CME 345: MODEL REDUCTION

Transcription

1 CME 345: MODEL REDUCTION Methods for Nonlinear Systems Charbel Farhat & David Amsallem Stanford University 1 / 65

2 Outline 1 Nested Approximations 2 Trajectory PieceWise Linear (TPWL) Method 3 Hyper-Reduction Methods 4 Local Approaches 5 References 2 / 65

3 Nested Approximations Nonlinear HDM HDM of interest d w(t; µ) dt = f (w(t; µ), u(t); t; µ) y(t; µ) = g (w(t; µ), u(t); t, µ) w R N : Vector of state variables u R p : Vector of input variables, typically p N y R q : Vector of output variables, typically q N µ R m : Vector of parameter variables, typically m N f : Nonlinear function Usually, there is no closed form solution for w(t; µ) 3 / 65

4 Nested Approximations Model Order Reduction by Petrov-Galerkin Projection Approximation of the state using a right ROB Resulting nonlinear ODE w(t; µ) Vq(t; µ) V d q(t; µ) = f(vq(t; µ), u(t); t; µ) + r(t; µ) dt Enforcement of the orthogonality of the residual r to a left ROB W W T V d dt q(t; µ) = WT f(vq(t; µ), u(t); t; µ) If W T V is nonsingular, the above equation can be re-written as d dt q(t; µ) = (WT V) 1 W T f(vq(t; µ), u(t); t; µ) 4 / 65

5 Nested Approximations An Issue with the Reduction of Nonlinear Models Petrov-Galerkin projection-based ROM d dt q(t; µ) = (WT V) 1 W T f(vq(t; µ), u(t); t; µ) k equations in terms of k unknowns For a given reduced state q(t; µ), the evaluation of f k (q(t; µ), u(t); t, µ) = (W T V) 1 W T f(vq(t; µ), u(t); t; µ) at a given time t and parameter µ is performed in 3 steps 1 compute w(t; µ) = Vq(t; µ) 2 evaluate f(vq(t; µ), u(t); t; µ) 3 left-multiply the result by (W T V) 1 W T to obtain (W T V) 1 W T f(vq(t), t) The computational cost associated with these three steps scales linearly with the dimension N of the HDM Hence, for nonlinear problems, dimensional reduction as described above does not necessarily lead to significant CPU time reduction 5 / 65

6 Nested Approximations Nested Approximations In this case, an additional level of approximation is required to ensure that the online cost associated with solving the reduced nonlinear equations does not scale with the dimension N of the HDM This leads to nested approximations state approximation nonlinear function approximation There are two main classes of nonlinear function approximation linearization approaches (TPWL, ManiMOR,...) hyper-reduction approaches (DEIM, ECSW, GNAT,...) 6 / 65

7 Trajectory PieceWise Linear (TPWL) Method Linear Approximation of the Nonlinear Function Consider a nonlinear HDM of the form d w(t) = f(w(t)) + Bu(t) dt stationary system no parametric dependence for now separable linear input For linear HDMs, reduced-order operators of the type A r = (W T V) 1 W T AV can be pre-computed offline once for all Idea: linearize f around an operating point w 1 f(w) f(w 1 ) + f w (w 1)(w w 1 ) = f(w 1 ) + A(w 1 )(w w 1 ) Then, the resulting approximated system is linear in the state w(t) d dt w(t) = A(w 1)w(t) + [Bu(t) + (f(w 1 ) A(w 1 )w 1 )] 7 / 65

8 Trajectory PieceWise Linear (TPWL) Method Model Order Reduction Approximated HDM system d dt w(t) = A(w 1)w(t) + [Bu(t) + (f(w 1 ) A(w 1 )w 1 )] Reduced-order system after Petrov-Galerkin projection d dt q(t) = (WT V) 1 W T A(w 1 )Vq(t) +(W T V) 1 W T [Bu(t) + (f(w 1 ) A(w 1 )w 1 )] The following linear time-invariant operators can be pre-computed A r = (W T V) 1 W T A(w 1)V R k k B r = (W T V) 1 W T B R k p F r = (W T V) 1 W T [(f(w 1) A(w 1)w 1)] R k 8 / 65

9 Trajectory PieceWise Linear (TPWL) Method Piecewise Linear Approximation of the Nonlinear Function Idea: Linearize the nonlinear function at multiple locations in the state space Extend the domain of validity of the linearization assumptions Approximated high-dimensional dynamical system d s dt w(t) = ω i (w(t))(f(w i ) + A i (w(t) w i )) + Bu(t) i=1 y(t) = g(w(t), u(t); t) the s points {w i } s i=1 are linearization points the s coefficients {ω i } s i=1 are weights such that s ω i (w) = 1, w R N i=1 9 / 65

10 Trajectory PieceWise Linear (TPWL) Method Reduced-Order Modeling For simplicity, assume W T V = I k : In this case, the ROM obtained via Petrov-Galerkin projection is d s dt q(t) = ω i (q(t))(w T f(w i ) + W T A i (Vq(t) w i )) + W T Bu(t) where i=1 y(t) = g(vq(t), u(t); t) Equivalently d dt q(t) = s ω i (q) = 1, q R k i=1 ( s ) ( s ) ω i (q(t))a ri q(t) + ω i (q(t)) F ri + B r u(t) i=1 A ri = W T A i V, i = 1,, s B r = W T B F ri = W T (f(w i ) A i w i ), i = 1,, s i=1 10 / 65

11 Trajectory PieceWise Linear (TPWL) Method Reduced-Order Modeling In this context, a complete model order reduction method should provide algorithms for selecting the linearization points {w i } s i=1 selecting the ROBs V and W Determining the weights { ω i (q)} s i=1, q R k 11 / 65

12 Trajectory PieceWise Linear (TPWL) Method Selection of the Linearization Points Note that each linear approximation of the nonlinear function f is valid only in a neighborhood of each w i Note also that, in practice, it is impossible to cover the entire state-space R N by local linear approximations The Trajectory PieceWise Linear (TPWL) model order reduction method (2001) uses pre-computed trajectories of the HDM (offline) to select the linearization regions selects an additional linearization point from the HDM trajectory if it is sufficiently far away from the previously selected points 12 / 65

13 Trajectory PieceWise Linear (TPWL) Method Selection of the ROBs Possible methods for constructing a global basis V include if the input function is linear in u, constructing Krylov subspaces K i = K(A 1 i, A 1 i B) = range(v i ) at each linearization point w i and assembling a global basis V such that range(v) = range ([V 1 V s]) ad-hoc methods (Balanced truncation, POD...) The left ROB W can be chosen based on the output of interest (two-sided Krylov moment matching), or simply as W = V (Galerkin projection) 13 / 65

14 Trajectory PieceWise Linear (TPWL) Method Determination of the Weights {ω i } The weights are used to characterize in the reduced space R k the distance of the current point q(t) to the projection of the linearization points onto range(v) that is, {q i = ( V T V ) 1 V T w i } s i=1 one possible choice is ω i (q) = ( ) exp βd2 i m ( 2 ) s exp βd2 j m 2 where β is a constant, d i = q q i 2, and m = min s j=1 d j other choices can be found in the literature j=1 14 / 65

15 Trajectory PieceWise Linear (TPWL) Method Further Developments A posteriori error estimators are available when f is negative monotone Stability guarantee is possible under some assumptions on f and specific choices for V and the weights { ω i (q)} s i=1 Passivity preservation (i.e. no energy creation in a passive system) is possible under similar assumptions TPWL using local ROBs (ManiMOR) 15 / 65

16 Trajectory PieceWise Linear (TPWL) Method Analysis of the TPWL Method Strengths The cost of the online phase does not scale with the size N of the HDM The online phase is not software-intrusive Weaknesses It is essential to choose good linearization points offline Requires the extraction of Jacobians from the HDM software Many parameters to adjust (number of linearization points, weights,...) 16 / 65

17 Hyper-Reduction Methods The Gappy POD First applied to face recognition (Emerson and Sirovich, Karhunen-Loeve Procedure for Gappy Data, 1996) Other applications flow sensing and estimation flow reconstruction nonlinear model order reduction 17 / 65

18 Hyper-Reduction Methods The Gappy POD Face recognition Procedure 1 build a database of N s faces (snapshots) 2 construct a POD basis V f for the database 3 for a new face f, record a small number k i of pixels f i1,, f iki 4 using the POD basis V f, approximately reconstruct the new face f (in the least-squares sense) 18 / 65

19 Hyper-Reduction Methods Nonlinear Function Approximation by Gappy POD The gappy approach can also be used to approximate the nonlinear function f in the reduced equations d dt q(t) = WT f(vq(t); t) (for simplicity, the input function u(t) is not considered here) The evaluation of all the entries of f( ; t) is computationally expensive (scales with N) Gappy approach only a small subset of these entries is evaluated the other entries are reconstructed either by interpolation or a least-squares strategy using a pre-computed ROB V f The dimension of the solution space is still reduced using any preferred model order reduction method (for example, POD) 19 / 65

20 Hyper-Reduction Methods Nonlinear Function Approximation by Gappy POD A complete model order reduction method based on the Gappy approach should then provide algorithms for selecting the evaluation entries I = {i 1,, i ki } selecting a reduced-order basis V f for the nonlinear function f reconstructing the complete approximated nonlinear function ˆf( ; t) 20 / 65

21 Hyper-Reduction Methods Construction of a POD Basis for f Construction of a POD basis V f of dimension k f 1 collect snapshots for the nonlinear function f from one or several transient simulations F = [f(w(t 1); t 1) f(w(t mf ); t mf )] R N m f 2 compute a thin SVD F = U f Σ f Z T f 3 truncate the ROB to a dimension k f m f by selecting the first k f vectors in U f V f = [u f,1 u ] f,kf 21 / 65

22 Hyper-Reduction Methods Reconstruction of an Approximated Nonlinear Function Assume that k i indices (entries) have been chosen (the choice of indices will be specified later) I = {i 1,, i ki } Consider the N k i mask matrix ] P = [e i1 e iki At each time t, given a value of the state w(t) = Vq(t), evaluate only those entries of f corresponding to the above indices f i1 (w(t); t) P T f(w(t); t) =. f iki (w(t); t) This is computationally economical if k i N Usually, only a subset of the entries of w(t) are required to construct the above vector (case of a sparse Jacobian) 22 / 65

23 Hyper-Reduction Methods Discrete Empirical Interpolation Method (DEIM) Case where k i = k f interpolation idea: ˆf ij (w; t) = f ij (w; t), w R N, j = 1,, k i this means that P Tˆf(w(t); t) = P T f(w(t); t) recalling that ˆf( ; t) belongs to the range of V f that is, then ˆf(Vq(t); t) = V f f r (q(t); t), where f r (q(t); t) R k f P T V f f r (q(t); t) = P T f(vq(t); t) assuming that P T V f is nonsingular f r (q(t); t) = (P T V f ) 1 P T f(vq(t); t) interpolating the high-dimensional nonlinear function ˆf( ; t) as follows ˆf( ; t) = V f (P T V f ) 1 P T f( ; t) = Π Vf,Pf( ; t) the Discrete Empirical Interpolation Method (DEIM) results in an oblique projection of the high-dimensional nonlinear vector 23 / 65

24 Hyper-Reduction Methods Oblique Projection of the High-Dimensional Nonlinear Vector ˆf(, t) = Vf (P T V f ) 1 P T f(, t) = Π Vf,Pf(, t) Π V,W = V(W T V) 1 W T : Oblique projector onto V, orthogonally to W 24 / 65

25 Hyper-Reduction Methods Least-Squares Reconstruction Case where k i > k f least-squares reconstruction idea: ˆf ij (w; t) f ij (w; t), w R N, j = 1,, N, in the least-squares sense this leads to the minimization problem f r (q(t); t) = argmin P T V f y r P T f(vq(t); t) 2 y r R k f note that M = P T V f R k i k f is a skinny matrix its singular value decomposition can be written as M = UΣZ T then, the left inverse of M can be defined as M = ZΣ U T where Σ = diag( 1 σ 1,, 1 σ r, 0,, 0) if Σ = diag(σ 1,, σ r, 0,, 0), where σ 1 σ r > 0 and therefore ˆf(q(t); t) = Vf (ZΣ U T ) P T f(vq(t); t) = V f (P T V f ) P T f(vq(t); t) 25 / 65

26 Hyper-Reduction Methods Greedy Function Sampling This selection takes place after the matrix V f = [v f,1 v f,kf ] has been computed using, for example, POD Greedy algorithm 1: [s, i 1 ] = max{ v f,1 } 2: V f = [v f,1 ], P = [e i1 ] 3: for l = 2 : k f do 4: solve P T V f c = P T v f,l for c 5: r = v f,l V f c 6: [s, i l ] = max{ r } 7: V f = [V f, v f,l ], P = [P, e il ] 8: end for 26 / 65

27 Hyper-Reduction Methods Analysis of Hyper-Reduction Methods Strengths The cost of the online phase does not scale with the size N of the HDM The hyper-reduced function is usually robust with respect to deviations from the original training trajectory Weaknesses The online phase is software-intrusive Many parameters to adjust (ROB sizes, mask size,...) 27 / 65

28 Hyper-Reduction Methods Application to the Reduction of the Burgers Equation Consider the inviscid Burgers equation U(x, t) t + 1 (U(x, t)) 2 = g(x) 2 x source term g(x) = 0.02 exp(0.02x) initial condition U(x, 0) = 1 inlet boundary condition U(0, t) = 5 Discretize it by a Finite Volume (Godunov) method 28 / 65

29 Hyper-Reduction Methods Application to the Reduction of the Burgers Equation k = 15, k f = 40, k i = U x Similar results for k i > 40 (least-squares reconstruction) 29 / 65

30 Hyper-Reduction Methods Application to the Reduction of the Burgers Equation Results of the greedy algorithm Index Greedy Selection / 65

31 Hyper-Reduction Methods Application to the Reduction of the Burgers Equation The dimension k f of the ROB V f is reduced from 40 to 30 k = 15, k f = 30, k i = U x Similar results for k i = 100 (no gaps) k f is too small in that case 31 / 65

32 Hyper-Reduction Methods Model Reduction at the Fully Discrete Level d Semi-discrete level: w(t) = f(w(t); t) dt Subspace approximation: w(t) Vq(t) V d q(t) f(vq(t); t) dt Fully discrete level (implicit, backward Euler scheme) V qn+1 q n t n f ( Vq n+1 ; t n+1) Fully discrete residual r n+1 (q n+1 ) = V qn+1 q n t n f ( Vq n+1 ; t n+1) Residual minimization (a.k.a model order reduction by least-squares or Petrov-Galerkin projection) q n+1 = argmin y R k r n+1 (y) 2 r(q n+1 ) is nonlinear use the gappy POD hyper-reduction 32 / 65

33 Hyper-Reduction Methods Gappy POD at the Fully Discrete Level Gappy POD procedure for the fully discrete residual r Algorithm 1 build a reduced-order basis V r R N kr for r such that V T r V = I kr 2 construct a sample mesh I (indices i 1,, i ki ) using the greedy procedure 3 consider the gappy approximation r n+1 (q n+1 ) V rr kr (q n+1 ) V r (P T V r ) P T r n+1 (Vq n+1 ) 4 determine the vector of generalized coordinates at t n+1 q n+1 = argmin y R k V rr kr (y) 2 = argmin r kr (y) 2 y R k ( ) = argmin P T V r P T r n+1 (Vy) y R k 2 33 / 65

34 Hyper-Reduction Methods Gauss-Newton for Nonlinear Least-Squares Problems Nonlinear least-squares problem: min y r(y) 2, where r R N, y R k, and k N Equivalent function to be minimized: φ(y) = 1 2 r(y) 2 2 = r(y) T r(y) Gradient: φ(y) = J(y) T r(y), where J(y) = r y (y) Iterative solution using Newton s method where What is 2 φ(y)? Gauss-Newton method y (j+1) = y (j) + y (j+1) 2 φ(y (j) ) y (j+1) = φ(y (j) ) 2 φ(y) = J(y) T J(y) + N i=1 2 φ(y) J(y) T J(y) 2 r i y 2 (y)r i(y) 34 / 65

35 Hyper-Reduction Methods Gauss-Newton for Nonlinear Least-Squares Problems Gauss-Newton method where y (j+1) = y (j) + y (j+1) J(y (j) ) T J(y (j) ) y (j+1) = J(y (j) ) T r(y (j) ) This is the normal equation for y (j+1) = argmin z QR decomposition of the Jacobian J(y (j) )z + r(y (j) ) J(y (j) ) = Q (j) R (j) Equivalent solution using the QR decomposition (assuming that R (j) is full rank) y (j+1) = J(y (j) ) r(y (j) ) = (R (j)) 1 ( Q (j)) T r(y (j) ) 2 35 / 65

36 Hyper-Reduction Methods Gauss-Newton with Approximated Tensors GNAT (Gauss-Newton with Approximated Tensors) = Gauss-Newton + gappy POD Minimization problem ( min P T ) V r P T r n+1 (Vy) y R k 2 Jacobian: Ĵ(y) = ( P T V r ) P T J n+1 (Vy) Define a small dimensional operator (and construct it offline) A = ( P T V r ) Least-squares problem at Gauss-Newton iteration j of t n+1 y (j) = argmin AP T J n+1 (Vy (j) )Vz + AP T r n+1 (Vy (j) ) z R k GNAT solution using QR AP T J n+1 (Vy (j) )V = Q (j) R (j) y (j) = (R (j)) 1 ( Q (j)) T AP T r n+1 (Vy (j) ) 2 36 / 65

37 Hyper-Reduction Methods Gauss-Newton with Approximated Tensors Further developments concept of a reduced mesh concept of an output mesh error bounds GNAT using local reduced-order bases 37 / 65

38 Hyper-Reduction Methods Application: Compressible Navier-Stokes Equations Turbulent flow past the Ahmed body (CFD benchmark in the automotive industry) 3D compressible Navier-Stokes equations N = Re = , M = (216km/h) 38 / 65

39 Hyper-Reduction Methods Application: Compressible Navier-Stokes Equations Model order reduction (POD + GNAT): k = 283, k f = 1, 514, k i = 2, 268 Method CPU Time Number of CPUs Relative Error HDM h 512 ROM (GNAT) 3.88 h % 39 / 65

40 Hyper-Reduction Methods Application: Compressible Navier-Stokes Equations Comparison to other model order reduction techniques drag coefficient, CD full-order model GNAT(1) collocation + Gal collocation + LS DEIM-like time, seconds 40 / 65

41 Hyper-Reduction Methods Application: Design Optimization of a Nozzle HDM: N = 2, 048, m = 5 shape parameters Model order reduction (POD + DEIM): k = 8, k f = 20, k i = 20 Steady, parameterized problem p 1 p 2 p 3 p 4 p 5 0 x L min µ R 5 M(w(µ)) M target 2 s.t. f(w(µ)), µ) = 0 where M denotes the local Mach number 41 / 65

42 Hyper-Reduction Methods Application: Design Optimization of a Nozzle Method Offline CPU Time Online CPU Time Total CPU Time HDM 78.8 s 78.8 s ROM (DEIM) 5.08 s 4.87 s 9.96 s M (x) Target Initial Guess Optimization (Hermite RBF) x A(x) Target Initial Guess Optimization (Hermite RBF) x 42 / 65

43 Local Approaches Local Approximation of the State Approximating the solution manifold M by a single subspace S can lead to a large-dimensional such subspace Idea: Approximate M using local subspaces {S l } L i=1 S 3 S 2 S 1 M 43 / 65

44 Local Approaches Local Approximation of the State In practice, the local approximation of the state takes place at the fully discrete level Each local subspace S l is associated with a local ROB V l At each time-step n, the state w n is computed as w n = w n 1 + w n The increment w n is then approximated in a subspace S l,n = range(v l,n ) as w n V l,n q n The choice of the reduced-order basis V l,n is specified later By induction, the state w n is computed as w n = w 0 + n i=1 V l,i q n 44 / 65

45 Local Approaches Local Approximation of the State The state w n is computed as w n = w 0 + n i=1 V l,i q n In practice, the ROBs {V l,i } n i=1 are chosen among a finite set of local ROBs {V l } L l=1 Hence L w n = w 0 + V l q n l This shows that l=1 w n w 0 + range([v 1 V V ]) Note that each local ROB can be of a different dimension V l R N k l 45 / 65

46 Local Approaches Construction of the Local ROBs Intuitively, a given local subspace S l should approximate only a portion of the solution manifold M The solution manifold is a subset of the solution space R N M R N The solution space R N is partitioned into L subdomains, where each subdomain is associated with a local approximation subspace S l = range(v l ) In practice, a set of solution snapshots {w i } Ns i=1 can be partitioned into L subsets using the k-means clustering algorithm This leads to a Voronoi tessellation of R N The k-means clustering algorithm is distance dependent After clustering, each snapshot subset can be compressed into a local ROB, for example, using POD 46 / 65

47 Local Approaches Construction of the Local ROBs Local ROBs construction procedure w(t, µ 3 )! w(t, µ 1 )! w(t, µ 2 )! 47 / 65

48 Local Approaches Construction of the Local ROBs Local ROBs construction procedure w(t, µ 3 )! w(t, µ 1 )! w(t, µ 2 )! 48 / 65

49 Local Approaches Construction of the Local ROBs Local ROBs construction procedure 49 / 65

50 Local Approaches Construction of the Local ROBs Local ROBs construction procedure 50 / 65

51 Local Approaches Construction of the Local ROBs Local ROBs construction procedure 51 / 65

52 Local Approaches Construction of the Local ROBs Local ROBs construction procedure 52 / 65

53 Local Approaches Construction of the Local ROBs Local ROBs construction procedure 53 / 65

54 Local Approaches Construction of the Local ROBs Local ROBs construction procedure 54 / 65

55 Local Approaches Construction of the Local ROBs Local ROBs construction procedure ROB #4! ROB #1! ROB #3! ROB #2! 55 / 65

56 Local Approaches Construction of the Local ROBs Local ROBs construction procedure ROB #4! ROB #1! ROB #3! ROB #2! 56 / 65

57 Local Approaches Online Selection of the Local ROB Online, at time-step n, a local ROB V l,n needs to be chosen The selection is based on the current location of w n 1 on the solution manifold M The local approximation subspace is selected as that associated with the cluster whose center is the closest to w n 1 l, n = argmin d(w n 1, wl c ) l {1,,L} Consider the case of the distance based on a weighted Euclidian norm d(w, z) = w z H = (w z) T H(w z) where H R N N is a symmetric positive definite matrix 57 / 65

58 Local Approaches Online Selection of the Local ROB Choice of the local approximation subspace at time-step n l, n = argmin d(w n 1, wl c ) l {1,,L} For a distance based on a weighted Euclidian norm, the solution of the above problem can be computed efficiently at a cost that does not depend on the large dimension N To show this, consider the special form of the solution w n 1 = w 0 + L l=1 V l q n 1 l Then, one needs to compare the distances d(w n 1, w c i l ) and d(w n 1, w c j l ) for 1 i j L 58 / 65

59 Local Approaches Online Selection of the Local ROB The two distances d(w n 1, w c i l ) and d(w n 1, w c j l ) can be compared as follows i,j = d(w n 1, w c i l ) 2 d(w n 1, w c j l ) 2 = w n 1 w c i l 2 H w n 1 w c j l 2 H = L l=1 = w c i l V l q n 1 l 2 H + w c i l w 0 2 H 2 L l=1 L [w c i ] T V l q n 1 l l=1 V l q n 1 l 2 H w c j l w 0 2 H + 2 w 0 2 H w c j l w 0 2 H + 2 L [w c j ] T V l q n 1 l l=1 L [w c i w c j ] T V l q n 1 l The following small quantities can be pre-computed offline and used online to compute economically i,j, 1 i j L a i,j = w c i l w 0 2 H w c j l l=1 w 0 2 H R, g i,j = [w c i w c j ] T V l R k l 59 / 65

60 Local Approaches Extension to Hyper-Reduction The local approach to nonlinear model reduction can be easily extended to hyper-reduction as follows The hyper-reduction approach is applied independently to each subset of snapshots It leads to the definition of the local ROBs for the state: V l, l = 1,, L the local ROBs for the residual: V r,l, l = 1,, L the local masks: I l, l = 1,, L The choice of the local ROBs and masks is still dictated by the location of the current time-iterate in the state space 60 / 65

61 Local Approaches Application Flow past the CRM (Common Research Model) (CFD benchmark in the aeronautical industry) 3D compressible Euler equations N = Constant acceleration of 2.5 m/s 2, from M = 0.8 to M = / 65

62 Local Approaches Application Model order reduction using a global ROB 62 / 65

63 Local Approaches Application Model order reduction using 5 local ROBs Very good accuracy can be obtained with k l 17 as opposed to k = 50 with a global ROB 63 / 65

64 References M. Rewienski, J. White. Model order reduction for nonlinear dynamical systems based on trajectory piecewise-linear approximations. Linear Algebra and its Applications 2006; 415(2-3): R. Everson, L. Sirovich. Karhunen Loeve procedure for gappy data. Journal of the Optical Society of America A 1995; 12(8): M. Barrault et al. An empirical interpolation method: application to efficient reduced basis discretization of partial differential equations. Comptes Rendus de lacademie des Sciences Paris 2004; 339: K. Willcox. Unsteady flow sensing and estimation via the gappy proper orthogonal decomposition. Computers and Fluids 2006; 35: S. Chaturantabut, D.C. Sorensen. Nonlinear model reduction via Discrete Empirical Interpolation. SIAM Journal on Scientific Computing 2010; 32: / 65

65 References K. Carlberg, C. Farhat, J. Cortial, D. Amsallem. The GNAT method for nonlinear model reduction: effective implementation and application to computational fluid dynamics and turbulent flows. Journal of Computational Physics 2013; 242: K. Carlberg, C. Bou-Mosleh, C. Farhat. Efficient nonlinear model reduction via a least-squares Petrov Galerkin projection and compressive tensor approximations. International Journal for Numerical Methods in Engineering 2011; 86(2): D. Amsallem, M. Zahr, Y. Choi, C. Farhat. Design optimization using hyper-reduced-order models. Structural and Multidisciplinary Optimization 2015; 51: D. Amsallem, M. Zahr, C. Farhat. Nonlinear model reduction based on local reduced order bases. International Journal for Numerical Methods in Engineering 2012; 92(12): K. Washabaugh, M. Zahr, C. Farhat, On the use of discrete nonlinear reduced-order models for the prediction of steady-state flows past parametrically deformed complex geometries. AIAA , AIAA SciTech 2016, San Diego, CA, January 4-8, / 65