Proper Orthogonal Decomposition applied Introduction for in - - PowerPoint PPT Presentation

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

Proper Orthogonal Decomposition applied in Parallel Solution to Large System of Stiff ODEs

• T. Pham & D. Tromeur Dervout

CDCSP/ICJ-UMR5208-CNRS Université Lyon 1

June-19-2009 Conference on Scientific Computing Conference in honour of E. Hairer’s 60th birthday

MS8 - Parallel methods for solving ODEs Funded by: National Research Agency - technologie logicielle 2006-2009 PARADE UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 1/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

Outline

–

1

Introduction for general POD method applied in model reduction POD and SVD POD in model reduction POD in decoupling dynamical system

2

Analysis and Algorithm Analysis Algorithm

3

Numerical tests Non-uniform time steps

4

Outlines and Conclusions

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 2/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

Motivation

– Reduced order model and parallel algorithm

Resolution of Full Order Model (FOM) takes very long time, even unrealistic in the case of huge number of unknowns Stiff ODE: implicit method - Newton iterations to solve the resulting non-linear system For large scale problems: huge number of jacobian evaluations - ill-conditioned matrices. Different scale of components: unstable numerical algorithm. POD process (Principal Component Analysis, Karhunen-Loeve expansion): overall behavior of a physical

• system. Reduced Order Model(ROM) widely used has

incredible performance Application of ROM for decoupling dynamical system: a new process in parallelism

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 3/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

Motivation

– Reduced order model and parallel algorithm

Resolution of Full Order Model (FOM) takes very long time, even unrealistic in the case of huge number of unknowns Stiff ODE: implicit method - Newton iterations to solve the resulting non-linear system For large scale problems: huge number of jacobian evaluations - ill-conditioned matrices. Different scale of components: unstable numerical algorithm. POD process (Principal Component Analysis, Karhunen-Loeve expansion): overall behavior of a physical

• system. Reduced Order Model(ROM) widely used has

incredible performance Application of ROM for decoupling dynamical system: a new process in parallelism

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 3/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

Motivation

– Reduced order model and parallel algorithm

Resolution of Full Order Model (FOM) takes very long time, even unrealistic in the case of huge number of unknowns Stiff ODE: implicit method - Newton iterations to solve the resulting non-linear system For large scale problems: huge number of jacobian evaluations - ill-conditioned matrices. Different scale of components: unstable numerical algorithm. POD process (Principal Component Analysis, Karhunen-Loeve expansion): overall behavior of a physical

• system. Reduced Order Model(ROM) widely used has

incredible performance Application of ROM for decoupling dynamical system: a new process in parallelism

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 3/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

Motivation

– Reduced order model and parallel algorithm

Resolution of Full Order Model (FOM) takes very long time, even unrealistic in the case of huge number of unknowns Stiff ODE: implicit method - Newton iterations to solve the resulting non-linear system For large scale problems: huge number of jacobian evaluations - ill-conditioned matrices. Different scale of components: unstable numerical algorithm. POD process (Principal Component Analysis, Karhunen-Loeve expansion): overall behavior of a physical

• system. Reduced Order Model(ROM) widely used has

incredible performance Application of ROM for decoupling dynamical system: a new process in parallelism

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 3/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

Motivation

– Reduced order model and parallel algorithm

Resolution of Full Order Model (FOM) takes very long time, even unrealistic in the case of huge number of unknowns Stiff ODE: implicit method - Newton iterations to solve the resulting non-linear system For large scale problems: huge number of jacobian evaluations - ill-conditioned matrices. Different scale of components: unstable numerical algorithm. POD process (Principal Component Analysis, Karhunen-Loeve expansion): overall behavior of a physical

• system. Reduced Order Model(ROM) widely used has

incredible performance Application of ROM for decoupling dynamical system: a new process in parallelism

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 3/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

Motivation

– Reduced order model and parallel algorithm

Resolution of Full Order Model (FOM) takes very long time, even unrealistic in the case of huge number of unknowns Stiff ODE: implicit method - Newton iterations to solve the resulting non-linear system For large scale problems: huge number of jacobian evaluations - ill-conditioned matrices. Different scale of components: unstable numerical algorithm. POD process (Principal Component Analysis, Karhunen-Loeve expansion): overall behavior of a physical

• system. Reduced Order Model(ROM) widely used has

incredible performance Application of ROM for decoupling dynamical system: a new process in parallelism

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 3/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

POD in the minimization problem

– POD process

Given a collection of functions: Y = {y1(x), . . . , yn(x)}, x ∈ Ω, yi(x) ∈ Rm. Goal : find an “optimal” set of basis functions Ψ = ψi(x) for the space V = span(yi). i.e finding a space satisfying: min

Ψ n

• j=1

yj −

l

• i=1

(yT

j ψi)ψi2

(1) w.r.t the Euclidean norm y =

• yTy and subjects to :

ψT

i ψj = δij

The optimality condition can be derived to the eigen value problem: Rψi(x) = λiψi(x), i = 1, . . . , l (2) where R denotes the correlation matrix,e.g: R = YY T in the discrete case.

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 4/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

POD in the minimization problem

– POD process

Given a collection of functions: Y = {y1(x), . . . , yn(x)}, x ∈ Ω, yi(x) ∈ Rm. Goal : find an “optimal” set of basis functions Ψ = ψi(x) for the space V = span(yi). i.e finding a space satisfying: min

Ψ n

• j=1

yj −

l

• i=1

(yT

j ψi)ψi2

(1) w.r.t the Euclidean norm y =

• yTy and subjects to :

ψT

i ψj = δij

The optimality condition can be derived to the eigen value problem: Rψi(x) = λiψi(x), i = 1, . . . , l (2) where R denotes the correlation matrix,e.g: R = YY T in the discrete case.

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 4/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

POD in the minimization problem

– POD process

Given a collection of functions: Y = {y1(x), . . . , yn(x)}, x ∈ Ω, yi(x) ∈ Rm. Goal : find an “optimal” set of basis functions Ψ = ψi(x) for the space V = span(yi). i.e finding a space satisfying: min

Ψ n

• j=1

yj −

l

• i=1

(yT

j ψi)ψi2

(1) w.r.t the Euclidean norm y =

• yTy and subjects to :

ψT

i ψj = δij

The optimality condition can be derived to the eigen value problem: Rψi(x) = λiψi(x), i = 1, . . . , l (2) where R denotes the correlation matrix,e.g: R = YY T in the discrete case.

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 4/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

POD in the minimization problem

– POD and SVD

Eigen value problem: YY Tψi = λiψi, i = 1, . . . , l Solution by SVD for Y ∈ Rm×n, m > n, d = rank(Y): UTYV = diag(σ1, . . . , σd), σ1 ≥ · · · ≥ σd > 0 where U = [u1, . . . , um] ∈ Rm×n and V = [v1, . . . , vn] ∈ Rn×n

• rthogonal containing singular vectors.

SVD and POD-from Yvi = σiui, Y Tui = σvi: YY Tui = σ2

i ui

POD basis: ψi = ui and λi = σ2

i , of rank l for i = 1, . . . , l ≤ d

Properties:

n

• j=1

yj −

l

• i=1

(yT

j ψi)ψi2 = d

• i=l+1

λi

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 5/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

POD in the minimization problem

– POD and SVD

Eigen value problem: YY Tψi = λiψi, i = 1, . . . , l Solution by SVD for Y ∈ Rm×n, m > n, d = rank(Y): UTYV = diag(σ1, . . . , σd), σ1 ≥ · · · ≥ σd > 0 where U = [u1, . . . , um] ∈ Rm×n and V = [v1, . . . , vn] ∈ Rn×n

• rthogonal containing singular vectors.

SVD and POD-from Yvi = σiui, Y Tui = σvi: YY Tui = σ2

i ui

POD basis: ψi = ui and λi = σ2

i , of rank l for i = 1, . . . , l ≤ d

Properties:

n

• j=1

yj −

l

• i=1

(yT

j ψi)ψi2 = d

• i=l+1

λi

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 5/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

POD in the minimization problem

– POD and SVD

Eigen value problem: YY Tψi = λiψi, i = 1, . . . , l Solution by SVD for Y ∈ Rm×n, m > n, d = rank(Y): UTYV = diag(σ1, . . . , σd), σ1 ≥ · · · ≥ σd > 0 where U = [u1, . . . , um] ∈ Rm×n and V = [v1, . . . , vn] ∈ Rn×n

• rthogonal containing singular vectors.

SVD and POD-from Yvi = σiui, Y Tui = σvi: YY Tui = σ2

i ui

POD basis: ψi = ui and λi = σ2

i , of rank l for i = 1, . . . , l ≤ d

Properties:

n

• j=1

yj −

l

• i=1

(yT

j ψi)ψi2 = d

• i=l+1

λi

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 5/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

POD in the minimization problem

– POD and SVD

Eigen value problem: YY Tψi = λiψi, i = 1, . . . , l Solution by SVD for Y ∈ Rm×n, m > n, d = rank(Y): UTYV = diag(σ1, . . . , σd), σ1 ≥ · · · ≥ σd > 0 where U = [u1, . . . , um] ∈ Rm×n and V = [v1, . . . , vn] ∈ Rn×n

• rthogonal containing singular vectors.

SVD and POD-from Yvi = σiui, Y Tui = σvi: YY Tui = σ2

i ui

POD basis: ψi = ui and λi = σ2

i , of rank l for i = 1, . . . , l ≤ d

Properties:

n

• j=1

yj −

l

• i=1

(yT

j ψi)ψi2 = d

• i=l+1

λi

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 5/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

POD in the minimization problem

– POD and SVD

Eigen value problem: YY Tψi = λiψi, i = 1, . . . , l Solution by SVD for Y ∈ Rm×n, m > n, d = rank(Y): UTYV = diag(σ1, . . . , σd), σ1 ≥ · · · ≥ σd > 0 where U = [u1, . . . , um] ∈ Rm×n and V = [v1, . . . , vn] ∈ Rn×n

• rthogonal containing singular vectors.

SVD and POD-from Yvi = σiui, Y Tui = σvi: YY Tui = σ2

i ui

POD basis: ψi = ui and λi = σ2

i , of rank l for i = 1, . . . , l ≤ d

Properties:

n

• j=1

yj −

l

• i=1

(yT

j ψi)ψi2 = d

• i=l+1

λi

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 5/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

POD in model reduction

– Computation of the POD basis

Dynamical system on a Hilbert space X: ˙ y(t) = f(t, y(t)), t ∈ (0, T) and y(0) = y0 (3) with continuous f: [0, T] × X → X Available or known snapshots (experimental or numerical simulation): Y = {yj}, ; yj = y(tj) ∈ Rm, 1 ≤ j ≤ n, 0 ≤ t1 < t2 < · · · < tn ≤ T Ensemble of snapshots: V = span{y1, . . . , yn} ∈ Rm. Where d = dimV Computation of the POD via the SVD, set: λi = σ2

i , ψi = ui,

keep l ≤ d modes

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 6/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

POD in model reduction

– Computation of the POD basis

Dynamical system on a Hilbert space X: ˙ y(t) = f(t, y(t)), t ∈ (0, T) and y(0) = y0 (3) with continuous f: [0, T] × X → X Available or known snapshots (experimental or numerical simulation): Y = {yj}, ; yj = y(tj) ∈ Rm, 1 ≤ j ≤ n, 0 ≤ t1 < t2 < · · · < tn ≤ T Ensemble of snapshots: V = span{y1, . . . , yn} ∈ Rm. Where d = dimV Computation of the POD via the SVD, set: λi = σ2

i , ψi = ui,

keep l ≤ d modes

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 6/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

POD in model reduction

– Computation of the POD basis

Dynamical system on a Hilbert space X: ˙ y(t) = f(t, y(t)), t ∈ (0, T) and y(0) = y0 (3) with continuous f: [0, T] × X → X Available or known snapshots (experimental or numerical simulation): Y = {yj}, ; yj = y(tj) ∈ Rm, 1 ≤ j ≤ n, 0 ≤ t1 < t2 < · · · < tn ≤ T Ensemble of snapshots: V = span{y1, . . . , yn} ∈ Rm. Where d = dimV Computation of the POD via the SVD, set: λi = σ2

i , ψi = ui,

keep l ≤ d modes

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 6/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

POD in model reduction

– Computation of the POD basis

Dynamical system on a Hilbert space X: ˙ y(t) = f(t, y(t)), t ∈ (0, T) and y(0) = y0 (3) with continuous f: [0, T] × X → X Available or known snapshots (experimental or numerical simulation): Y = {yj}, ; yj = y(tj) ∈ Rm, 1 ≤ j ≤ n, 0 ≤ t1 < t2 < · · · < tn ≤ T Ensemble of snapshots: V = span{y1, . . . , yn} ∈ Rm. Where d = dimV Computation of the POD via the SVD, set: λi = σ2

i , ψi = ui,

keep l ≤ d modes

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 6/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

POD in model reduction

– Reduced order system

Introduce the reduced variable: ξ(t) ∈ Vl = span{ψ1, . . . , ψl} ⊂ V The dynamical system under the reduced form: ξ′(t) = UTf(t, Uξ(t)) Error formula for the POD basis of rank l:

n

• j=1

yj −

l

• j=1

(yT

j ψi)ψi2 = d

• i=l+1

λi In general case d ≪ m: a few number of modes can represent with accuracy the FOM Drawbacks:

Snapshots avaible only when the FOM are solved Positions of snapshots influence the accuracy of the model

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 7/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

POD in model reduction

– Reduced order system

Introduce the reduced variable: ξ(t) ∈ Vl = span{ψ1, . . . , ψl} ⊂ V The dynamical system under the reduced form: ξ′(t) = UTf(t, Uξ(t)) Error formula for the POD basis of rank l:

n

• j=1

yj −

l

• j=1

(yT

j ψi)ψi2 = d

• i=l+1

λi In general case d ≪ m: a few number of modes can represent with accuracy the FOM Drawbacks:

Snapshots avaible only when the FOM are solved Positions of snapshots influence the accuracy of the model

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 7/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

POD in model reduction

– Reduced order system

Introduce the reduced variable: ξ(t) ∈ Vl = span{ψ1, . . . , ψl} ⊂ V The dynamical system under the reduced form: ξ′(t) = UTf(t, Uξ(t)) Error formula for the POD basis of rank l:

n

• j=1

yj −

l

• j=1

(yT

j ψi)ψi2 = d

• i=l+1

λi In general case d ≪ m: a few number of modes can represent with accuracy the FOM Drawbacks:

Snapshots avaible only when the FOM are solved Positions of snapshots influence the accuracy of the model

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 7/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

POD in model reduction

– Reduced order system

Introduce the reduced variable: ξ(t) ∈ Vl = span{ψ1, . . . , ψl} ⊂ V The dynamical system under the reduced form: ξ′(t) = UTf(t, Uξ(t)) Error formula for the POD basis of rank l:

n

• j=1

yj −

l

• j=1

(yT

j ψi)ψi2 = d

• i=l+1

λi In general case d ≪ m: a few number of modes can represent with accuracy the FOM Drawbacks:

Snapshots avaible only when the FOM are solved Positions of snapshots influence the accuracy of the model

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 7/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

POD in model reduction

– Reduced order system

Introduce the reduced variable: ξ(t) ∈ Vl = span{ψ1, . . . , ψl} ⊂ V The dynamical system under the reduced form: ξ′(t) = UTf(t, Uξ(t)) Error formula for the POD basis of rank l:

n

• j=1

yj −

l

• j=1

(yT

j ψi)ψi2 = d

• i=l+1

λi In general case d ≪ m: a few number of modes can represent with accuracy the FOM Drawbacks:

Snapshots avaible only when the FOM are solved Positions of snapshots influence the accuracy of the model

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 7/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

POD in decoupling dynamical systems

– Decouple a dynamical system

Dynamical system: x(t)′ = f(t, x(t)) =

• f(t, x1(t), x2(t))

g(t, x1(t), x2(t))

• , x(0) = x0

(4) where x(t) = (x1(t), x2(t)) ∈ Rn1 × Rn2 Some q snapshots: X = X1 X2

• =

x1,1 . . . x1,q x2,1 . . . x2,q

• (5)

POD basis by the SVD of X1 and X2: X1 = U1S1V ′

1

X2 = U2S2V ′

2

Singular values of Xi decay rapidly: truncate the POD basis at the first l1 and l2 vectors (l1, l2 < q): U1,l1, S1,l1, V1,l1 and U2,l2, S2,l2, V2,l2.

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 8/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

POD in decoupling dynamical systems

– Decouple a dynamical system

Dynamical system: x(t)′ = f(t, x(t)) =

• f(t, x1(t), x2(t))

g(t, x1(t), x2(t))

• , x(0) = x0

(4) where x(t) = (x1(t), x2(t)) ∈ Rn1 × Rn2 Some q snapshots: X = X1 X2

• =

x1,1 . . . x1,q x2,1 . . . x2,q

• (5)

POD basis by the SVD of X1 and X2: X1 = U1S1V ′

1

X2 = U2S2V ′

2

Singular values of Xi decay rapidly: truncate the POD basis at the first l1 and l2 vectors (l1, l2 < q): U1,l1, S1,l1, V1,l1 and U2,l2, S2,l2, V2,l2.

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 8/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

POD in decoupling dynamical systems

– Decouple a dynamical system

Dynamical system: x(t)′ = f(t, x(t)) =

• f(t, x1(t), x2(t))

g(t, x1(t), x2(t))

• , x(0) = x0

(4) where x(t) = (x1(t), x2(t)) ∈ Rn1 × Rn2 Some q snapshots: X = X1 X2

• =

x1,1 . . . x1,q x2,1 . . . x2,q

• (5)

POD basis by the SVD of X1 and X2: X1 = U1S1V ′

1

X2 = U2S2V ′

2

Singular values of Xi decay rapidly: truncate the POD basis at the first l1 and l2 vectors (l1, l2 < q): U1,l1, S1,l1, V1,l1 and U2,l2, S2,l2, V2,l2.

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 8/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

POD in decoupling dynamical systems

– Decouple a dynamical system

Dynamical system: x(t)′ = f(t, x(t)) =

• f(t, x1(t), x2(t))

g(t, x1(t), x2(t))

• , x(0) = x0

(4) where x(t) = (x1(t), x2(t)) ∈ Rn1 × Rn2 Some q snapshots: X = X1 X2

• =

x1,1 . . . x1,q x2,1 . . . x2,q

• (5)

POD basis by the SVD of X1 and X2: X1 = U1S1V ′

1

X2 = U2S2V ′

2

Singular values of Xi decay rapidly: truncate the POD basis at the first l1 and l2 vectors (l1, l2 < q): U1,l1, S1,l1, V1,l1 and U2,l2, S2,l2, V2,l2.

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 8/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

POD in decoupling dynamical systems

– ROM for subsystems

Introduce α1 and α2: x1(t) ≈ UT

1,l1α1(t), α1 ∈ span{U1,l1}

x2(t) ≈ UT

2,l2α2(t), α2 ∈ span{U2,l2}

Independent sub-systems: (S1) x′

1

α′

2

• =
• f(t, x1, U2,l2α2)

UT

2,l2g(t, x1, U2,l2α2)

• (S2)

α′

1

x′

2

• =

UT

1,l1f(t, U1,l1α1, x2)

g(t, U1,l1α1, , x2)

• (S1), (S2) solved on parallel computers

Solve (S1) and (S2) by updating POD basis after tq

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 9/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

POD in decoupling dynamical systems

– ROM for subsystems

Introduce α1 and α2: x1(t) ≈ UT

1,l1α1(t), α1 ∈ span{U1,l1}

x2(t) ≈ UT

2,l2α2(t), α2 ∈ span{U2,l2}

Independent sub-systems: (S1) x′

1

α′

2

• =
• f(t, x1, U2,l2α2)

UT

2,l2g(t, x1, U2,l2α2)

• (S2)

α′

1

x′

2

• =

UT

1,l1f(t, U1,l1α1, x2)

g(t, U1,l1α1, , x2)

• (S1), (S2) solved on parallel computers

Solve (S1) and (S2) by updating POD basis after tq

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 9/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

POD in decoupling dynamical systems

– ROM for subsystems

Introduce α1 and α2: x1(t) ≈ UT

1,l1α1(t), α1 ∈ span{U1,l1}

x2(t) ≈ UT

2,l2α2(t), α2 ∈ span{U2,l2}

Independent sub-systems: (S1) x′

1

α′

2

• =
• f(t, x1, U2,l2α2)

UT

2,l2g(t, x1, U2,l2α2)

• (S2)

α′

1

x′

2

• =

UT

1,l1f(t, U1,l1α1, x2)

g(t, U1,l1α1, , x2)

• (S1), (S2) solved on parallel computers

Solve (S1) and (S2) by updating POD basis after tq

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 9/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

POD in decoupling dynamical systems

– ROM for subsystems

Introduce α1 and α2: x1(t) ≈ UT

1,l1α1(t), α1 ∈ span{U1,l1}

x2(t) ≈ UT

2,l2α2(t), α2 ∈ span{U2,l2}

Independent sub-systems: (S1) x′

1

α′

2

• =
• f(t, x1, U2,l2α2)

UT

2,l2g(t, x1, U2,l2α2)

• (S2)

α′

1

x′

2

• =

UT

1,l1f(t, U1,l1α1, x2)

g(t, U1,l1α1, , x2)

• (S1), (S2) solved on parallel computers

Solve (S1) and (S2) by updating POD basis after tq

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 9/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

Error analysis

– Error formula in reduced models

ROM and FOM in (S1) linearized to the first order: e′

1

= Jf(x1)e1 + Jf(x2)e2 e′

2

= (U2UT

2 − I)g(x1, x2, t) + U2UT 2 (Jg(x1)e1 + Jg(y2)e2)

U2 is truncated, J - jacobian matrix Error in decoupling system: e′(t) = M(t)e(t) + N(t) (6) where e(t) = (e1(t), e2(t))T, M(t) =

• Jf(y1)

Jf(y2) U2UT

2 Jg(y1)

U2UT

2 Jg(y2),

• and

N(t) =

• 0n1

(U2UT

2 − I)g

• UCBL, Cdcsp

Pham, Tromeur-Dervout POD June-19-2009 10/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

Error analysis

– Error formula in reduced models

ROM and FOM in (S1) linearized to the first order: e′

1

= Jf(x1)e1 + Jf(x2)e2 e′

2

= (U2UT

2 − I)g(x1, x2, t) + U2UT 2 (Jg(x1)e1 + Jg(y2)e2)

U2 is truncated, J - jacobian matrix Error in decoupling system: e′(t) = M(t)e(t) + N(t) (6) where e(t) = (e1(t), e2(t))T, M(t) =

• Jf(y1)

Jf(y2) U2UT

2 Jg(y1)

U2UT

2 Jg(y2),

• and

N(t) =

• 0n1

(U2UT

2 − I)g

• UCBL, Cdcsp

Pham, Tromeur-Dervout POD June-19-2009 10/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

Error analysis

– Residual in decoupling

e(t) is “small” when: the residual ǫ(t) = N(t) < TOL When the residual increases, the POD basis is no longer appropriated to represent the ROM: y′

2 = g ∈ Ker(U2UT 2 − I)

(S1) and (S2) have to update the POD basis each other

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 11/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

Error analysis

– Residual in decoupling

e(t) is “small” when: the residual ǫ(t) = N(t) < TOL When the residual increases, the POD basis is no longer appropriated to represent the ROM: y′

2 = g ∈ Ker(U2UT 2 − I)

(S1) and (S2) have to update the POD basis each other

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 11/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

Error analysis

– Residual in decoupling

e(t) is “small” when: the residual ǫ(t) = N(t) < TOL When the residual increases, the POD basis is no longer appropriated to represent the ROM: y′

2 = g ∈ Ker(U2UT 2 − I)

(S1) and (S2) have to update the POD basis each other

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 11/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

Algorithm with uniform time steps

– 1/2

Algorithm 1 Parallel algorithm with POD: for each system (SI) Initialization q snapshots of the full system up to the first node T1 Compute the POD basis for each separated states Initialize uI,i(T1) and put the initial condition for uI(T1) for Tj: j=1 to J do for tit=Tj to Tj+1 do Take a step uI(tit) Update the snapshot matrix X with the local uI,I end for Compute the POD basis from the latest snapshot matrix X Make the global communication between SI to exchange the last POD basis Received the new basis, update uI,i end for

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 12/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

Algorithm with uniform time steps

– 2/2

S1 t0 t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 Send (U1, α1) S2

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 13/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

Algorithm of variable time steps

– An overall simplified algorithm

Algorithm 2 Parallel algorithm with POD: exchange based on residual with variable time step

1: Compute the POD basis for each separated states 2: Initialize uI(tI,1) and put the initial condition 3: for tI,i = tI,1 to end of the simulation do 4:

Take a step uI(tI,i)

5:

Update uI,I(tI,i) to the XI,I - the matrix of snapshots

6:

Compute the residual ǫI,J

7:

if ǫI,J > TOL then

8:

Ask for (SJ) for an update of the POD basis

9:

end if

10: end for

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 14/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

Remarks on algorithms

– Advantages and drawbacks

Subsystem under the ROM: smaller size, less amount of computation effort Execution of solver on parallel computers ODE solver: same amount of RHS evaluations, but saving on jacobian computation (evaluation and linear solving for implicit methods) Drawbacks:

POD basis for ROM is avaible only up to the current time Dependence of the reduced space on the time grid

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 15/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

Remarks on algorithms

– Advantages and drawbacks

Subsystem under the ROM: smaller size, less amount of computation effort Execution of solver on parallel computers ODE solver: same amount of RHS evaluations, but saving on jacobian computation (evaluation and linear solving for implicit methods) Drawbacks:

POD basis for ROM is avaible only up to the current time Dependence of the reduced space on the time grid

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 15/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

Remarks on algorithms

– Advantages and drawbacks

Subsystem under the ROM: smaller size, less amount of computation effort Execution of solver on parallel computers ODE solver: same amount of RHS evaluations, but saving on jacobian computation (evaluation and linear solving for implicit methods) Drawbacks:

POD basis for ROM is avaible only up to the current time Dependence of the reduced space on the time grid

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 15/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

Remarks on algorithms

– Advantages and drawbacks

Subsystem under the ROM: smaller size, less amount of computation effort Execution of solver on parallel computers ODE solver: same amount of RHS evaluations, but saving on jacobian computation (evaluation and linear solving for implicit methods) Drawbacks:

POD basis for ROM is avaible only up to the current time Dependence of the reduced space on the time grid

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 15/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

Remarks on algorithms

– Advantages and drawbacks

Subsystem under the ROM: smaller size, less amount of computation effort Execution of solver on parallel computers ODE solver: same amount of RHS evaluations, but saving on jacobian computation (evaluation and linear solving for implicit methods) Drawbacks:

POD basis for ROM is avaible only up to the current time Dependence of the reduced space on the time grid

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 15/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

Remarks on algorithms

– Advantages and drawbacks

Subsystem under the ROM: smaller size, less amount of computation effort Execution of solver on parallel computers ODE solver: same amount of RHS evaluations, but saving on jacobian computation (evaluation and linear solving for implicit methods) Drawbacks:

POD basis for ROM is avaible only up to the current time Dependence of the reduced space on the time grid

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 15/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

Diurnal Kinetic advection-diffusion

– Problem overview

Large stiff two-dimensional system of PDE models a chemical mechanism Evolution and concentration of two chemical species: ∂ci ∂t = Kh ∂2ci ∂x2 +V ∂ci ∂x + ∂ ∂y Kv(y)∂ci ∂y +Ri(c1, c2, t) (i = 1, 2) 0 ≤ x ≤ 20, 30 ≤ y ≤ 50 (in km) Kh = 4.0 · 10−6,V = 10−3, Kv = 10−8exp(y/5) q1 = 1.63 · 10−16, q2 = 4.66 · 10−16, c3 = 3.7 · 1016 R1(c1, c2, t) = −q1c1c3 − q2c1c2 + 2q3(t)c3 + q4(t)c2 R2(c1, c2, t) = q1c1c3 − q2c1c2 − q4(t)c2 Solver: CVODE-SUNDIALS

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 16/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

Diurnal Kinetic advection-diffusion

– Problem overview

Large stiff two-dimensional system of PDE models a chemical mechanism Evolution and concentration of two chemical species: ∂ci ∂t = Kh ∂2ci ∂x2 +V ∂ci ∂x + ∂ ∂y Kv(y)∂ci ∂y +Ri(c1, c2, t) (i = 1, 2) 0 ≤ x ≤ 20, 30 ≤ y ≤ 50 (in km) Kh = 4.0 · 10−6,V = 10−3, Kv = 10−8exp(y/5) q1 = 1.63 · 10−16, q2 = 4.66 · 10−16, c3 = 3.7 · 1016 R1(c1, c2, t) = −q1c1c3 − q2c1c2 + 2q3(t)c3 + q4(t)c2 R2(c1, c2, t) = q1c1c3 − q2c1c2 − q4(t)c2 Solver: CVODE-SUNDIALS

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 16/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

Diurnal Kinetic advection-diffusion

– Problem overview

Large stiff two-dimensional system of PDE models a chemical mechanism Evolution and concentration of two chemical species: ∂ci ∂t = Kh ∂2ci ∂x2 +V ∂ci ∂x + ∂ ∂y Kv(y)∂ci ∂y +Ri(c1, c2, t) (i = 1, 2) 0 ≤ x ≤ 20, 30 ≤ y ≤ 50 (in km) Kh = 4.0 · 10−6,V = 10−3, Kv = 10−8exp(y/5) q1 = 1.63 · 10−16, q2 = 4.66 · 10−16, c3 = 3.7 · 1016 R1(c1, c2, t) = −q1c1c3 − q2c1c2 + 2q3(t)c3 + q4(t)c2 R2(c1, c2, t) = q1c1c3 − q2c1c2 − q4(t)c2 Solver: CVODE-SUNDIALS

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 16/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

Choice of snapshots

– Analysis of the POD basis 1/2

case 4 case 3 case 2 case 1 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000

Figure: Position of time snapshots

1e+11 2e+11 3e+11 4e+11 5e+11 6e+11 7e+11 8e+11 10000 20000 30000 40000 50000 60000 70000 80000 90000 100000 case 1 case 2 case 3 case 4 reference solution

(a)

1e-04 0.01 1 100 10000 1e+06 1e+08 1e+10 1e+12 1e+14 5 10 15 20 25 30 35 40 case 1 case 2 case 3 case 4

(b) Figure: The singular values for different snapshot positions and the sample solution. Time interval [0, 1.0 · 105], Left: the solutions, Right: the SVD taken from snapshots

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 17/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

Choice of snapshots

– Analysis of the POD basis 2/2

Small modes ignored in the ROM represent noise to the trajectory Snapshot have to be taken regulary on the time simulation

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 18/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

Choice of snapshots

– Analysis of the POD basis 2/2

Small modes ignored in the ROM represent noise to the trajectory Snapshot have to be taken regulary on the time simulation

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 18/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

Reference solution

– At sample points

1e+07 2e+07 3e+07 1e+04 2e+04 3e+04 4e+04 t C1 BL MD TR 1e+11 2e+11 3e+11 4e+11 5e+11 1e+04 2e+04 3e+04 4e+04 t C2 BL MD TR

Figure: Reference solution C1 and C2 in time at some sample points: BL bottom left, MD: middle, TR: top right on the space grid

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 19/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

Solution from the fixed time step version

– 1/3

S1 t0 t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13 t14 t15 t16 t17 t18 t19 t20 S2 Send POD1 q=5 Send POD2 q = 5 r = 8

Figure: Data exchange scheme: fixed time steps with q = 5 and r = 8

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 20/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

Solution from the fixed time step version

– 2/3

0.00 0.01 0.03 0.04 1.0e+04 2.0e+04 3.0e+04 4.0e+04 BL r=40 r=80 r=160 0.00 0.01 0.02 0.03 0.04 1.0e+04 2.0e+04 3.0e+04 4.0e+04 BL r=40 r=80 r=160

Figure: Relative error at sample points in time t for different values of r, q = 20, step size fixed at δt = 1.0,ROM built with l modes Pq

i=l+1 λi

Pq

i=1 λi

< 10−6, λl+1 < 10−2

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 21/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

Solution from the fixed time step version

– 2/3

1e-03 1e-02 1e-01 1e+00 160 120 80 40 r (S1) q=20 q=30 q=40 1e-03 1e-02 1e-01 1e+00 160 120 80 40 r (S2) q=20 q=30 q=40

Figure: Relative error in max norm, relative tolerance is fixed at 1 · 10−5 Table: Relative error from the decoupled model compared to the error due to the numerical solver

Eref∞ ES0∞ ES1∞ 1.55.10−2 7.73.10−3 9.14.10−3

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 22/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

Solution from the fixed time step version

– Performance on parallel computer 3/3 Table: Compare a multi-subsystems version vs single solver version, the grid in space is MX = 20, MY = 20 N◦ procs Single 2 3 system S S1 S2 S1 S2 S3 Size 800 420 420 307 307 306 Fevals 2651 3957 5455 4102 6175 6281 LUs 771 1152 1336 1053 1627 1614 Times (s) 667.51 200.5 200.5 31.41 31.46 31.45 Single solver: tolerance at 10−5. POD basis truncated at 10−2

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 23/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

Numerical result with non-uniform steps

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0.00e+00 1.00e+07 2.00e+07 3.00e+07 0e+00 2e+04 4e+04 6e+04 8e+04 1e+05 t C1 Ref S2 S1 3.00e+11 3.20e+11 3.40e+11 3.60e+11 3.80e+11 0e+00 2e+04 4e+04 6e+04 8e+04 1e+05 t C2 Ref S1 S2 0.00e+00 5.00e+06 1.00e+07 0e+00 2e+04 4e+04 6e+04 8e+04 1e+05 t C2 Ref S2 S1 2.00e+11 4.00e+11 6.00e+11 0e+00 2e+04 4e+04 6e+04 8e+04 1e+05 t C2 Ref S1 S2 0.00e+00 1.00e+07 2.00e+07 3.00e+07 0e+00 2e+04 4e+04 6e+04 8e+04 1e+05 t C2 Ref S2 S1 3.00e+11 3.20e+11 3.40e+11 3.60e+11 3.80e+11 4.00e+11 4.20e+11 0e+00 2e+04 4e+04 6e+04 8e+04 1e+05 t C2 Ref S1 S2

Figure: Compared the solution from the algorithm to the reference solution at some sample points

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 24/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

Outlines and Perspectives

–

New application of ROM to solve dynamical system in parallel High performance decoupling algorithm Coming method: improve the POD basis by using incremental POD formula [Gu and Eisenstat, 1995]

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 25/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

Outlines and Perspectives

–

New application of ROM to solve dynamical system in parallel High performance decoupling algorithm Coming method: improve the POD basis by using incremental POD formula [Gu and Eisenstat, 1995]

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 25/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

Outlines and Perspectives

–

New application of ROM to solve dynamical system in parallel High performance decoupling algorithm Coming method: improve the POD basis by using incremental POD formula [Gu and Eisenstat, 1995]

UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 25/26

POD Pham, Tromeur-Dervout Introduction for general POD method applied in model reduction

POD and SVD POD in model reduction POD in decoupling dynamical system

Analysis and Algorithm

Analysis Algorithm

Numerical tests

Non-uniform time steps

Outlines and Conclusions References

References I

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Gu, M. and Eisenstat, S. C. (1995). Downdating the singular value decomposition. SIAM J. Matrix Anal. Appl., 16(3):793–810. Homescu, C., Petzold, L. R., and Serban, R. (2007). Error estimation for reduced-order models of dynamical systems. SIAM Rev., 49(2):277–299. Kunisch, K. and Volkwein, S. (2001). Galerkin proper orthogonal decomposition methods for parabolic problems.

• Numer. Math., 90(1):117–148.

Kunisch, K. and Volkwein, S. (2002). Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J. Numer. Anal., 40(2):492–515 (electronic).

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