint problem is introduced troug Lagrange multipliers Ψ T A T Ψ = 0, in Ω \ Σ, ϕ n S = j, on S wall \ x b, Ψ T A n S ) = 0, on S in S out, [ ] [ ] Ψ T Σ = 0, tgψ T F t Σ = 0, on Σ, [ ] Ψ T x b ) F t Σ = [j] xb, at x x n b. b S t Σ 2.30) Te procedure is as follows: first te linearized Euler equations and Rankine-Hugoniot conditions are multiplied by te adjoint variables. Ten, te result is integrated over te domain taking care of te discontinuities), and finally te result is added to 2.28). Upon identifying te corresponding terms in tat final expression wit te adjoint equations, we can write te variation of te functional as: [ ) j j δj = S P np + t tg κ n S [ ) )] + n v n S )ϑ + tg v t S ϑ δs ds S\x b + j + j )] n S δs ds n S S in Ψ T A n S )δu in ds + [jp)] xb n S n Σ n S t Σ δs x b ), 2.31) were κ is te surface curvature, ϕ = ψ 2, ψ 3 ) and ϑ = ρψ 1 + ρ v S ϕ + ρhψ 4. Using 2.30) and 2.31) we are able to compute any variation of J due to a geometrical or inlet flow cange, even wen te computation of J includes a sock discontinuity impinging on te surface of integration S. Note tat te importance of some of tese terms depends on te actual flow condition and, in particular, on te relative angle of impingement of te discontinuity on te surface S. Wile not explicitly used in te results presented ere, tis derivation is meant to suggest tat care must be exercised to ensure tat terms in adjoint solutions tat include discontinuities are accounted for carefully. Witin te context of actual flow discretizations, discontinuities are represented by regions of ig gradients. Terefore, te importance of all terms must be carefully assessed. 3. Numerical experiments In tis section a selection of preliminary numerical results are sown. Two geometries involving single and multiple sock reflections in a constant area cannel) ave been selected as baseline configurations. Inviscid simulations at ig Mac number and

9 Adjoint based metods for computation of response surfaces 403 Figure 1. Density contours for nominal conditions Mac number 4.0 and ramp angle 6.0 ) Figure 2. Adjoint density contours for nominal conditions and pressure sensor located in te lower wall Figure 3. Grid adaptation using a gradient-based metod upper), adjoint-based computable error middle) and te new robust metod lower) functionals defined over small patces of te solid wall will be common aspects of te simulations tat we ave performed Robust grid adaptation To study te applicability of tis metod, a two-dimensional Euler supersonic test case as been selected. In particular, we are interested in creating a robust grid for te Clemens configuration Wagner et al. 2007). Te nominal conditions as in te experiments) are an inlet Mac number of 5.0 and a ramp angle 6.0 Fig.1). Te sensitivity computation Fig.2) as been performed using a second-order continuous adjoint solution. Te objective of tis test is to verify if te grid adapted using te new robust tecnique) for tose nominal conditions also provides a good value of te objective function for te complete response surface. Te baseline grid as a total of 1, 356 nodes and 2, 450 triangular elements. Te total lengt of te setup is mm, and te pressure sensor is located on te lower wall at mm from te inlet te size of te pressure sensor is 5.08 mm). In order to compare te performance of different grid adaptation tecniques Fig.3)

10 404 F. Palacios, K. Duraisamy and J. J. Alonso Figure 4. Comparison between different adaptation strategies grey color signifies an error < 4.0%) Figure 5. Direct comparison between computable error based adaptation and robust grid adaptation we ave used tree different error indicators: gradient based, classical adjoint based computable error), and te new robust adaptation metod. Once te adaptation is completed, te final step consists of computing te entire response surface Fig.4) and comparing te results wit te response surface tat as been computed using a very fine reference grid Fig.5). In conclusion, for te complete response surface computation, te robust grid strategy provides a lower error compared wit oter tecniques, despite te fact tat only one adaptation around Mac 5.0 and ramp angle 6.0 is used.

11 Adjoint based metods for computation of response surfaces 405 Figure 6. Density contours of te supersonic ramp Figure 7. Adjoint density contours of te supersonic ramp; note tat te sensor is located were te adjoint solution reaces it maximum value at te solid wall Figure 8. Comparison between continuous adjoint sensitivity and finite-difference tecnique on a baseline grid continuous adjoint ; finite differences ; finite differences wit bi-grid ) Figure 9. Comparison between continuous adjoint sensitivity and finite-differences tecnique on a very fine grid continuous adjoint ; finite differences bi-grid ) ; finite differences wit 3.2. Bi-grid filtering tecnique on a supersonic ramp To study te applicability of te bi-grid filtering tecnique, a two-dimensional Euler supersonic ramp of 25 Fig.6) is selected. In particular, we are interested in te derivative, wit respect to te inlet Mac number, of te pressure on a small sensor located approximately in te middle of te ramp. Wit te use of a fourt-order dissipation-based discretization, te continuous adjoint Fig.7) sensitivity was confirmed to be witin 0.1% of te analytical sensitivity. Similarly, te discrete adjoint sensitivity was confirmed to exactly matc te finite-difference value for a small enoug finite difference step size. For comparison purposes, we use finite differences to approximate te discrete sensitivity, and te bi-grid tecnique are applied to te solution of te direct problem. Te convergence of te finite-difference sensitivity is sown for two different grids: baseline grid Fig.8) wit 5, 680 nodes, Mac number 4.3, and a very fine grid Fig.9) wit 22, 472 nodes, Mac number 4.1. It is interesting to note tat te finite-difference approac sows a strange beavior near te analytical value of te gradient, beyond wic it does not converge to te analytical value of te gradient represented by te continuous adjoint solution). Once te bi-grid tecnique is applied to te flow solution, we obtain a value for te discrete sensitivity tat is very similar to te analytic one.

12 406 F. Palacios, K. Duraisamy and J. J. Alonso 4. Future plans and conclusions In tis work we proposed and sowed preliminary results for two new adjoint-based tecniques tat were aimed at enancing te computation of response surfaces and flow sensitivities. We ave demonstrated tat by adapting te computational grid using a residualbased strategy for te flow and te adjoint solutions, one can obtain a robust grid wit respect to small variations of te flow parameters. Furter work will be required to incorporate anisotropic adaptation, develop sensors to estimate te strengt of adaptation, and obtain rigorous upper bounds for te error estimation. We ave also proposed a metod to filter te flow solution wit te aim of obtaining noise-free sensitivities using discrete tecniques. Tis bi-grid strategy is expected to be especially useful for flows wit strong socks and for objective functions tat are defined over a small portion of te computational boundary. Furter work must be done in order to understand te beavior of te filtering operator. Current tests are underway in testing tis witin te framework of Automatic Differentiation AD). Acknowledgments Tis work was supported by te U.S. Department of Energy under te Predictive Science Academic Alliance Program PSAAP). Te first autor was visiting te Aero/Astro Department at Stanford University wile tis work was performed. He also tanks Prof. E. Zuazua for elpful discussions. REFERENCES Anderson, W. K. & Venkatakrisnan, V Design optimization on unstructured grids wit a continuous adjoint formulation. AIAA Paper Baeza, A., Castro, C., Palacios, F. & Zuazua, E D Euler sape design on nonregular flows using adjoint Rankine-Hugoniot relations. AIAA J. 47 3), Bardos, C. & Pironneau, O A formalism for te differentiation of conservation laws. C. R. Acad. Sci., Paris I 335), Boroucaki, H., Villard, J., Laug, P. & George, P. L Surface mes enancement wit geometric singularities identification. Comput. Metods Appl. Mec. Engrg. 194, Castro, C., Lozano, C., Palacios, F. & Zuazua, E A systematic continuous adjoint approac to viscous aerodynamic design on unstructured grids. AIAA J. 45 9), Cantrasmi, T. & Iaccarino, G Computing sock interactions under uncertainty. AIAA Paper Cung, H. S. & Alonso, J. J Using gradients to construct response surface models for ig-dimensional design optimization problems. AIAA Paper Eldred, M. S Recent advances in non-intrusive polynomial caos and stocastic collocation metods for uncertainty analysis and design. AIAA Paper Fidkowski, K. J. & Darmofal, D. L Output-based error estimation and mes adaptation in computational fluid dynamics: overview and recent results. AIAA Paper Gauger, N. R., Walter, A., Moldenauer, C. & Widalm, M Automatic

13 Adjoint based metods for computation of response surfaces 407 differentiation of an entire design cain for aerodynamic sape optimization. New Res. in Num. and Exp. Fluid Mec. 96, Giles, M. B., Duta, M. C., Müller, J.-D. & Pierce, N. A Algoritm developments for discrete adjoint metods. AIAA J. 41 2), Giles, M. B. & Pierce, N. A Adjoint error correction for integral outputs. Tec. Rep. 01/18. Oxford University Computing Laboratory, Numerical Analysis Group, Oxford, England. Glowinski, R Ensuring well-posedness by analogy; Stokes problem and boundary control for te wave equation. J. of Comp. Pys. 103, Griewank, A Evaluating derivatives, principles and tecniques of algoritmic diferentiation. Society for Industrial and Applied Matematics, Piladelpia. Jameson, A Aerodynamic design via control teory. J. Sci. Comput. 3, Kim, S., Alonso, J. J. & Jameson, A Design optimization of ig-lift configurations using a viscous continuous adjoint metod. AIAA Paper Mader, C. A., Martins, J. R. R. A., Alonso, J. J. & van der Weide, E ADjoint: an approac for rapid development of discrete adjoint solvers. AIAA J. 46 4), Martins, J. R. R. A., Mader, C. A. & Alonso, J. J ADjoint: an approac for rapid development of discrete adjoint solvers. AIAA Paper Mavriplis, D. J Multigrid solution of te discrete adjoint for optimization problems on unstructured meses. AIAA J. 44 1), Nadaraja, S. K. & Jameson, A A comparison of te continuous and discrete adjoint approac to automatic aerodynamic optimization. AIAA Paper Park, M. A Tree-dimensional turbulent rans adjointbased error correction. AIAA Paper Pierce, N. A. & Giles, M. B Adjoint and defect error bounding and correction for functional estimates. J. of Comp. Pys. 200, Venditti, D. A. & Darmofal, D. L Grid adaptation for functional outputs: application to two-dimensional inviscid flows. J. of Comp. Pys. 176, Wagner, J. L., Valdivia, A., Yuceil, K. B., Clemens, N. T. & Dolling, D. S An experimental investigation of supersonic inlet unstart. AIAA Paper Zuazua, E Propagation, observation, control and numerical approximation of waves approximated by finite difference metod. SIAM Review 47 2),