Adaptive Coarse Spaces and Multiple Search Directions: Tools for - PowerPoint PPT Presentation

Adaptive Coarse Spaces and Multiple Search Directions: Tools for Robust Domain Decomposition Algorithms Nicole Spillane Center for Mathematical Modelling at the Universidad de Chile in Santiago. July 9th, 2015 – DD23 Jeju Island, Korea

Acknowledgements Collaborators ◮ Pierre Gosselet (CNRS, ENS Cachan) ◮ Frédéric Nataf (CNRS, Université Pierre et Marie Curie) ◮ Daniel J. Rixen (University of Munich) ◮ François-Xavier Roux (ONERA, Université Pierre et Marie Curie) Funding Financial support by CONICYT through project Fondecyt 3150090. Nicole Spillane (U. de Chile) 2 / 33

Balancing Domain Decomposition (BDD): Ku ∗ = f ( K spd) BDD reduces the problem to the interface Γ : Au ∗ , Γ = b , where A := K ΓΓ − K Γ I K − 1 II K I Γ , Ω s Γ s Ω Γ b := f Γ − K Γ I K − 1 II f I . The operator A is a sum of local contributions : N R s ⊤ S s R s , S s := K s I s I s ) − 1 K s Γ s Γ s − K s Γ s I s ( K s I s Γ s , � A = R s : Γ → Γ s . s = 1 The preconditioner H also: J. Mandel. Balancing domain decomposition. N N Comm. Numer. Methods � � R s ⊤ D s R s = I . Engrg. , 9(3):233–241, 1993. R s ⊤ D s S s † D s R s , with H := s = 1 s = 1 N � R s ⊤ D s Ker ( S s ) . The coarse space is range ( U ) := s = 1 Nicole Spillane (U. de Chile) 3 / 33

Illustration of the Problem: Heterogeneous Elasticity N = 81 subdomains, ν = 0 . 4, E 1 = 10 7 and E 2 = 10 12 Young’s modulus E 10 0 10 − 1 Error (log scale) 10 − 2 10 − 3 10 − 4 Metis Partition 10 − 5 10 − 6 0 20 40 60 80 100 120 140 Iterations Problem: We want to design new DD methods with three objectives: ◮ Reliability : robustness and scalability. ◮ Efficiency : adapt automatically to difficulty. ◮ Simplicity : non invasive implementation. Nicole Spillane (U. de Chile) 4 / 33

Illustration of the Problem: Heterogeneous Elasticity N = 81 subdomains, ν = 0 . 4, E 1 = 10 7 and E 2 = 10 12 Young’s modulus E 10 0 10 − 1 Error (log scale) 10 − 2 Global 10 − 3 Local 10 − 4 Simultaneous PPCG 10 − 5 Metis Partition GenEO 10 − 6 10 − 7 0 20 40 60 80 100 120 140 Iterations Problem: We want to design new DD methods with three objectives: ◮ Reliability : robustness and scalability. ◮ Efficiency : adapt automatically to difficulty. ◮ Simplicity : non invasive implementation. Nicole Spillane (U. de Chile) 4 / 33

Contents Adaptive Coarse Spaces (GenEO) Multi Preconditioned CG (Simultaneous BDD) Adaptive Multi Preconditioned CG Nicole Spillane (U. de Chile) 5 / 33

Adaptive Coarse Spaces (GenEO) Multi Preconditioned CG (Simultaneous BDD) Adaptive Multi Preconditioned CG Projected PCG [Nicolaides, 1987 – Dostál, 1988] for Ax ∗ = b preconditioned by H and projection Π ◮ Assume that A , H ∈ R n × n are spd and U ∈ R n × n 0 is full rank, ◮ Define Π := I − U ( U ⊤ AU ) − 1 U ⊤ A . Convergence 1 x 0 = U ( U ⊤ AU ) − 1 U ⊤ b ; � Initial Guess 2 r 0 = b − Ax 0 ; � Initial residual [Kaniel, 66 – Meinardus, 63] 3 z 0 = Hr 0 ; 4 p 0 = Π z 0 ; � Initial search direction � √ κ − 1 � i 5 for i = 0 , 1 , . . . , convergence do � x ∗ − x i � A � 2 √ κ + 1 q i = Ap i ; 6 � x ∗ − x 0 � A α i = ( q ⊤ i p i ) − 1 ( p ⊤ i r i ) ; 7 x i + 1 = x i + α i p i ; � Update approximate solution ◮ κ = λ max 8 , r i + 1 = r i − α i q i ; � Update residual 9 λ min z i + 1 = Hr i + 1 ; � Precondition 10 i p i ) − 1 ( q ⊤ β i = ( q ⊤ ◮ λ max and λ min : i z i + 1 ) ; 11 p i + 1 = Π z i + 1 − β i p i ; � Project and orthogonalize extreme eigenvalues 12 13 end of HA Π excluding 0. 14 Return x i + 1 ; → GenEO is a choice of range ( U ) that guarantees fast convergence. Nicole Spillane (U. de Chile) 6 / 33

Adaptive Coarse Spaces (GenEO) Multi Preconditioned CG (Simultaneous BDD) Adaptive Multi Preconditioned CG Bibliography (1/2) (coarse spaces based on generalized eigenvalue problems) Multigrid M. Brezina, C. Heberton, J. Mandel, and P. Vaněk. Technical Report 140, University of Colorado Denver , April 1999. Additive Schwarz J. Galvis and Y. Efendiev. Multiscale Model. Simul. , 2010. Y. Efendiev, J. Galvis, R. Lazarov, and J. Willems. ESAIM Math. Model. Numer. Anal. , 2012. F. Nataf, H. Xiang, V. Dolean, and N. S. SIAM J. Sci. Comput. , 2011. V. Dolean, F. Nataf, R. Scheichl, and N. S. Comput. Methods Appl. Math. , 2012. N. S., V. Dolean, P. Hauret, F. Nataf, C. Pechstein, and R. Scheichl. Numer. Math. , 2014. Optimized Schwarz Methods S. Loisel, H. Nguyen, and R. Scheichl. Technical report, submited, 2015. R. Haferssas, P. Jolivet, and F. Nataf. Technical report, Submited, hal-01100926, 2015. Nicole Spillane (U. de Chile) 7 / 33

Adaptive Coarse Spaces (GenEO) Multi Preconditioned CG (Simultaneous BDD) Adaptive Multi Preconditioned CG Bibliography (2/2) (coarse spaces based on generalized eigenvalue problems) Substructuring Methods P.E. Bjørstad, J. Koster and P. Krzyżanowski. Applied Parallel Computing , 2001. J. Mandel and B. Sousedík. Comput. Methods Appl. Mech. Engrg. , 2007. B. Sousedík, J. Šístek, and J. Mandel. Computing , 2013 N. S. and D. J. Rixen. Int. J. Numer. Meth. Engng. , 2013. H. H. Kim and E. T. Chung, Model. Simul., 2015. A. Klawonn, P. Radtke, and O. Rheinbach. DD22 Proceedings , 2014. A. Klawonn, P. Radtke, and O. Rheinbach. SIAM J. Numer. Anal. , 2015. And many other talks at this conference: ◮ Minisymposium on monday: Olof B. Widlund, Clark R. Dohrmann (with Clemens Pechstein), ◮ Pierre Jolivet (HPDDM https://github.com/hpddm ), ◮ Frédéric Nataf’s plenary talk tomorrow ! ◮ Session on friday morning (CT 7) ! Nicole Spillane (U. de Chile) 8 / 33

Adaptive Coarse Spaces (GenEO) Multi Preconditioned CG (Simultaneous BDD) Adaptive Multi Preconditioned CG Adaptive Coarse Space: Strategy (1/5) Relative error bounded w.r.t C / c Prove ( 2 ) Prove ( 1 ) Problem dependant Use trace theorems, Poincaré in- equalities, Korn inequalities... ( 1 ) holds ( 2 ) holds DD method dependant FETI Additive BDD Schwarz ( 1 ) λ min > c if stable decomposition Abstract Framework ( 2 ) λ max < C if local solver is stable ([Toselli, Widlund (2005)]) [N. S., D. J. Rixen, 2013] [ N. S., V. Dolean, P. Hauret, F. Nataf, C. Pechstein, and R. Scheichl, 2014] Nicole Spillane (U. de Chile) 9 / 33

Adaptive Coarse Spaces (GenEO) Multi Preconditioned CG (Simultaneous BDD) Adaptive Multi Preconditioned CG Adaptive Coarse Space: Strategy (2/5) Relative error bounded w.r.t C / c Prove ( 2 ) Prove ( 1 ) Problem dependant Use trace theorems, Poincaré in- equalities, Korn inequalities... ( 1 ) holds ( 2 ) holds DD method dependant FETI Additive BDD Schwarz ( 1 ) λ min > c if stable decomposition Abstract Framework ( 2 ) λ max < C if local solver is stable ([Toselli, Widlund (2005)]) [N. S., D. J. Rixen, 2013] [ N. S., V. Dolean, P. Hauret, F. Nataf, C. Pechstein, and R. Scheichl, 2014] Nicole Spillane (U. de Chile) 10 / 33

Adaptive Coarse Spaces (GenEO) Multi Preconditioned CG (Simultaneous BDD) Adaptive Multi Preconditioned CG Adaptive Coarse Space: Strategy (3/5) Relative error bounded w.r.t C / c Prove ( 2 ) Prove ( 1 ) Make ( 1 ) or ( 2 ) local: In x s ⊤ M s x s � τ x s ⊤ N s x s for all x s . � Problem dependant Use trace theorems, Poincaré in- equalities, Korn inequalities... (Botleneck Estimate) ( 1 ) holds ( 2 ) holds DD method dependant FETI Additive BDD Schwarz ( 1 ) λ min > c if stable decomposition Abstract Framework ( 2 ) λ max < C if local solver is stable ([Toselli, Widlund (2005)]) [N. S., D. J. Rixen, 2013] [ N. S., V. Dolean, P. Hauret, F. Nataf, C. Pechstein, and R. Scheichl, 2014] Nicole Spillane (U. de Chile) 11 / 33

Adaptive Coarse Spaces (GenEO) Multi Preconditioned CG (Simultaneous BDD) Adaptive Multi Preconditioned CG Adaptive Coarse Space: Strategy (4/5) Relative error bounded w.r.t C / c Solve M s x s k N s x s k = λ s k . Prove ( 2 ) Prove ( 1 ) In � botleneck estimate x s ⊤ M s x s � τ x s ⊤ N s x s � Problem dependant ◮ holds in span { x s Use trace theorems, Poincaré in- k ; λ s k � τ } , equalities, Korn inequalities... ◮ does not hold in span { x s k ; λ s k > τ } . ( 1 ) holds ( 2 ) holds DD method dependant FETI Additive BDD Schwarz ( 1 ) λ min > c if stable decomposition Abstract Framework ( 2 ) λ max < C if local solver is stable ([Toselli, Widlund (2005)]) [N. S., D. J. Rixen, 2013] [ N. S., V. Dolean, P. Hauret, F. Nataf, C. Pechstein, and R. Scheichl, 2014] Nicole Spillane (U. de Chile) 12 / 33