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POD Pham, Tromeur-Dervout Proper Orthogonal Decomposition applied Introduction for in Parallel Solution to Large System of Stiff general POD method applied in model reduction ODEs POD and SVD POD in model reduction POD in decoupling dynamical system Analysis and T. Pham & D. Tromeur Dervout Algorithm Analysis Algorithm CDCSP/ICJ-UMR5208-CNRS Université Lyon 1 Numerical tests Non-uniform time steps June-19-2009 Outlines and Conclusions References 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

Outline – POD Pham, Introduction for general POD method applied in model 1 Tromeur-Dervout reduction Introduction for general POD POD and SVD method applied in model reduction POD in model reduction POD and SVD POD in decoupling dynamical system POD in model reduction POD in decoupling dynamical system Analysis and Algorithm 2 Analysis and Algorithm Analysis Analysis Algorithm Algorithm Numerical tests Non-uniform time steps Numerical tests 3 Outlines and Conclusions Non-uniform time steps References Outlines and Conclusions 4 UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 2/26

Motivation – Reduced order model and parallel algorithm POD Pham, Tromeur-Dervout Resolution of Full Order Model (FOM) takes very long time, Introduction for even unrealistic in the case of huge number of unknowns general POD method applied in Stiff ODE: implicit method - Newton iterations to solve the model reduction POD and SVD resulting non-linear system POD in model reduction For large scale problems: huge number of jacobian POD in decoupling dynamical system evaluations - ill-conditioned matrices. Analysis and Algorithm Different scale of components: unstable numerical algorithm. Analysis Algorithm POD process (Principal Component Analysis, Numerical tests Karhunen-Loeve expansion): overall behavior of a physical Non-uniform time steps system. Reduced Order Model(ROM) widely used has Outlines and Conclusions incredible performance References Application of ROM for decoupling dynamical system: a new process in parallelism UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 3/26

Motivation – Reduced order model and parallel algorithm POD Pham, Tromeur-Dervout Resolution of Full Order Model (FOM) takes very long time, Introduction for even unrealistic in the case of huge number of unknowns general POD method applied in Stiff ODE: implicit method - Newton iterations to solve the model reduction POD and SVD resulting non-linear system POD in model reduction For large scale problems: huge number of jacobian POD in decoupling dynamical system evaluations - ill-conditioned matrices. Analysis and Algorithm Different scale of components: unstable numerical algorithm. Analysis Algorithm POD process (Principal Component Analysis, Numerical tests Karhunen-Loeve expansion): overall behavior of a physical Non-uniform time steps system. Reduced Order Model(ROM) widely used has Outlines and Conclusions incredible performance References Application of ROM for decoupling dynamical system: a new process in parallelism UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 3/26

Motivation – Reduced order model and parallel algorithm POD Pham, Tromeur-Dervout Resolution of Full Order Model (FOM) takes very long time, Introduction for even unrealistic in the case of huge number of unknowns general POD method applied in Stiff ODE: implicit method - Newton iterations to solve the model reduction POD and SVD resulting non-linear system POD in model reduction For large scale problems: huge number of jacobian POD in decoupling dynamical system evaluations - ill-conditioned matrices. Analysis and Algorithm Different scale of components: unstable numerical algorithm. Analysis Algorithm POD process (Principal Component Analysis, Numerical tests Karhunen-Loeve expansion): overall behavior of a physical Non-uniform time steps system. Reduced Order Model(ROM) widely used has Outlines and Conclusions incredible performance References Application of ROM for decoupling dynamical system: a new process in parallelism UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 3/26

Motivation – Reduced order model and parallel algorithm POD Pham, Tromeur-Dervout Resolution of Full Order Model (FOM) takes very long time, Introduction for even unrealistic in the case of huge number of unknowns general POD method applied in Stiff ODE: implicit method - Newton iterations to solve the model reduction POD and SVD resulting non-linear system POD in model reduction For large scale problems: huge number of jacobian POD in decoupling dynamical system evaluations - ill-conditioned matrices. Analysis and Algorithm Different scale of components: unstable numerical algorithm. Analysis Algorithm POD process (Principal Component Analysis, Numerical tests Karhunen-Loeve expansion): overall behavior of a physical Non-uniform time steps system. Reduced Order Model(ROM) widely used has Outlines and Conclusions incredible performance References Application of ROM for decoupling dynamical system: a new process in parallelism UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 3/26

Motivation – Reduced order model and parallel algorithm POD Pham, Tromeur-Dervout Resolution of Full Order Model (FOM) takes very long time, Introduction for even unrealistic in the case of huge number of unknowns general POD method applied in Stiff ODE: implicit method - Newton iterations to solve the model reduction POD and SVD resulting non-linear system POD in model reduction For large scale problems: huge number of jacobian POD in decoupling dynamical system evaluations - ill-conditioned matrices. Analysis and Algorithm Different scale of components: unstable numerical algorithm. Analysis Algorithm POD process (Principal Component Analysis, Numerical tests Karhunen-Loeve expansion): overall behavior of a physical Non-uniform time steps system. Reduced Order Model(ROM) widely used has Outlines and Conclusions incredible performance References Application of ROM for decoupling dynamical system: a new process in parallelism UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 3/26

Motivation – Reduced order model and parallel algorithm POD Pham, Tromeur-Dervout Resolution of Full Order Model (FOM) takes very long time, Introduction for even unrealistic in the case of huge number of unknowns general POD method applied in Stiff ODE: implicit method - Newton iterations to solve the model reduction POD and SVD resulting non-linear system POD in model reduction For large scale problems: huge number of jacobian POD in decoupling dynamical system evaluations - ill-conditioned matrices. Analysis and Algorithm Different scale of components: unstable numerical algorithm. Analysis Algorithm POD process (Principal Component Analysis, Numerical tests Karhunen-Loeve expansion): overall behavior of a physical Non-uniform time steps system. Reduced Order Model(ROM) widely used has Outlines and Conclusions incredible performance References Application of ROM for decoupling dynamical system: a new process in parallelism UCBL, Cdcsp Pham, Tromeur-Dervout POD June-19-2009 3/26

POD in the minimization problem – POD process POD Given a collection of functions: Pham, Y = { y 1 ( x ) , . . . , y n ( x ) } , x ∈ Ω , y i ( x ) ∈ R m . Tromeur-Dervout Goal : find an “optimal” set of basis functions Ψ = ψ i ( x ) for Introduction for general POD the space V = span ( y i ) . i.e finding a space satisfying: method applied in model reduction POD and SVD n l POD in model reduction � � ( y T j ψ i ) ψ i � 2 min � y j − (1) POD in decoupling dynamical system Ψ j = 1 i = 1 Analysis and Algorithm Analysis � w.r.t the Euclidean norm � y � = y T y and subjects to : Algorithm Numerical tests ψ T Non-uniform time i ψ j = δ ij steps Outlines and Conclusions The optimality condition can be derived to the eigen value References problem: R ψ i ( x ) = λ i ψ i ( 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 in the minimization problem – POD process POD Given a collection of functions: Pham, Y = { y 1 ( x ) , . . . , y n ( x ) } , x ∈ Ω , y i ( x ) ∈ R m . Tromeur-Dervout Goal : find an “optimal” set of basis functions Ψ = ψ i ( x ) for Introduction for general POD the space V = span ( y i ) . i.e finding a space satisfying: method applied in model reduction POD and SVD n l POD in model reduction � � ( y T j ψ i ) ψ i � 2 min � y j − (1) POD in decoupling dynamical system Ψ j = 1 i = 1 Analysis and Algorithm Analysis � w.r.t the Euclidean norm � y � = y T y and subjects to : Algorithm Numerical tests ψ T Non-uniform time i ψ j = δ ij steps Outlines and Conclusions The optimality condition can be derived to the eigen value References problem: R ψ i ( x ) = λ i ψ i ( 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