Scalable Nonlinear Domain Decomposition Methods Martin Lanser Mathematical Institute, University of Cologne Based on joint work with Axel Klawonn (University of Cologne) Oliver Rheinbach (TU Bergakademie Freiberg) SPPEXA Symposium 2016 Munich 01/25/2016 - 01/27/2016
M. Lanser, A. Klawonn, O. Rheinbach Scalable Nonlinear Domain Decomposition Methods EXASTEEL - Bridging Scales for Multiphase Steels Principal Investigators • A. Klawonn, U Cologne • O. Rheinbach, TU Freiberg • J. Schr¨ oder, U Duisburg-Essen • D. Balzani, TU Dresden • G. Wellein, U Nuremberg-Erlangen • O. Schenk, U Lugano • Challenging 3D multiscale problems from nonlinear structural mechanics with plasticity. • Highly concurrent computational scale bridging in continuum mechanics (FE2) • Parallel FE2 implementation FE2TI based on PETSc and BoomerAMG • Hybrid domain decomposition/multigrid implicit solvers for nonlinear problems
M. Lanser, A. Klawonn, O. Rheinbach Scalable Nonlinear Domain Decomposition Methods Efficient Parallel Solver: FETI-DP F inite E lement T earing and I nterconnecting - D ual- P rimal Divide and Conquer Algorithm: Decompose computational domain into N nonoverlapping subdomains. FETI-DP coarse space: Strong coupling in few degrees of freedom. K (1) K (1) T � � � BB Π B . ... . . K T � K BB Π B =: . K ( N ) K ( N ) T � � � K Π B K ΠΠ BB Π B K (1) K ( N ) � � � K ΠΠ · · · Π B Π B Introduce Lagrange multipliers and enforce zero jump between subdomains: B B u B = 0 K T � B T K BB u B f B Π B B ˜ = � � u Π ˜ f Π K Π B K ΠΠ O λ 0 B B O O In compact form: � � � ˜ � � � ˜ � B T u f K = 0 B O λ
M. Lanser, A. Klawonn, O. Rheinbach Scalable Nonlinear Domain Decomposition Methods Classical FETI-DP Algorithm First reducing to the Lagrange multipliers: F λ = d F = B B K − 1 BB B T + B B K − 1 S − 1 K Π B K − 1 BB B T BB � K B Π � ΠΠ � . B B � �� � � �� � local solvers coarse problem; coupled! B B : Communication over the interface. K − 1 BB : Local direct solvers. � S − 1 ΠΠ := � K ΠΠ − � K ΠB K − 1 BB � K T ΠB : Exact solution of a global problem ⇒ scaling bottleneck The Preconditioner Preconditioner: M − 1 := B D, ∆ SB T (Sum of local operators) D, ∆ 1. S Schur complement of K (Interior variables eliminated). Local solvers. 2. B D, ∆ appropriately scaled jump operator (scaling depends on pde coeff.) FETI-DP is PCG solving M − 1 F λ = M − 1 d
M. Lanser, A. Klawonn, O. Rheinbach Scalable Nonlinear Domain Decomposition Methods Newton-Krylov FETI-DP Classical use of FETI-DP in the context of nonlinear finite element problems: For a nonlinear problem arising from a discretization of a nonlinear partial differential equation A ( u ) = 0 we linearize first with a Newton method u ( k +1) = u ( k ) − α ( k ) δu ( k ) with a step length α ( k ) , and the update δu ( k ) is given by: DA ( u ( k ) ) δu ( k ) = A ( u ( k ) ) . (1) Newton-Krylov FETI-DP is decomposing the computational domain and using a FETI-DP type method in order to solve ( 1 ). Linearize first Decomposition Elimination
M. Lanser, A. Klawonn, O. Rheinbach Scalable Nonlinear Domain Decomposition Methods Nonlinear FETI-DP Methods Linearization Elimination Decomposition first Nonlinear Elimination Linearization • Decomposition of the discretized nonlinear problem before linearization • ⇒ local nonlinear problems ⇒ Increased local work • Reduced number of Newton steps, Krylov iterations, and communication • Combinable with hybrid FETI-DP/Multigrid methods All nonlinear FETI-DP methods are based on the nonlinear FETI-DP saddlepoint system: u ) + B T λ − ˜ � K (˜ f = 0 B ˜ u = 0
M. Lanser, A. Klawonn, O. Rheinbach Scalable Nonlinear Domain Decomposition Methods Nonlinear FETI-DP Methods - Linearize First Linearization Elimination Decomposition first Nonlinear Elimination Linearization • Decomposition of the discretized nonlinear problem before linearization • ⇒ local nonlinear problems ⇒ Increased local work • Reduced number of Newton steps, Krylov iterations, and communication • Combinable with hybrid FETI-DP/Multigrid methods All nonlinear FETI-DP methods are based on the nonlinear FETI-DP saddlepoint system: u ) + B T λ − ˜ � K (˜ f = 0 B ˜ u = 0
M. Lanser, A. Klawonn, O. Rheinbach Scalable Nonlinear Domain Decomposition Methods Nonlinear FETI-DP Methods - Linearize First Based on the nonlinear master system u ) + B T λ − ˜ � K (˜ f = 0 B ˜ u = 0 the Newton linearization with respect to (˜ u, λ ) results in the linear system � � � δ ˜ � � � D � � u ) + B T λ − ˜ B T u K (˜ u ) K (˜ f = . (2) δλ B 0 B ˜ u u = ( δu T u T Π ) T : With splitting up δ ˜ B , δ ˜ D � K B + B T K T B T B λ − f B DK BB δu B Π B B = K Π − ˜ � D � D � δ ˜ u Π . f Π K Π B K ΠΠ 0 δλ B B 0 0 B ˜ u Linearized system can be solved using any FETI-DP type method. We consider hybrid FETI-DP/Multigrid variants: inexact (reduced) FETI-DP
M. Lanser, A. Klawonn, O. Rheinbach Scalable Nonlinear Domain Decomposition Methods Inexact Reduced Nonlinear FETI-DP Considering the linearized system D � K B + B T K T B T B λ − f B DK BB δu B Π B B = K Π − ˜ � D � D � δ ˜ u Π K Π B K ΠΠ 0 f Π δλ B B 0 0 B ˜ u we perform an elimination of δu B , which yields � � � δ ˜ � � − D � K Π B DK − 1 BB B T S ΠΠ u Π B = r.h.s. (3) − B B DK − 1 BB D � K T − B B DK − 1 BB B T δλ Π B B with � S ΠΠ := D � K ΠΠ − D � K Π B DK − 1 BB D � K T Π B . Exact solution of � S ΠΠ not necessary. Solution of coarse problem is moved to the preconditioner ⇒ Inexact solution possible. . See Klawonn, Lanser, Rheinbach (SISC, 2015) for details.
M. Lanser, A. Klawonn, O. Rheinbach Scalable Nonlinear Domain Decomposition Methods Inexact Reduced Nonlinear FETI-DP We solve ( 3 ) iteratively (GMRES) using the block-triangular preconditioner � � ˆ S − 1 0 B − 1 ˆ ΠΠ r,L = Π B ˆ − M − 1 B B DK − 1 BB D � K T S − 1 − M − 1 ΠΠ fi nest grid • M − 1 : one of the standard FETI-DP preconditioners restricting • ˆ S − 1 ΠΠ : some cycles of an AMG (algebraic multigrid) smoothing interpolating method, applied to � second grid S ΠΠ . • If ˆ S − 1 ΠΠ is a good preconditioner of � S ΠΠ , inexact reduced FETI-DP has convergence bounds of the same quality as classical FETI-DP. ccoarsest grid solving One V-cycle of an AMG method. See Klawonn, Rheinbach (IJNME 2007, ZAMM 2010) for details.
M. Lanser, A. Klawonn, O. Rheinbach Scalable Nonlinear Domain Decomposition Methods Inexact Nonlinear FETI-DP We solve the linearized system � � � δ ˜ � � � D � K + B T λ − ˜ � B T u K f = (4) δλ B 0 B ˜ u iteratively (GMRES) using the block-triangular preconditioner � � ˆ K − 1 0 B − 1 ˆ = − M − 1 B ˆ K − 1 − M − 1 L fi nest grid K − 1 : ˆ some cycles of an AMG (algebraic multigrid) • restricting method, applied to D � smoothing g K . n i t a second grid l o p K − 1 is a good preconditioner of D � • If ˆ r e K , inexact FETI- t n i DP has convergence bounds of the same quality as classical FETI-DP. solving ccoarsest grid One V-cycle of an AMG method.
M. Lanser, A. Klawonn, O. Rheinbach Scalable Nonlinear Domain Decomposition Methods Inexact Nonlinear FETI-DP We present two different choices for M − 1 . Standard Dirichlet preconditioner: N � M − 1 := M − 1 B ( i ) ∆ ,D S ( i ) ∆∆ B ( i ) T FETID := ∆ ,D , i =1 � � − 1 where S ( i ) ∆∆ := DK ( i ) ∆∆ − DK ( i ) DK ( i ) DK ( i ) I ∆ is the Schur complement of the tangential ∆ I II � � − 1 DK ( i ) matrix on the interface of subdomain Ω i . A sparse direct solver is used for . II Preconditioner without sparse direct solvers: M − 1 := M − 1 FETID / AMG , � � − 1 DK ( i ) in M F ET ID is replaced by some applications of sequential AMG to DK ( i ) where II . II
M. Lanser, A. Klawonn, O. Rheinbach Scalable Nonlinear Domain Decomposition Methods Implementation Remarks • Parallelization Strategy: MPI based • Nonlinear FETI-DP is written in C/C++ using PETSc, Umfpack, MUMPS, PARDISO, BoomerAMG • Efficient direct solver packages for local FETI-DP subdomain problems (Umfpack or MUMPS) • If available, the thread parallel direct solver PARDISO can also be used for all local FETI-DP subdomain problems • Parallel AMG implementation BoomerAMG is used as a preconditioner for the global FETI-DP coarse problem � S ΠΠ Nonlinear Domain Decomposition Nonlinear FETI-DP and Nonlinear BDDC: Klawonn, Lanser, Rheinbach (2012, 2013, 2014, 2015) ASPIN: Cai, Keyes 2002; Cai, Keyes, Marcinkowski 2002; Hwang, Cai 2005, 2007; Groß, Krause 2010,13; MSPIN: Keyes, Liu, 2015 Nonlinear Neumann-Neumann: Bordeu, Boucard, Gosselet 2009; Nonlinear FETI-1: Pebrel, Rey, Gosselet 2008; Other DD work reversing linearization and decomposition: Ganis, Juntunen, Pencheva, Wheeler, Yotov 2014; Ganis, Kumar, Pencheva, Wheeler, Yotov 2014
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