REDUCED ORDER METHODS

Transcription

1 REDUCED ORDER METHODS Elisa SCHENONE Legato Team Computational Mechanics Elisa SCHENONE RUES Seminar April, 6th 5

2 team at and of Numerical simulation of biological flows Respiration modelling Blood flow modelling Cardiac electrophysiology avr I 6 6 Elisa SCHENONE 6 6 RUES Seminar V V V5 V avf III avl II V V6 6 April, 6th 5

3 team at and of Numerical simulation of biological flows Respiration modelling Blood flow modelling Cardiac electrophysiology avr I 6 6 Elisa SCHENONE 6 6 RUES Seminar V V V5 V avf III avl II V V6 6 April, 6th 5

4 PhD project: Inverse problems and reduced models in cardiac electrophysiology Supervisors: Muriel BOULAKIA and Jean-Frédéric GERBEAU Subjects: Simulation of realistic Electrocardiograms Parameters estimation Reduced order models (RB, POD, ALP, EIM) Inverse problems Post-Doc project: Reduced models in computational mechanics Supervisors: Stéphane BORDAS and Lars BEEX Subjects: Simulation of hyperelastic materials with reduced order models Elisa SCHENONE RUES Seminar April, 6th 5

5 PhD project: Inverse problems and reduced models in cardiac electrophysiology Supervisors: Muriel BOULAKIA and Jean-Frédéric GERBEAU Subjects: Simulation of realistic Electrocardiograms Parameters estimation Reduced order models (RB, POD, ALP, EIM) Inverse problems Post-Doc project: Reduced models in computational mechanics Supervisors: Stéphane BORDAS and Lars BEEX Subjects: Simulation of hyperelastic materials with reduced order models Elisa SCHENONE RUES Seminar April, 6th 5

6 Summary PDEs numerical approximation Finite Element Methods (overview) Reduced Order Methods Reduced-Basis Proper Orthogonal Decomposition Numerical Results Elisa SCHENONE RUES Seminar April, 6th 5

7 A numerical approximation Overview on Finite Element Method (FEM) FEM space u = u(x) Elisa SCHENONE RUES Seminar April, 6th 5 5

8 A numerical approximation Overview on Finite Element Method (FEM) FEM space u i u i+ u i u = u(x) x i x i x i+ Elisa SCHENONE RUES Seminar April, 6th 5 5

9 A numerical approximation Overview on Finite Element Method (FEM) FEM space NX u i u i+ u(x) ' i u i v i (x) u = u(x) u i x i x i x i+ v i v i v i+ x i x i x i+ Elisa SCHENONE RUES Seminar April, 6th 5 5

10 A numerical approximation Overview on Finite Element Method (FEM) FEM space NX u i u i+ u(x) ' i u i v i (x) u = u(x) u i ' (x) ' (x) u(x) ' ũ ' (x) +ũ ' (x) x i x i x i+ Elisa SCHENONE RUES Seminar April, 6th 5 5

11 FEM versus ROM FEM space V N span{v,...,v N } V dim(v N )=N NX ROM space V N span{',...,' N } V N dim(v N )=N N NX u = u i v i (x), u i = hu, v i i eu = ũ i ' i (x), ũ i = hu, ' i i i i x x N =['...' N ] R N N eu = T N u x Elisa SCHENONE RUES Seminar April, 6th 5 6

12 General ROM formulation Diffusion problem K R N N Linear case r ( ru) =f, R d FEM Ku = f u R N f R N =[',...,' N ] R N N Weak formulation Z 8v V Z rurv +B.C.= fv ROM ekeu = e f ek = T K, e K R N N eu = T u, eu R N e f = T f, e f R N Elisa SCHENONE RUES Seminar April, 6th 5 7

13 Reduced-Basis Method A classical approach to Reduced Order Models Use information given by FEM solution(s) to build a suitable basis Elisa SCHENONE RUES Seminar April, 6th 5 8

14 Reduced-Basis Method A classical approach to Reduced Order Models Use information given by FEM solution(s) to build a suitable basis.solve with FEM the (linear) t u = F (n) x, ), in [,T] R d R + Elisa SCHENONE RUES Seminar April, 6th 5 8

15 Reduced-Basis Method A classical approach to Reduced Order Models Use information given by FEM solution(s) to build a suitable basis.solve with FEM the (linear) t u = F (n) x, ), in [,T] R d R +.Collect snapshots of FEM solution(s) in a matrix S =(u,...,u p ) R N p u i = 6 u(x,t i ) R N, 8i =,...,N u(x N,t i ) Elisa SCHENONE RUES Seminar April, 6th 5 8

16 Reduced-Basis Method A classical approach to Reduced Order Models Use information given by FEM solution(s) to build a suitable basis.solve with FEM the (linear) t u = F (n) x, ), in [,T] R d R +.Collect snapshots of FEM solution(s) in a matrix S =(u,...,u p ) R N p u i = 6 u(x,t i ) R N, 8i =,...,N u(x N,t i ).Orthonormalization of the matrix = orthonorm(s) Can reproduce events described by the FEM solution(s) Elisa SCHENONE RUES Seminar April, 6th 5 8

17 RB application Diffusion problem.solve with FEM the (linear) problem(s) for different f u = f, =(, ) u snapshots of FEM solution(s).orthonormalization of the matrix =[,..., ]. 6. x - Elisa SCHENONE RUES Seminar April, 6th 5 9

18 RB application Diffusion problem RB basis functions Elisa SCHENONE RUES Seminar April, 6th 5

19 RB application Diffusion problem FEM RB (N=) ERROR. 6. x x x -7 NB the solution is in the basis Elisa SCHENONE RUES Seminar April, 6th 5

20 RB application Diffusion problem FEM RB (N=) ERROR. 6. x -.. x -.. NB the solution is NOT in the basis Elisa SCHENONE RUES Seminar April, 6th 5

21 RB application Diffusion problem FEM snapshots 5 sources. 6. x - Elisa SCHENONE RUES Seminar April, 6th 5

22 RB application Diffusion problem FEM RB (N=5) ERROR. 6. x x -..5 NB the solution is NOT in the basis Elisa SCHENONE RUES Seminar April, 6th 5

23 Proper Orthogonal Decomposition (POD) Another classical approach to Reduced Order Models Use information given by FEM solution(s) to build a suitable basis Elisa SCHENONE RUES Seminar April, 6th 5 5

24 Proper Orthogonal Decomposition (POD) Another classical approach to Reduced Order Models Use information given by FEM solution(s) to build a suitable basis.solve with FEM the t u = F (n) x, ), in [,T] R d R + Elisa SCHENONE RUES Seminar April, 6th 5 5

25 Proper Orthogonal Decomposition (POD) Another classical approach to Reduced Order Models Use information given by FEM solution(s) to build a suitable basis.solve with FEM the t u = F (n) x, ), in [,T] R d R +.Collect snapshots of FEM solution(s) in a matrix S =(u,...,u p ) R N p u i = 6 u(x,t i ) R N, 8i =,...,N u(x N,t i ) Elisa SCHENONE RUES Seminar April, 6th 5 5

26 Proper Orthogonal Decomposition (POD) Another classical approach to Reduced Order Models Use information given by FEM solution(s) to build a suitable basis.solve with FEM the t u = F (n) x, ), in [,T] R d R +.Collect snapshots of FEM solution(s) in a matrix S =(u,...,u p ) R N p u i = 6 u(x,t i ) R N, 8i =,...,N u(x N,t i ).Compute Singular Value Decomposition (SVD) of the matrix S = T, =diag( i ).Keep first eigenvectors as ROM basis functions Can reproduce events described by the FEM solution(s) Elisa SCHENONE RUES Seminar April, 6th 5 5

27 RB application Diffusion problem.solve with FEM the (linear) problem(s) for different f u = f, =(, ) u snapshots of FEM solution(s).svd of the matrix S = T, =diag( i ).Keep first eigenvectors =[,..., ]. 6. x - Elisa SCHENONE RUES Seminar April, 6th 5 6

28 POD application Diffusion problem POD basis functions -.. Elisa SCHENONE RUES Seminar April, 6th 5 7

29 POD application Diffusion problem FEM POD (N=) ERROR. 6. x x x -7 NB the solution is in the basis Elisa SCHENONE RUES Seminar April, 6th 5 8

30 POD application Diffusion problem FEM POD (N=) ERROR. 6. x -.. x -.. NB the solution is NOT in the basis Elisa SCHENONE RUES Seminar April, 6th 5 9

31 POD application Diffusion problem FEM snapshots 5 sources. 6. x - Elisa SCHENONE RUES Seminar April, 6th 5

32 POD application Diffusion problem POD basis functions. Elisa SCHENONE RUES Seminar.5 April, 6th 5

33 POD application Diffusion problem FEM POD (N=) ERROR. 6. x x -.. NB the solution is NOT in the basis Elisa SCHENONE RUES Seminar April, 6th 5

34 POD application Diffusion problem FEM POD (N=5) ERROR. 6. x x -.. NB the solution is NOT in the basis Elisa SCHENONE RUES Seminar April, 6th 5

35 POD application Diffusion problem FEM POD (N=) ERROR. 6. x x -.. NB the solution is NOT in the basis Elisa SCHENONE RUES Seminar April, 6th 5

36 POD application Diffusion problem FEM POD (N=5) ERROR. 6. x x -.. NB the solution is NOT in the basis Elisa SCHENONE RUES Seminar April, 6th 5 5

37 POD application Diffusion problem EIGENVALUES ERROR NX N T X i i ku FEM u POD k ku FEM k Elisa SCHENONE RUES Seminar April, 6th 5 6

38 POD application in Cardiac Electrophysiology Simulation of a myocardial infarction Infarcted tissue: damaged area which cannot be activated to build the POD basis, we use a FEM solution with NO infarction [Boulakia, Schenone, Gerbeau - ] Elisa SCHENONE RUES Seminar April, 6th 5 7

39 POD application in Cardiac Electrophysiology Simulation of a myocardial infarction Infarcted tissue: damaged area which cannot be activated to build the POD basis, we use a FEM solution with NO infarction [Boulakia, Schenone, Gerbeau - ] FEM (79,57 basis) POD ( basis) Elisa SCHENONE RUES Seminar April, 6th 5 7

40 POD application in Cardiac Electrophysiology Simulation of a myocardial infarction I avr V V Infarcted tissue: damaged area which cannot be activated to build the POD basis, we use a FEM solution with II avl V V5 III avf V V6 FEM [Boulakia, Schenone, Gerbeau - ] FEM (79,57 basis) POD ( basis) Elisa SCHENONE RUES Seminar April, 6th 5 7

41 POD application in Cardiac Electrophysiology Simulation of a myocardial infarction An efficient POD to simulate an infarction in any point of the heart to build the POD basis, we use many FEM solutions with infarction [Boulakia, Schenone, Gerbeau - ] Elisa SCHENONE RUES Seminar April, 6th 5 8

42 POD application in Cardiac Electrophysiology Simulation of a myocardial infarction An efficient POD to simulate an infarction in any point of the heart to build the POD basis, we use many FEM solutions with infarction healthy FEM solution infarct FEM solution infarct FEM solution infarct m FEM solution S h = u h u h... u N T h S I S I S Im = u I u I... u N T I = u u... u N T I I I R N N T compute the SVD on an enlarged matrix hs i S = h S S... S R N (m+)n T I I Im R N N T R N N T = u I m u I m... u N T I m R N N T [Boulakia, Schenone, Gerbeau - ] Elisa SCHENONE RUES Seminar April, 6th 5 8

43 POD application in Cardiac Electrophysiology Simulation of a myocardial infarction An efficient POD to simulate an infarction in any point of the heart to build the POD basis, we use many FEM solutions with infarction [Boulakia, Schenone, Gerbeau - ] FEM (79,57 basis) POD ( basis) Elisa SCHENONE RUES Seminar April, 6th 5 8

44 POD application in Cardiac Electrophysiology Simulation of a myocardial infarction I avr V V An efficient POD to simulate an infarction in any point of the heart to build the POD basis, we use II III avl avf V V V5 V6 [Boulakia, Schenone, Gerbeau - ] FEM (79,57 basis) POD ( basis) Elisa SCHENONE RUES Seminar April, 6th 5 8

45 Non-linear problems Example r ( (u)ru) =f, R d ZWeak formulation Z n (u)rurv +B.C.= K(u)u fv, 8v V possible solution Empirical Interpolation Methods Ingredients: a reduced- basis approximation space, e.g. RB or POD an interpolation procedure tho choose some points where the PDE is solved in its strong form Elisa SCHENONE RUES Seminar April, 6th 5 9

46 References Reduced Basis Y. Maday, A. T. Patera, D. V. Rovas. A blackbox reduced-basis output bound method for noncoercive linear problems. Studies in Math. and its Appl., :5 569, Y. Maday, E.M. Rønquist. A reduced-basis element method. J. of scient. comp., 7():7 59, Proper Orthogonal Decomposition L. Sirovich. Low dimensional description of complicated phenomena. Contemp. Math., 99:77 5, 989 L. Lumley. The structure of inhomogeneous turbulence. Atmospheric turbulence and radio wave propagation, pp , 967 Empirical Intermpolation Method M. Barrault, Y. Maday, N. C. Nguyen, A. T. Patera. An empirical interpolation method: Application to efficient reduced-basis discretization of partial differential equations. C. R. Math. Acad. Sci. Paris, 9:667 67, S. Chaturantabut, D. Sorensen. Nonlinear model reduction via discrete empirical interpolation. SIAM J. of Scient. Comp., (5):77 76, Elisa SCHENONE RUES Seminar April, 6th 5