POD Model Reduction of Large Scale Geophysical Models. Ionel M. Navon

Transcription

1 POD Model Reduction of Large Scale Geophysical Models Ionel M. Navon School of Computational Science Florida State University Tallahassee, Florida

2 Thanks to Prof. D. N. Daescu Dept. of Mathematics and Statistics, Portland State University, USA Prof. J. Zhu and Dr. Yanhua Cao Institute of Atmospheric Physics Chinese Academy of Science, Beijing, China and Prof. Z. D. Luo et al. Department of Mathematics Beijing Jiaotong University,Beijing, China Prof Chris. C. Pain, Dr. Fangxin Fang et al. Applied Modelling and Computation Group Imperial College, London, UK

3 Thanks to Prof. D. N. Daescu Dept. of Mathematics and Statistics, Portland State University, USA Prof. J. Zhu and Dr. Yanhua Cao Institute of Atmospheric Physics Chinese Academy of Science, Beijing, China and Prof. Z. D. Luo et al. Department of Mathematics Beijing Jiaotong University,Beijing, China Prof Chris. C. Pain, Dr. Fangxin Fang et al. Applied Modelling and Computation Group Imperial College, London, UK

4 Thanks to Prof. D. N. Daescu Dept. of Mathematics and Statistics, Portland State University, USA Prof. J. Zhu and Dr. Yanhua Cao Institute of Atmospheric Physics Chinese Academy of Science, Beijing, China and Prof. Z. D. Luo et al. Department of Mathematics Beijing Jiaotong University,Beijing, China Prof Chris. C. Pain, Dr. Fangxin Fang et al. Applied Modelling and Computation Group Imperial College, London, UK

5 Outline

6 ROM: Principles Dimension reduction means representing a vector in high dimensional space x R n with a corresponding vector in a much lower dimensional space x R m. Consider the state equations S : ẋ(t) = f (x(t), u(t)), y(t) = h(x(t), u(t)) x( ) R n : state vector y( ) R p : observation vector u( ) R m : input vector n m, p Find S := ( f, h) with x(t) R k, k n assuring Preservation of stability Computational stability and efficient Approximation error small-global error bound

7 ROM: Principles Dimension reduction means representing a vector in high dimensional space x R n with a corresponding vector in a much lower dimensional space x R m. Consider the state equations S : ẋ(t) = f (x(t), u(t)), y(t) = h(x(t), u(t)) x( ) R n : state vector y( ) R p : observation vector u( ) R m : input vector n m, p Find S := ( f, h) with x(t) R k, k n assuring Preservation of stability Computational stability and efficient Approximation error small-global error bound

8 POD Generalities: POD is related to the principal component analysis, Karhunen-Loève expansion in the stochastic process theory, and principal of empirical orthogonal eigenfunctions. POD is the most used technique for the reduced-order modeling of nonlinear PDEs. POD proceeds by retaining the characteristics of the data set that contribute most to its variance.

9 POD Generalities: POD is related to the principal component analysis, Karhunen-Loève expansion in the stochastic process theory, and principal of empirical orthogonal eigenfunctions. POD is the most used technique for the reduced-order modeling of nonlinear PDEs. POD proceeds by retaining the characteristics of the data set that contribute most to its variance.

10 POD Generalities: POD is related to the principal component analysis, Karhunen-Loève expansion in the stochastic process theory, and principal of empirical orthogonal eigenfunctions. POD is the most used technique for the reduced-order modeling of nonlinear PDEs. POD proceeds by retaining the characteristics of the data set that contribute most to its variance.

11 POD Principles: Find a projection P r : R n R n of fixed rank r that minimizes the error T 0 x(t) P r x(t) 2 dt, x(t) R n, with 0 t T Introduce the symmetric, positive-semi-definite n n matrix C = T Solve the eigenvalue problem with 0 x(t) [x(t)] T dt C φ k = λ k φ k, k = 1,, n λ 1 λ 2 λ n, and The optimal projection is P r = T 0 r φ k [φ k ] T. k=1 φ i φ j dt = δ i,j

12 POD Principles: Find a projection P r : R n R n of fixed rank r that minimizes the error T 0 x(t) P r x(t) 2 dt, x(t) R n, with 0 t T Introduce the symmetric, positive-semi-definite n n matrix C = T Solve the eigenvalue problem with 0 x(t) [x(t)] T dt C φ k = λ k φ k, k = 1,, n λ 1 λ 2 λ n, and The optimal projection is P r = T 0 r φ k [φ k ] T. k=1 φ i φ j dt = δ i,j

13 POD Principles: Find a projection P r : R n R n of fixed rank r that minimizes the error T 0 x(t) P r x(t) 2 dt, x(t) R n, with 0 t T Introduce the symmetric, positive-semi-definite n n matrix C = T Solve the eigenvalue problem with 0 x(t) [x(t)] T dt C φ k = λ k φ k, k = 1,, n λ 1 λ 2 λ n, and The optimal projection is P r = T 0 r φ k [φ k ] T. k=1 φ i φ j dt = δ i,j

14 POD Principles: Find a projection P r : R n R n of fixed rank r that minimizes the error T 0 x(t) P r x(t) 2 dt, x(t) R n, with 0 t T Introduce the symmetric, positive-semi-definite n n matrix C = T Solve the eigenvalue problem with 0 x(t) [x(t)] T dt C φ k = λ k φ k, k = 1,, n λ 1 λ 2 λ n, and The optimal projection is P r = T 0 r φ k [φ k ] T. k=1 φ i φ j dt = δ i,j

15 POD Steps: Replace the set of data x(t) by the snapshots x(t j ) at discrete times t 1, t 2,, t m. Transform the n n eigenvalue problem into the m m eigenvalue problem with C replaced by C = m x(t j ) [x(t j )] T ω i, ω i quadrature weights j=1 Define the n m matrix X = [x(t 1 ) x(t 2 ),..., x(t m )] and the m m matrix W = diag{ω(t 1 ), ω(t 2 ),..., ω(t m )} The reduced eigenvalue problem becomes X T WX u k = λ k u k, u k R m

16 POD Steps: Replace the set of data x(t) by the snapshots x(t j ) at discrete times t 1, t 2,, t m. Transform the n n eigenvalue problem into the m m eigenvalue problem with C replaced by C = m x(t j ) [x(t j )] T ω i, ω i quadrature weights j=1 Define the n m matrix X = [x(t 1 ) x(t 2 ),..., x(t m )] and the m m matrix W = diag{ω(t 1 ), ω(t 2 ),..., ω(t m )} The reduced eigenvalue problem becomes X T WX u k = λ k u k, u k R m

17 POD Steps: Replace the set of data x(t) by the snapshots x(t j ) at discrete times t 1, t 2,, t m. Transform the n n eigenvalue problem into the m m eigenvalue problem with C replaced by C = m x(t j ) [x(t j )] T ω i, ω i quadrature weights j=1 Define the n m matrix X = [x(t 1 ) x(t 2 ),..., x(t m )] and the m m matrix W = diag{ω(t 1 ), ω(t 2 ),..., ω(t m )} The reduced eigenvalue problem becomes X T WX u k = λ k u k, u k R m

18 POD Steps: Replace the set of data x(t) by the snapshots x(t j ) at discrete times t 1, t 2,, t m. Transform the n n eigenvalue problem into the m m eigenvalue problem with C replaced by C = m x(t j ) [x(t j )] T ω i, ω i quadrature weights j=1 Define the n m matrix X = [x(t 1 ) x(t 2 ),..., x(t m )] and the m m matrix W = diag{ω(t 1 ), ω(t 2 ),..., ω(t m )} The reduced eigenvalue problem becomes X T WX u k = λ k u k, u k R m

19 POD Shortcomings: POD is sensitive to details of snapshots used POD is sensitive to the choice of inner products POD depends on how well the data ensemble captures the relevant system behavior POD sometimes yields unstable models despite the original system being stable POD does not take account of system outputs when performing the reduction, and hence the reduced-order models produced may be inefficient

20 POD Shortcomings: POD is sensitive to details of snapshots used POD is sensitive to the choice of inner products POD depends on how well the data ensemble captures the relevant system behavior POD sometimes yields unstable models despite the original system being stable POD does not take account of system outputs when performing the reduction, and hence the reduced-order models produced may be inefficient

21 POD Shortcomings: POD is sensitive to details of snapshots used POD is sensitive to the choice of inner products POD depends on how well the data ensemble captures the relevant system behavior POD sometimes yields unstable models despite the original system being stable POD does not take account of system outputs when performing the reduction, and hence the reduced-order models produced may be inefficient

22 POD Shortcomings: POD is sensitive to details of snapshots used POD is sensitive to the choice of inner products POD depends on how well the data ensemble captures the relevant system behavior POD sometimes yields unstable models despite the original system being stable POD does not take account of system outputs when performing the reduction, and hence the reduced-order models produced may be inefficient

23 POD Shortcomings: POD is sensitive to details of snapshots used POD is sensitive to the choice of inner products POD depends on how well the data ensemble captures the relevant system behavior POD sometimes yields unstable models despite the original system being stable POD does not take account of system outputs when performing the reduction, and hence the reduced-order models produced may be inefficient

24 4D-Var Principles: Four dimensional variational data assimilation is in principle a least-squares fit in 4 dimensions between the predicted state of the atmosphere and the observations. The adjustment to the predicted state is made at the initial time t 0, which ensures that the analysis state (4-dimensional) is a model trajectory. 4D-Var is a method of estimating a set of parameters by optimizing the fit between the solution of the model and a set of observations which the model is meant to predict. In this context, the procedure of adjusting the parameters until the model best predicts the observables, is known as optimization.

25 4D-Var Principles: Four dimensional variational data assimilation is in principle a least-squares fit in 4 dimensions between the predicted state of the atmosphere and the observations. The adjustment to the predicted state is made at the initial time t 0, which ensures that the analysis state (4-dimensional) is a model trajectory. 4D-Var is a method of estimating a set of parameters by optimizing the fit between the solution of the model and a set of observations which the model is meant to predict. In this context, the procedure of adjusting the parameters until the model best predicts the observables, is known as optimization.

26 4D-Var data assimilation Results: Given x, y, x b state, observation and background vectors, M, H model and observation operators, B, R background and observational error covariance matrices, Find an optimal estimate (analysis) state vector x a solution of min J (x); x R n xa = arg min J where the cost function J is J = 1 2 (x x b) T B(x x b ) [y H(x)]T R 1 [y H(x)] or the discrete form J = 1 2 (x x b) T B(x x b )+ 1 2 p k=1 [y k H k (M k (x))] T R 1 k [y k H k (M k (x))]

27 4D-Var data assimilation Results: Given x, y, x b state, observation and background vectors, M, H model and observation operators, B, R background and observational error covariance matrices, Find an optimal estimate (analysis) state vector x a solution of min J (x); x R n xa = arg min J where the cost function J is J = 1 2 (x x b) T B(x x b ) [y H(x)]T R 1 [y H(x)] or the discrete form J = 1 2 (x x b) T B(x x b )+ 1 2 p k=1 [y k H k (M k (x))] T R 1 k [y k H k (M k (x))]

28 2 Four-dimensional variational data assimilation state x b x 0 a y(t) x(t) = M(x b ) x(t) = M(x 0 a ) min J (x 0) x 0 R m x a 0 = Arg min J time J (x 0 ) = 1 2 x 0 x b 2 B N k=1 H k x k y k 2 R 1 k

29 2 Four-dimensional variational data assimilation state x b x 0 a y(t) x(t) = M(x b ) x(t) = M(x 0 a ) min J (x 0) x 0 R m x a 0 = Arg min J time J (x 0 ) = 1 2 x 0 x b 2 B N k=1 H k x k y k 2 R 1 k Typical dimension is of order m Reduced Order 4D-Var Data Assimilation

30 Framework : Given an ensemble data set collected from observational data at various instants in time t 1, t 2,..., t p snapshots [ x (1), x (2),..., x (p)] x (k) R n Define the weighted ensemble average of the data and the perturbation X x = p ω k x (k), with 0 ω k 1, and k=1 X = the weighted covariance matrix p ω k = 1 k=1 [ ] x (1) x, x (2) x,..., x (p) x C = X T WX where W = diag{ω 1, ω 2,..., ω p } the norm x 2 A =< x, x > A= x T Ax, A R n n is an SPD matrix, { Id for the Euclidean norm A = Λ a diagonal matrix for the total energy metric.

31 Framework : Given an ensemble data set collected from observational data at various instants in time t 1, t 2,..., t p snapshots [ x (1), x (2),..., x (p)] x (k) R n Define the weighted ensemble average of the data and the perturbation X x = p ω k x (k), with 0 ω k 1, and k=1 X = the weighted covariance matrix p ω k = 1 k=1 [ ] x (1) x, x (2) x,..., x (p) x C = X T WX where W = diag{ω 1, ω 2,..., ω p } the norm x 2 A =< x, x > A= x T Ax, A R n n is an SPD matrix, { Id for the Euclidean norm A = Λ a diagonal matrix for the total energy metric.

32 POD basis: Find a projection operator P r : R n R p of fixed rank r that minimizes the error p ω k (x (k) x) P r (x (k) x) 2 A k=1 Solve the eigenvalue problem C Aφ k = σ 2 k φ k, k = 1,, p with < φ i, φ j > A = δ i,j, 1 i, j p The optimal r-dimensional subspace is {φ 1, φ 2,, φ r }, and the optimal projection is P r = r [φ k ] T Aφ k k=1

33 POD basis: Find a projection operator P r : R n R p of fixed rank r that minimizes the error p ω k (x (k) x) P r (x (k) x) 2 A k=1 Solve the eigenvalue problem C Aφ k = σ 2 k φ k, k = 1,, p with < φ i, φ j > A = δ i,j, 1 i, j p The optimal r-dimensional subspace is {φ 1, φ 2,, φ r }, and the optimal projection is P r = r [φ k ] T Aφ k k=1

34 POD basis: Find a projection operator P r : R n R p of fixed rank r that minimizes the error p ω k (x (k) x) P r (x (k) x) 2 A k=1 Solve the eigenvalue problem C Aφ k = σ 2 k φ k, k = 1,, p with < φ i, φ j > A = δ i,j, 1 i, j p The optimal r-dimensional subspace is {φ 1, φ 2,, φ r }, and the optimal projection is P r = r [φ k ] T Aφ k k=1

35 Computational aspects: Solve the eigenvalue problem W 1/2 X T AX W 1/2 ψ k = σk 2 Use the singular value decomposition (SVD) A 1/2 X W 1/2 = UΣV T Compute the POD modes φ k = 1 X W 1/2 ψ k σ k Test of the fraction of total information captured for 0 < γ 1 l r select l such that { σk 2 } { σk 2 } γ k=1 k=1

36 Computational aspects: Solve the eigenvalue problem W 1/2 X T AX W 1/2 ψ k = σk 2 Use the singular value decomposition (SVD) A 1/2 X W 1/2 = UΣV T Compute the POD modes φ k = 1 X W 1/2 ψ k σ k Test of the fraction of total information captured for 0 < γ 1 l r select l such that { σk 2 } { σk 2 } γ k=1 k=1

37 Computational aspects: Solve the eigenvalue problem W 1/2 X T AX W 1/2 ψ k = σk 2 Use the singular value decomposition (SVD) A 1/2 X W 1/2 = UΣV T Compute the POD modes φ k = 1 X W 1/2 ψ k σ k Test of the fraction of total information captured for 0 < γ 1 l r select l such that { σk 2 } { σk 2 } γ k=1 k=1

38 Computational aspects: Solve the eigenvalue problem W 1/2 X T AX W 1/2 ψ k = σk 2 Use the singular value decomposition (SVD) A 1/2 X W 1/2 = UΣV T Compute the POD modes φ k = 1 X W 1/2 ψ k σ k Test of the fraction of total information captured for 0 < γ 1 l r select l such that { σk 2 } { σk 2 } γ k=1 k=1

39 Reduced order control Results: Set the following Φ = [φ 1, φ 2,, φ r ] the R n r matrix of POD basis, and η = (η 1, η 2,..., η r ) the coordinates vector in R r Project x x onto the r-dimensional subspace {φ 1, φ 2,, φ r } P r (x x) = r η k (t)φ k, where η k = φ T k AP r (x x) or η = Φ T AP r (x x) k=1 Find an optimal estimate (analysis) state vector η a R r solution of min J ˆ(η); η a = arg min J ˆ η R r where the reduced cost function Jˆ is given by k=1 [ J ˆ(x) = 1 2 Pr (x x b ) T] P T r BP r [P r (x x b )] r [ Pr {y k H k (M k (x))} T] P T r R 1 k P r [P r {y k H k (M k (x))}]

40 Reduced order control Results: Set the following Φ = [φ 1, φ 2,, φ r ] the R n r matrix of POD basis, and η = (η 1, η 2,..., η r ) the coordinates vector in R r Project x x onto the r-dimensional subspace {φ 1, φ 2,, φ r } P r (x x) = r η k (t)φ k, where η k = φ T k AP r (x x) or η = Φ T AP r (x x) k=1 Find an optimal estimate (analysis) state vector η a R r solution of min J ˆ(η); η a = arg min J ˆ η R r where the reduced cost function Jˆ is given by k=1 [ J ˆ(x) = 1 2 Pr (x x b ) T] P T r BP r [P r (x x b )] r [ Pr {y k H k (M k (x))} T] P T r R 1 k P r [P r {y k H k (M k (x))}]

41 Evaluation of the weights ω i using the 4D-Var cost function : From the tangent linear model M(t i, t) and M T (t, t i ) δj x(t) J (x(t)), δx(t) = x(t) J (x(t)), M(t i, t)δx(t i ) = M T (t, t i ) x(t) J (x(t)), δx(t i ) = A 1 M T (t, t i ) x(t) J (x(t)), δx(t i ) A The dual-weights ω i to the snapshots are the normalized values α i = A 1 M T (t, t i ) x(t) J (x(t)) A, ω k = α k r j=1 α, k = 1, 2,... r j

42 Evaluation of the weights ω i using the 4D-Var cost function : From the tangent linear model M(t i, t) and M T (t, t i ) δj x(t) J (x(t)), δx(t) = x(t) J (x(t)), M(t i, t)δx(t i ) = M T (t, t i ) x(t) J (x(t)), δx(t i ) = A 1 M T (t, t i ) x(t) J (x(t)), δx(t i ) A The dual-weights ω i to the snapshots are the normalized values α i = A 1 M T (t, t i ) x(t) J (x(t)) A, ω k = α k r j=1 α, k = 1, 2,... r j

43 Evaluation of the weights ω i using the 4D-Var cost function : From the tangent linear model M(t i, t) and M T (t, t i ) δj x(t) J (x(t)), δx(t) = x(t) J (x(t)), M(t i, t)δx(t i ) = M T (t, t i ) x(t) J (x(t)), δx(t i ) = A 1 M T (t, t i ) x(t) J (x(t)), δx(t i ) A The dual-weights ω i to the snapshots are the normalized values α i = A 1 M T (t, t i ) x(t) J (x(t)) A, ω k = α k r j=1 α, k = 1, 2,... r j

44 3 Reduced order 4D-Var - general framework R m k x Px R k ˆ J (η) := J (x + Ψη) min η R k ˆ J (η) x a 0 x + Ψη a Projection on Span{ψ 1, ψ 2,..., ψ k }: P ψ,k = ΨΨ T A, k m P ψ,k (x 0 x) = ΨΨ T A(x 0 x) = Ψη, η R k

45 3 Reduced order 4D-Var - general framework R m k x Px R k ˆ J (η) := J (x + Ψη) min η R k ˆ J (η) x a 0 x + Ψη a Projection on Span{ψ 1, ψ 2,..., ψ k }: P ψ,k = ΨΨ T A, k m P ψ,k (x 0 x) = ΨΨ T A(x 0 x) = Ψη, η R k Low-order optimization problem k Reduced Order 4D-Var Data Assimilation

46 4 The Proper Orthogonal Decomposition Method Empirical Orthogonal Functions, Karhunen-Loève decomposition R m k (i) x Px (i) (i) Px R k x (i) Optimal order k representation min {ψ} n ω j x (i) P ψ,k x (i) 2 i=1 ψ i, ψ j A = δ ij, 1 i j k A Ensemble data {x (i) }, i = 1, n

47 4 The Proper Orthogonal Decomposition Method Empirical Orthogonal Functions, Karhunen-Loève decomposition R m k (i) x Px (i) (i) Px R k x (i) Optimal order k representation min {ψ} n ω j x (i) P ψ,k x (i) 2 i=1 ψ i, ψ j A = δ ij, 1 i j k A Ensemble data {x (i) }, i = 1, n Dependence on the metric A and the weights ω Reduced Order 4D-Var Data Assimilation

48 Problem Description: Model: A two-dimensional global shallow-water (SW) model Two data assimilation experiments are set up: DAS-I, is a model inversion problem where data is provided for all discrete state components and no background term is included in the cost functional DAS-II, the background term is included in the cost and data is provided at every 4th grid point on the longitudinal and latitudinal directions (i.e. only 6% of the state is observed every six hours). Algorithms & schemes the explicit flux-form semi-lagrangian (FFSL) scheme of Lin and Rood (1997) The adjoint model to the SW-FFSL scheme of Akella and Navon (2006) and TAMC software (Giering and Kaminski 1998).

49 Problem Description: Model: A two-dimensional global shallow-water (SW) model Two data assimilation experiments are set up: DAS-I, is a model inversion problem where data is provided for all discrete state components and no background term is included in the cost functional DAS-II, the background term is included in the cost and data is provided at every 4th grid point on the longitudinal and latitudinal directions (i.e. only 6% of the state is observed every six hours). Algorithms & schemes the explicit flux-form semi-lagrangian (FFSL) scheme of Lin and Rood (1997) The adjoint model to the SW-FFSL scheme of Akella and Navon (2006) and TAMC software (Giering and Kaminski 1998).

50 Problem Description: Model: A two-dimensional global shallow-water (SW) model Two data assimilation experiments are set up: DAS-I, is a model inversion problem where data is provided for all discrete state components and no background term is included in the cost functional DAS-II, the background term is included in the cost and data is provided at every 4th grid point on the longitudinal and latitudinal directions (i.e. only 6% of the state is observed every six hours). Algorithms & schemes the explicit flux-form semi-lagrangian (FFSL) scheme of Lin and Rood (1997) The adjoint model to the SW-FFSL scheme of Akella and Navon (2006) and TAMC software (Giering and Kaminski 1998).

51 Problem Description: continued Input: the ECMWF ERA-40 atmospheric data to specify the SW model state variables at the initial time Resolution & time step : ( grid cells) such that the dimension of the discrete state vector x = (h, u, v) is The time step t = 450 s Reference initial state x ref 0 : the 500mb ECMWF ERA-40 data valid for 06h UTC 15 March Snapshots: from small random perturbations δx 0 in the reference initial conditions and a full model integration starting with x ref 0 + δx 0.

52 Problem Description: continued Input: the ECMWF ERA-40 atmospheric data to specify the SW model state variables at the initial time Resolution & time step : ( grid cells) such that the dimension of the discrete state vector x = (h, u, v) is The time step t = 450 s Reference initial state x ref 0 : the 500mb ECMWF ERA-40 data valid for 06h UTC 15 March Snapshots: from small random perturbations δx 0 in the reference initial conditions and a full model integration starting with x ref 0 + δx 0.

53 Problem Description: continued Input: the ECMWF ERA-40 atmospheric data to specify the SW model state variables at the initial time Resolution & time step : ( grid cells) such that the dimension of the discrete state vector x = (h, u, v) is The time step t = 450 s Reference initial state x ref 0 : the 500mb ECMWF ERA-40 data valid for 06h UTC 15 March Snapshots: from small random perturbations δx 0 in the reference initial conditions and a full model integration starting with x ref 0 + δx 0.

54 Problem Description: continued Input: the ECMWF ERA-40 atmospheric data to specify the SW model state variables at the initial time Resolution & time step : ( grid cells) such that the dimension of the discrete state vector x = (h, u, v) is The time step t = 450 s Reference initial state x ref 0 : the 500mb ECMWF ERA-40 data valid for 06h UTC 15 March Snapshots: from small random perturbations δx 0 in the reference initial conditions and a full model integration starting with x ref 0 + δx 0.

55 Results: Figure: Isopleths of the geopotential height (m) for the reference run configuration at the initial time specified from ECMWF ERA-40 data sets Bottom:the 24h forecast of the shallow water model

56 Results: Figure: Isopleths of the geopotential height (m) for the reference run the 24h forecast of the shallow water model

57 Results: 1 FRACTION OF THE VARIANCE CAPTURED DWPOD POD REDUCED BASIS DIMENSION Figure: The fraction of the variance captured by the POD and DWPOD modes from the snapshot data as a function of the dimension of the reduced space.

58 Results: FIRST POD MODE FIRST DWPOD MODE LATITUDE th POD MODE th DWPOD MODE LATITUDE th POD MODE th DWPOD MODE LATITUDE LONGITUDE LONGITUDE Figure: Isopleths of the POD and DWPOD modes of rank 1, 5, and 10. A total energy norm is used to provide point values.

59 Results: DWPOD POD error in functional J representation ( log 10 ) reduced space dimension Figure: Comparative results for the reduced-order POD and DWPOD forecasts for k = 5, 10, 15, 20, 25. Top figure: error (log 10) in the reduced-order representation of the time integrated total energy of the system

60 Results: 2.5 DWPOD 2 POD error in state forecast: log10 (energy norm) time step Figure: Comparative results for the reduced-order POD and DWPOD forecasts for k = 5, 10, 15, 20, 25. Total energy error (log 10) x ref i ˆx i A of the reduced-order representation of the forecast at each time step in the interval 0-24h.

61 Results: Figure: Zonal averaged errors in the background estimate to the initial conditions.

62 Results: FULL SPACE OPTIMIZATION: DAS I 4.5 FULL SPACE OPTIMIZATION: DAS II 5.5 LOG 10 (COST) LOG 10 (COST) ITERATION NUMBER ITERATION NUMBER Figure: The iterative minimization process in the full state space for DAS-I (left) and DAS-II (right).

63 Results: 11 x DAS I DAS II 9 8 snapshot weight time step Figure: The dual weights to the snapshot data determined by the adjoint model in DAS-I and in DAS-II

64 Results: 70 DAS I: DWPOD BASIS VECTOR OF RANK latitude longitude Figure: Isopleths of the 10 th mode in the DWPOD basis for DAS-I

65 Results: 70 DAS II: DWPOD BASIS VECTOR OF RANK latitude longitude Figure: Isopleths of the 10 th mode in the DWPOD basis for DAS-II. A distinct configuration it is noticed since the DWPOD basis is adjusted to the optimization problem at hand.

66 Results: DWPOD k = 5 DWPOD k = 10 DWPOD k = 15 POD k = 5 POD k = 10 POD k = log 10 ( cost ) iteration number Figure: The iterative minimization process in the reduced space for the POD and DWPOD spaces of dimension 5, 10, and 15. Optimization without background term and dense observations, corresponding to DAS-I

67 Results: DWPOD k = 5 DWPOD k = 10 DWPOD k = 15 POD k = 5 POD k = 10 POD k = log 10 ( cost ) iteration number Figure: The iterative minimization process in the reduced space for the POD and DWPOD spaces of dimension 5, 10, and 15.Optimization with background term and sparse observations, corresponding to DAS-II

68 Results: Figure: Zonal averaged errors in the analysis provided by the reduced order 4D-Var data assimilation Results for the DAS-I experiments with POD and DWPOD spaces of dimension 5, 10, and 15.

69 Results: Figure: Zonal averaged errors in the analysis provided by the reduced order 4D-Var data assimilation. Results for the DAS-II experiments with POD and DWPOD spaces of dimension 5, 10, and 15.

70 Conclusions: An adjoint-model approach is proposed to directly incorporate information from the DAS into the optimality criteria that defines the reduced space basis. The dual weighted POD method is novel in reduced order 4D-Var data assimilation and relies on a weighted ensemble data mean and weighted snapshots with weights determined by the adjoint DAS ( Data Assimilation System). The DWPOD space was found to increase the accuracy in the representation of a forecast aspect by as much as an order of magnitude versus the POD space representation.

71 Conclusions: An adjoint-model approach is proposed to directly incorporate information from the DAS into the optimality criteria that defines the reduced space basis. The dual weighted POD method is novel in reduced order 4D-Var data assimilation and relies on a weighted ensemble data mean and weighted snapshots with weights determined by the adjoint DAS ( Data Assimilation System). The DWPOD space was found to increase the accuracy in the representation of a forecast aspect by as much as an order of magnitude versus the POD space representation.

72 Conclusions: An adjoint-model approach is proposed to directly incorporate information from the DAS into the optimality criteria that defines the reduced space basis. The dual weighted POD method is novel in reduced order 4D-Var data assimilation and relies on a weighted ensemble data mean and weighted snapshots with weights determined by the adjoint DAS ( Data Assimilation System). The DWPOD space was found to increase the accuracy in the representation of a forecast aspect by as much as an order of magnitude versus the POD space representation.

73 Conclusions: continued The benefit gained from the dual-weighted procedure diminishes as the dimension of the reduced space increases from 10 to 15, indicating that most of the information provided by the snapshot data is captured by the reduced basis. In 4D-Var data assimilation twin experiments, optimization in the DWPOD space provided a reduction in the analysis errors by as much as a factor of three when compared to the POD-based optimization. Use of dual weighted goal oriented criteria may also serve also as goal-oriented a posteriori error estimate to drive grid adaptivity.

74 Conclusions: continued The benefit gained from the dual-weighted procedure diminishes as the dimension of the reduced space increases from 10 to 15, indicating that most of the information provided by the snapshot data is captured by the reduced basis. In 4D-Var data assimilation twin experiments, optimization in the DWPOD space provided a reduction in the analysis errors by as much as a factor of three when compared to the POD-based optimization. Use of dual weighted goal oriented criteria may also serve also as goal-oriented a posteriori error estimate to drive grid adaptivity.

75 Conclusions: continued The benefit gained from the dual-weighted procedure diminishes as the dimension of the reduced space increases from 10 to 15, indicating that most of the information provided by the snapshot data is captured by the reduced basis. In 4D-Var data assimilation twin experiments, optimization in the DWPOD space provided a reduction in the analysis errors by as much as a factor of three when compared to the POD-based optimization. Use of dual weighted goal oriented criteria may also serve also as goal-oriented a posteriori error estimate to drive grid adaptivity.

76 The reduced-gravity model: A reduced order approach to four-dimensional variational data assimilation using proper orthogonal decomposition. Yanhua Cao, Jiang Zhu, and Zhendong Luo International Journal for Numerical Methods in Fluids, Volume 53, Issue 10, (2007)

77 The reduced-gravity model: u t fv = g h x + τ x ρ 0 H + A 2 u αu v h fu = g t y + τ y ρ 0 H + A 2 v αv h t + H( u x + v y ) = 0 (u, v) : the horizontal velocity components of the depth-averaged currents h : the total layer thickness f : the Coriolis force H : the mean depth of the layer ρ 0 : the density of water A : the horizontal eddy coefficient α : the friction coefficient (τ x, τ y ) : the wind stress

78 POD 4D-Var: The dynamic model governing the ocean flow U(t, x) du = F(t, U) dt U(0, x) = U 0 (x) the reduced dynamic model: the forward model dc k dt = F(t, Ū + p c i φ i ), φ k i=1 c k (t = 0) = c k (0)

79 POD 4D-Var: The dynamic model governing the ocean flow U(t, x) du = F(t, U) dt U(0, x) = U 0 (x) the reduced dynamic model: the forward model dc k dt = F(t, Ū + p c i φ i ), φ k i=1 c k (t = 0) = c k (0)

80 POD 4D-Var: U POD 0, V, U b, Ū state, observation, background and mean vectors, M, H model and observation operators, B, R background and observational error covariance matrices Find an optimal estimate (analysis) state vector {U0 POD } a solution of J (U0 POD ) = (U0 POD U b ) T B(U0 POD U b ) + with + [ V H(U POD ) ] T R 1 [ V H(U POD ) ] (U POD 0 )(x) = (U POD 0 )(0, x) = Ū(x) + (U POD )(x) = (U POD 0 )(t, x) = Ū(x) + p c k (0)φ k (x) k=1 p c k (t)φ k (x) k=1

81 POD 4D-Var: U POD 0, V, U b, Ū state, observation, background and mean vectors, M, H model and observation operators, B, R background and observational error covariance matrices Find an optimal estimate (analysis) state vector {U0 POD } a solution of J (U0 POD ) = (U0 POD U b ) T B(U0 POD U b ) + with + [ V H(U POD ) ] T R 1 [ V H(U POD ) ] (U POD 0 )(x) = (U POD 0 )(0, x) = Ū(x) + (U POD )(x) = (U POD 0 )(t, x) = Ū(x) + p c k (0)φ k (x) k=1 p c k (t)φ k (x) k=1

82 Computational aspects : The adjoint model of the reduced forward model is used to calculate the gradient of the cost function J (U POD 0 ) The initial value of the cost function in the full model space is different from the initial value of the cost function in the POD space. U 0 = U b for the full model β 0 = P T r (U b Ū) for the reduced model Use of an adaptive POD 4D-Var

83 Computational aspects : The adjoint model of the reduced forward model is used to calculate the gradient of the cost function J (U POD 0 ) The initial value of the cost function in the full model space is different from the initial value of the cost function in the POD space. U 0 = U b for the full model β 0 = P T r (U b Ū) for the reduced model Use of an adaptive POD 4D-Var

84 Computational aspects : The adjoint model of the reduced forward model is used to calculate the gradient of the cost function J (U POD 0 ) The initial value of the cost function in the full model space is different from the initial value of the cost function in the POD space. U 0 = U b for the full model η 0 = P T r (U b Ū) for the reduced model Use of an adaptive POD 4D-Var

85 Adaptive POD 4D-Var : 1 Establish POD model using background initial conditions 2 Perform optimization iterations to obtain the optimal solution 3 Generate a new set of snapshots if after a preset number of iterations, the cost function cannot be reduced. 4 Establish a new POD model using the new snapshots and continue the optimization process 5 Return back to 2 if the optimality conditions are not reached

86 Adaptive POD 4D-Var : 1 Establish POD model using background initial conditions 2 Perform optimization iterations to obtain the optimal solution 3 Generate a new set of snapshots if after a preset number of iterations, the cost function cannot be reduced. 4 Establish a new POD model using the new snapshots and continue the optimization process 5 Return back to 2 if the optimality conditions are not reached

87 Adaptive POD 4D-Var : 1 Establish POD model using background initial conditions 2 Perform optimization iterations to obtain the optimal solution 3 Generate a new set of snapshots if after a preset number of iterations, the cost function cannot be reduced. 4 Establish a new POD model using the new snapshots and continue the optimization process 5 Return back to 2 if the optimality conditions are not reached

88 Adaptive POD 4D-Var : 1 Establish POD model using background initial conditions 2 Perform optimization iterations to obtain the optimal solution 3 Generate a new set of snapshots if after a preset number of iterations, the cost function cannot be reduced. 4 Establish a new POD model using the new snapshots and continue the optimization process 5 Return back to 2 if the optimality conditions are not reached

89 Adaptive POD 4D-Var : 1 Establish POD model using background initial conditions 2 Perform optimization iterations to obtain the optimal solution 3 Generate a new set of snapshots if after a preset number of iterations, the cost function cannot be reduced. 4 Establish a new POD model using the new snapshots and continue the optimization process 5 Return back to 2 if the optimality conditions are not reached

90 Results: x y Wind Stress ( τ, τ ) Surface Layer Density ρ H h Deep Ocean Density ρ + Δ ρ u=v=0

91 Results:

92 Results: Figure: Evolution of the cost function and gradient in 4DVAR experiment. (a) cost function; (b) gradient as a function of the number of minimization iterations.

93 Results:

94 Results: Figure: Evolution of the cost function and gradient in the POD 4D-Var

95 Results:

96 Results: Figure: Evolution of the cost function and gradient in adaptive POD 4D-Var

97 Results:

98 Results: Figure: RMSE of the results compared to the true state for upper layer thickness

99 Results: Figure: Errors between the true state and the numerical approximations for upper layer thickness at the initial time

100 Results: Figure: Upper layer thickness Feb.,May,Aug.and Nov. in case of 5,20,30 snapshots

101 Results: (a) (b) (c) (d) Figure: Figure 4 Upper layer thickness in February, May, August and November in case of 5 snapshots, 20 snapshots, 30 snapshots, energy capture 95%, the full model approximation and the reduced order approximation. Black isoline: full order approximation, red isoline: 5 snapshots, green isoline: 20 snapshots, blue isoline: 30 snapshots.

102 Summary of ICOM Developing a new (open source, community) modeling framework using a range of powerful and novel numerical methods Wide range of applicability from process/laboratory scales to regional/global Build upon a fluids code used for atmosphere/ocean modeling, coastal engineering, impact cratering, engineering fluids, multiphase flows, computational biology etc.

103 Summary of ICOM Developing a new (open source, community) modeling framework using a range of powerful and novel numerical methods Wide range of applicability from process/laboratory scales to regional/global Build upon a fluids code used for atmosphere/ocean modeling, coastal engineering, impact cratering, engineering fluids, multiphase flows, computational biology etc.

104 Summary of ICOM Developing a new (open source, community) modeling framework using a range of powerful and novel numerical methods Wide range of applicability from process/laboratory scales to regional/global Build upon a fluids code used for atmosphere/ocean modeling, coastal engineering, impact cratering, engineering fluids, multiphase flows, computational biology etc.

105 Summary of ICOM-continued Take advantage of cross-fertilization of computational techniques between disciplines The target problems have in common complex geometries and typically multi-scale and anisotropic solution structures which makes unstructured meshes and dynamic mesh adaptivity a natural choice with which to tackle these problems

106 Summary of ICOM-continued Take advantage of cross-fertilization of computational techniques between disciplines The target problems have in common complex geometries and typically multi-scale and anisotropic solution structures which makes unstructured meshes and dynamic mesh adaptivity a natural choice with which to tackle these problems

107 Summary of ICOM-continued Take advantage of cross-fertilization of computational techniques between disciplines The target problems have in common complex geometries and typically multi-scale and anisotropic solution structures which makes unstructured meshes and dynamic mesh adaptivity a natural choice with which to tackle these problems

108 Motivation for a new ocean model utilizing unstructured and adaptive mesh methods Need to resolve a wide range of spatial and temporal scales Model internal waves, boundary currents, eddies, overflows, convection events etc., accurately and efficiently within a global and coupled context Need for accurate and efficient representation of highly complex domains

109 Motivation for a new ocean model utilizing unstructured and adaptive mesh methods Need to resolve a wide range of spatial and temporal scales Model internal waves, boundary currents, eddies, overflows, convection events etc., accurately and efficiently within a global and coupled context Need for accurate and efficient representation of highly complex domains

110 Motivation for a new ocean model utilizing unstructured and adaptive mesh methods Need to resolve a wide range of spatial and temporal scales Model internal waves, boundary currents, eddies, overflows, convection events etc., accurately and efficiently within a global and coupled context Need for accurate and efficient representation of highly complex domains

111 Motivation for a new ocean model utilizing unstructured and adaptive mesh methods-continued: Ability to model interaction of flow with small scale topography, shelf seas, coastal regions, islands, estuaries, harbors,etc. Exploit anisotropy prevalent in this application area and attempt to take advantage of recent developments in CFD, numerical analysis, etc. Health in the genetic diversity of models and modeling approaches used

112 Motivation for a new ocean model utilizing unstructured and adaptive mesh methods-continued: Ability to model interaction of flow with small scale topography, shelf seas, coastal regions, islands, estuaries, harbors,etc. Exploit anisotropy prevalent in this application area and attempt to take advantage of recent developments in CFD, numerical analysis, etc. Health in the genetic diversity of models and modeling approaches used

113 Motivation for a new ocean model utilizing unstructured and adaptive mesh methods-continued: Ability to model interaction of flow with small scale topography, shelf seas, coastal regions, islands, estuaries, harbors,etc. Exploit anisotropy prevalent in this application area and attempt to take advantage of recent developments in CFD, numerical analysis, etc. Health in the genetic diversity of models and modeling approaches used

114 A quick overview of the numerical technology: Start with Fluidity:an open source control volume finite element solver for 3D compressible multi-phase fluids. Has been developed by AMCG for more than a decade and is the basis for a range of multi-physics multi-scale applications Add an adaptivity library which performs topological operations on the mesh, and mesh movement, to optimize the size and shape of elements in response to error measures Working robustly and efficiently in parallel

115 A quick overview of the numerical technology: Start with Fluidity:an open source control volume finite element solver for 3D compressible multi-phase fluids. Has been developed by AMCG for more than a decade and is the basis for a range of multi-physics multi-scale applications Add an adaptivity library which performs topological operations on the mesh, and mesh movement, to optimize the size and shape of elements in response to error measures Working robustly and efficiently in parallel

116 A quick overview of the numerical technology: Start with Fluidity:an open source control volume finite element solver for 3D compressible multi-phase fluids. Has been developed by AMCG for more than a decade and is the basis for a range of multi-physics multi-scale applications Add an adaptivity library which performs topological operations on the mesh, and mesh movement, to optimize the size and shape of elements in response to error measures Working robustly and efficiently in parallel

117 A quick overview of the numerical technology-continued Refer to the models oceanographic form as ICOM: note it is obviously non-hydrostatic Also building an adjoint to the model for data assimilation, sensitivity studies and goal-based error estimation Make use of open source solutions for solvers, preconditioners, I/O, visualization, etc

118 A quick overview of the numerical technology-continued Refer to the models oceanographic form as ICOM: note it is obviously non-hydrostatic Also building an adjoint to the model for data assimilation, sensitivity studies and goal-based error estimation Make use of open source solutions for solvers, preconditioners, I/O, visualization, etc

119 A quick overview of the numerical technology-continued Refer to the models oceanographic form as ICOM: note it is obviously non-hydrostatic Also building an adjoint to the model for data assimilation, sensitivity studies and goal-based error estimation Make use of open source solutions for solvers, preconditioners, I/O, visualization, etc

120 A quick overview of the numerical technology-continued A finite element discretization of the Boussinesq equations in 3-D Use of a non-hydrostatic solver to model dense water formation and flow over steep topography; Accurate and robust representation of hydrostatic and geostrophic balance; Inclusion of barotropic and baroclinic modes; Anisotropic unstructured meshes in the vertical as well as the horizontal to capture the details of local flow in all three directions.

121 A quick overview of the numerical technology-continued A finite element discretization of the Boussinesq equations in 3-D Use of a non-hydrostatic solver to model dense water formation and flow over steep topography; Accurate and robust representation of hydrostatic and geostrophic balance; Inclusion of barotropic and baroclinic modes; Anisotropic unstructured meshes in the vertical as well as the horizontal to capture the details of local flow in all three directions.

122 A quick overview of the numerical technology-continued A finite element discretization of the Boussinesq equations in 3-D Use of a non-hydrostatic solver to model dense water formation and flow over steep topography; Accurate and robust representation of hydrostatic and geostrophic balance; Inclusion of barotropic and baroclinic modes; Anisotropic unstructured meshes in the vertical as well as the horizontal to capture the details of local flow in all three directions.

123 A quick overview of the numerical technology-continued A finite element discretization of the Boussinesq equations in 3-D Use of a non-hydrostatic solver to model dense water formation and flow over steep topography; Accurate and robust representation of hydrostatic and geostrophic balance; Inclusion of barotropic and baroclinic modes; Anisotropic unstructured meshes in the vertical as well as the horizontal to capture the details of local flow in all three directions.

124 A quick overview of the numerical technology-continued A finite element discretization of the Boussinesq equations in 3-D Use of a non-hydrostatic solver to model dense water formation and flow over steep topography; Accurate and robust representation of hydrostatic and geostrophic balance; Inclusion of barotropic and baroclinic modes; Anisotropic unstructured meshes in the vertical as well as the horizontal to capture the details of local flow in all three directions.

125 A quick overview of the numerical technology-continued The second-order Crank-Nicolson scheme A semi-implicit projection method for pressure, ensuring the flow remains divergence-free while decoupling the computations for the momentum and continuity equations A fourth-order pressure filter is employed to aid stability Standard Petrov-Galerkin weightings are also used to improve numerical stability in the presence of advection dominated flows.

126 A quick overview of the numerical technology-continued The second-order Crank-Nicolson scheme A semi-implicit projection method for pressure, ensuring the flow remains divergence-free while decoupling the computations for the momentum and continuity equations A fourth-order pressure filter is employed to aid stability Standard Petrov-Galerkin weightings are also used to improve numerical stability in the presence of advection dominated flows.

127 A quick overview of the numerical technology-continued The second-order Crank-Nicolson scheme A semi-implicit projection method for pressure, ensuring the flow remains divergence-free while decoupling the computations for the momentum and continuity equations A fourth-order pressure filter is employed to aid stability Standard Petrov-Galerkin weightings are also used to improve numerical stability in the presence of advection dominated flows.

128 A quick overview of the numerical technology-continued The second-order Crank-Nicolson scheme A semi-implicit projection method for pressure, ensuring the flow remains divergence-free while decoupling the computations for the momentum and continuity equations A fourth-order pressure filter is employed to aid stability Standard Petrov-Galerkin weightings are also used to improve numerical stability in the presence of advection dominated flows.

129 Reduced order governing equations: The 3-D non-hydrostatic Boussinesq equations: u t u = 0, + u u + f k u = p ρgk + τ

130 Reduced order governing equations-continued: The variables can be expressed as an expansion of the POD basis functions for u, v, w, p M u u(t, x, y, z) = ū + α m,u (t)φ m,u (x, y, z), m=1 M v v(t, x, y, z) = v + α m,v (t)φ m,v (x, y, z), m=1 M w w(t, x, y, z) = w + α m,w (t)φ m,w (x, y, z), m=1 M p p(t, x, y, z) = p + α m,p (t)φ m,p (x, y, z) m=1

131 Geostrophic pressure: The pressure is divided into two parts: p = p ng + p g. The geostrophic pressure has to satisfy the geostrophic balance: p g = f k u Taking the divergence of the above equation, an elliptic equation for geostrophic pressure is obtained 2 p g = ( fv) x + (fu) y

132 Geostrophic pressure: The pressure is divided into two parts: p = p ng + p g. The geostrophic pressure has to satisfy the geostrophic balance: p g = f k u Taking the divergence of the above equation, an elliptic equation for geostrophic pressure is obtained 2 p g = ( fv) x + (fu) y

133 Geostrophic pressure-continued: To accurately represent geostrophic pressure its basis functions are split into two sets: Φ pgu and Φ pgv which are associated with the u- and v-velocity components. The geostrophic pressure can be obtained from a quadratic finite element representation while linear finite element representations are used for the velocity components

134 Geostrophic pressure-continued: Furthermore the geostrophic pressure can be represented by a summation of the two sets of geostrophic basis functions, which are calculated by solving the following elliptic equations using a conjugate gradient iterative method: 2 Φ pgu,m = (f Φ m,u) y 2 Φ pgv,m = ( f Φ m,v) x

135 Geostrophic pressure-continued: The geostrophic pressure can therefore be expressed as: M M p g = p g + α m,u Φ m,u + α m,v Φ m,v m=1 m=1 In addition the average geostrophic pressure is calculated from: 2 p g = ( f v) x + (f ū) y

136 Geostrophic pressure-continued: The geostrophic pressure can therefore be expressed as: M M p g = p g + α m,u Φ m,u + α m,v Φ m,v m=1 m=1 In addition the average geostrophic pressure is calculated from: 2 p g = ( f v) x + (f ū) y

137 Adaptive meshes in POD reduced order forward and adjoint models Challenges: The snapshots can be different length at different time levels The POD mesh of the forward model can diff from the POD mesh of the adjoint model Solution: To overcome these difficulties, a standard reference fixed mesh is adopted for the reduced model. The solutions from the original full model are interpolated from their own mesh onto the same reference fixed mesh at each time level, and then stored in the snapshots A goal-based error measurement approach is employed to find an optimal mesh for both reduced forward and adjoint models

138 Adaptive meshes in POD reduced order forward and adjoint models Challenges: The snapshots can be different length at different time levels The POD mesh of the forward model can diff from the POD mesh of the adjoint model Solution: To overcome these difficulties, a standard reference fixed mesh is adopted for the reduced model. The solutions from the original full model are interpolated from their own mesh onto the same reference fixed mesh at each time level, and then stored in the snapshots A goal-based error measurement approach is employed to find an optimal mesh for both reduced forward and adjoint models

139 Adaptive meshes in POD reduced order forward and adjoint models Challenges: The snapshots can be different length at different time levels The POD mesh of the forward model can diff from the POD mesh of the adjoint model Solution: To overcome these difficulties, a standard reference fixed mesh is adopted for the reduced model. The solutions from the original full model are interpolated from their own mesh onto the same reference fixed mesh at each time level, and then stored in the snapshots A goal-based error measurement approach is employed to find an optimal mesh for both reduced forward and adjoint models

140 Adaptive meshes in POD reduced order forward and adjoint models Challenges: The snapshots can be different length at different time levels The POD mesh of the forward model can diff from the POD mesh of the adjoint model Solution: To overcome these difficulties, a standard reference fixed mesh is adopted for the reduced model. The solutions from the original full model are interpolated from their own mesh onto the same reference fixed mesh at each time level, and then stored in the snapshots A goal-based error measurement approach is employed to find an optimal mesh for both reduced forward and adjoint models

141 Goal-based approach to choose an optimal mesh for both POD reduced order forward and adjoint models A function (defined below as the model reduction errors or the solution u which is of interest, say, in the target regions) is used to (1) determine an optimal for both reduced forward and adjoint models. Suppose that the functional whose accuracy is to be optimized is represented as I I (ψ), and I (ψ) = f (ψ) dv Ω In addition, the above functional can be also used to optimize uncertainties (inversion problem ) in models; optimize the POD bases and thus improve the accuracy of reduced models.

142 Goal-based approach to choose an optimal mesh for both POD reduced order forward and adjoint models A function (defined below as the model reduction errors or the solution u which is of interest, say, in the target regions) is used to (1) determine an optimal for both reduced forward and adjoint models. Suppose that the functional whose accuracy is to be optimized is represented as I I (ψ), and I (ψ) = f (ψ) dv Ω In addition, the above functional can be also used to optimize uncertainties (inversion problem ) in models; optimize the POD bases and thus improve the accuracy of reduced models.

143 Goal-based approach to choose an optimal mesh for both POD reduced order forward and adjoint models-continued The nodal Metric Tensor M i is obtained from the reduced forward model M i = γ H i ɛ i where H i is the forward Hessian matrix at node i, ɛ i is the forward interpolation error at node i The nodal Metric Tensor M i is obtained from the reduced forward model M i = γ H ɛ i i where H i is the adjoint Hessian matrix at node i, ɛ i is the adjoint interpolation error at node i

144 Goal-based approach to choose an optimal mesh for both POD reduced order forward and adjoint models-continued The nodal Metric Tensor M i is obtained from the reduced forward model M i = γ H i ɛ i where H i is the forward Hessian matrix at node i, ɛ i is the forward interpolation error at node i The nodal Metric Tensor M i is obtained from the reduced forward model M i = γ H ɛ i i where H i is the adjoint Hessian matrix at node i, ɛ i is the adjoint interpolation error at node i

145 Goal-based approach to choose an optimal mesh for both POD reduced order forward and adjoint models-continued: The interpolation error is related to the error contribution of the functional To satisfy the goal, the minimal ellipsoid is obtained by superscribing both ellipses and used to determine an optimal mesh ( M Gs i = G s M i, M i )

146 Goal-based approach to choose an optimal mesh for both POD reduced order forward and adjoint models-continued: The interpolation error is related to the error contribution of the functional To satisfy the goal, the minimal ellipsoid is obtained by superscribing both ellipses and used to determine an optimal mesh ( M Gs i = G s M i, M i )

147 Goal-based approach to choose an optimal mesh for both POD reduced order forward and adjoint models-continued Figure: An optimal mesh obtained by the goal-based error measure approach