Nonlinear Multigrid and Domain Decomposition Methods

Transcription

1 Nonlinear Multigrid and Domain Decomposition Methods Rolf Krause Institute of Computational Science Universit`a della Svizzera italiana 1 Stepping Stone Symposium, Geneva, 2017

2 Non-linear problems Non-linear problem H, W Banach spaces, F : D H W, D open, F C 1 (D). Find u D such that F (u ) = 0 Construct sequence if iterates x k X, in D, k > 0, via u k+1 = u k + α k c k with u k u. u k correction or step, α k > 0 steplength special case F = J solution may not be unique Efficiency and global convergence? R. Krause (Università della Svizzera italiana) Nonlinear Domain Decomposition Methods 2

3 Non-linear problems Newton s method x k X, Newton s method replaces F by the linear model F (u k + c k ) F (u k ) + F (u k )c k = 0 leading to the the Newton correction c k = F (u k ) 1 F (u k ) and the Newton update c k+1 = c k + α k c k, (1) α k 0 damping or line-search parameter for the Newton correction. F (u k ) Fréchet derivative / Jacobian. Here: H = R n Invariance under affine transformations [Ortega 70, Deuflhard, Heindl 79, Deuflhard..., 11] First linearize, then solve R. Krause (Università della Svizzera italiana) Nonlinear Domain Decomposition Methods 3

4 Non-linear problems (Inexact) Newton Method 1: procedure (I)N(F, u 0 D, TOL) Non-linearity, start value, and tolerance 2: k = 0 3: while F (u k ) > TOL or u k+1 u > TOL do 4: Solve (approximately) F (u k )c k = F (u k ) 5: Determine α k Damping / Line search 6: u k+1 = u k + α k c k Update 7: k k + 1 8: end while 9: return u k solution found 10: end procedure the direction of the correction c k is given by the (approximate) solution of F (u k )c k = F (u k ) Close to u, α k =1 and exact solution ensures quadratic convergence stopping criterion: residual based or error based use multigrid/domain decompositionfor solving For strong non-linearities, α k might deteriorate R. Krause (Università della Svizzera italiana) Nonlinear Domain Decomposition Methods 4

5 Non-linear problems Properties of the Newton Direction [Deuflhard 11] General level set function T (u A) = 1 AF 2 (u) 2 2, A regular: T (u A) = (AF (u)) T (AF (u)) The natural choice A = F (u) 1 leads to T (u F (u) 1 ) = F (u) 1 F (u) For J(u) = 1 (Au, u) (f, u) we get 2 F (u) 1 F (u) = 2 J(u) J(u) = A 1 (f Au) = A 1 (Au Au) The Newton correction is direction of steepest descent for T (u F (u) 1 ) Damping strategy for the exact Newton method can be derived using T ( A) J convex and quadratic: Newton step leads to minimizer u = u + A 1 (Au Au) Isolines of J and T ( I ) will form our energy-landscape R. Krause (Università della Svizzera italiana) Nonlinear Domain Decomposition Methods 5

6 Non-linear problems Parallel solution of non-linear problems handling the nonlinearity Newton first linearize, then decompose (multigrid, DD as inner solver) nonlinear DD first decompose, then linearize - choice of sub-space/sub-domain model global communication - choice of coarse space convergence control intermediate approach: inexact solution after linearization constraints, non-smooth energies,... R. Krause (Università della Svizzera italiana) Nonlinear Domain Decomposition Methods 6

7 Additive and Multiplicative Trust-Region Methods Nonlinear Domain Decomposition Scheme R. Krause (Università della Svizzera italiana) Nonlinear Domain Decomposition Methods 14

8 Trust-Region Methods NDM - Additive non-linear decomposition 1: procedure NDM((V p, I p, P p, F p) p=1,...,p, x 0 2 X,TOL). Decomposition, start value, and tolerance 2: k =0 3: while kf (x k )k > TOL do 4: for p 1, P do 5: Solve (approximately) F p(p px k + pc p)=0 6: end for 7: x k+1 = x k + P P p=1 Ip pcp 8: k 7! k +1. Update 9: end while 10: return x k. solution found 11: end procedure R. Krause (Università della Svizzera italiana) Nonlinear Domain Decomposition Methods 13

9 Non-linear problems Newton Path F (ū(λ)) = (1 λ)f (u 0 ), T (ū(λ) A) = (1 λ) 2 T (u 0 A), dū dλ = F (ū) 1 F (u 0 ), x(0) = x 0, ū(1) = u, dū = F (u 0 ) 1 F (u 0 ) c 0, dλ λ=0 connects start value and nearest solution, or collapses (F singular), or ends at D[Davidenko 53, Deuflhard 72, 11] Newton correction (in the first step) ist tangent to the Newton path R. Krause (Università della Svizzera italiana) Nonlinear Domain Decomposition Methods 8

10 Non-linear problems Newton-Path: DD Examples F 1(x 1, x 2) = (x 1 x ) 3 x 3 2, F 2(x 1, x 2) = x 1 +2x 2 3. The exact solution is u =[1, 1] T. INB: Inexact Newton with backtracking ASPIN: standard ASPIN Exact-ASPIN: ASPIN with analytical Jacobian for the preconditioned system 2 Initial guess: x0=(2,2) 2 Initial guess: x0=(0,2) Contours of F(x) Newton soln 0.4 Newton path for INB ASPIN soln Newton path for ASPIN 0.2 Exact-ASPIN soln Newton path for exact-aspin Contours of F(x) Newton soln 0.4 Newton path for INB ASPIN soln Newton path for ASPIN 0.2 Exact-ASPIN soln Newton path for exact-aspin R. Krause (Università della Svizzera italiana) Nonlinear Domain Decomposition Methods 9

11 State of the Art Globalization Strategies Trust Region Methods Globalization Strategy: Trust Region Method Newton Step: Solve s 2 R n : (s) = 1 hs, Bsi + hrj(u), si =min! 2 subject to ksk 1 apple, u + s 2B B symmetric approximation to the Hessian (Quasi Newton Method) Quadratic approximation to a nonlinear function Acceptance Criterion = J(u+s) J(u) (s) then u = u + s Update of Trust Region bymeansof Theorem: If (s) = min! is solved su ciently well + Compactness of Levelsets ) convergence to first-/second-order critical points. C. Groß, R. Krause Nonlinearly Preconditioned Globalization Strategies 5

12 Nonlinear Additive Preconditioning Non linear Additive Preconditioning Algorithms Idea: Instead of computing search directions s with a parallel solver, solve local nonlinear minimization problems employing globalization strategies After the parallel phase, the corrections must be combined to a global search direction Similar Approaches: Parallel Variable Distribution [Ferris,Mangasarian 94] : asynchronous solution of local minimization problems. But: Unduly expansive computation of damping parameters ASPIN method [Cai,Keyes 02] : asynchronous solution of local minimization problems + global post smoothing. But: Formulation based on first-order conditions C. Groß, R. Krause Nonlinearly Preconditioned Globalization Strategies 8

13 Nonlinear Additive Preconditioning Concept: Non linear Additive Preconditioning Algorithms [G Krause 2009] 1 Decompose R n such that R n = S k D k where D k R n. 2 Compute several s k 2 D k by means of H k (P k u G + s k ) < H k (P k u G ) where H k is a subset objective function, u G is the current global iterate and P k : R n! D k. 3 Combine the subset corrections s k as follows u G = u G + X k I k s k where I k : D k! R n. 4 Post smoothing employing a globalization strategy C. Groß, R. Krause Nonlinearly Preconditioned Globalization Strategies 9

14 Nonlinear Additive Preconditioning Local Objective Function Non linear local functions [G Krause 2009] H k (u) =J k (u)+hr k rj(u G ) rj k (P k u G ), u P k u G i where u G is the current global iterate J k is a arbitrary local objective function, givenapriori R k =(I k ) T Properties of the coupling term hr k rj(u G ) rj k (P k u G ), u P k u G i: first local Newton correction solves in direction of the restricted current global gradient! (cf., [Nash 00, Gratton et al. 06] ) C. Groß, R. Krause Nonlinearly Preconditioned Globalization Strategies 10

15 A Non-linear Additively Preconditioned Trust-Region Method Non linear Additively Preconditioned Trust-Region Strategy (APTS) [G Krause 2009] 1 Parallel Computation: Solve s k 2 D k : H k (P k u G + s k ) < H k (P k u G ), w.r.t. ki k s k k 1 apple G employing a Trust Region Method. G : global Trust-Region radius 2 Combination: if P k I ks k is good enough: u G = u G + X k I k s k 3 Smoothing compute some global Trust-Region steps and goto (1) C. Groß, R. Krause Nonlinearly Preconditioned Globalization Strategies 11

16 A Non-linear Additively Preconditioned Trust-Region Method Ensuring Convergence [G Krause 2009] Measuring the quality of the additively computed correction where P k s k = P k I ks k Acceptance if Increase G if J(u G ) J(u G + P k = I ks k ) P k (Hk (P k u G ) H k (P k u G + s k )) (otherwise P k s k is disposed!), otherwise decrease G C. Groß, R. Krause Nonlinearly Preconditioned Globalization Strategies 12

17 A Non-linear Additively Preconditioned Trust-Region Method Convergence to First order Critical Points Theorem If the respective objective functions are su ciently smooth and a local minimizer exists, then the APTS method to computes a sequence converging to a first order critical point. Moreover, for any domain decomposition, the APTS algorithm computes a first order critical point without global smoothing. C. Groß, R. Krause Nonlinearly Preconditioned Globalization Strategies 13

18 Examples Unconstrained Example - APTS Energy optimal deformation of the geometry first order su ciency conditions (krj(u)k 2) after each iteration, comparison APTS vs normal Trust-Region (4 preconditioning iterations, 4 global smoothing iterations) 140,000 unknowns 7 processors C. Groß, R. Krause Nonlinearly Preconditioned Globalization Strategies 17

19 Examples Constrained Example APTS Energy optimal deformation of the geometry first order su ciency conditions (kd(u)rj(u)k 2) after each iteration, comparison APTS vs normal Trust-Region (4 preconditioning iterations, 4 global smoothing iterations) 900,000 unknowns 7 processors C. Groß, R. Krause Nonlinearly Preconditioned Globalization Strategies 18

20 Examples Constrained Example APLS Energy optimal deformation of the geometry first order su ciency conditions (kd(u)rj(u)k 2) after each iteration, comparison APLS vs normal Linesearch (4 preconditioning iterations, 4 global smoothing iterations) 900,000 unknowns 8 processors C. Groß, R. Krause Nonlinearly Preconditioned Globalization Strategies 20

21 Examples Unconstrained Example APLS Energy optimal deformation of the geometry first order su ciency conditions (krj(u)k 2) after each iteration, comparison APLS vs normal Linesearch (4 preconditioning iterations, 4 global smoothing iterations) 140,000 unknowns 8 processors C. Groß, R. Krause Nonlinearly Preconditioned Globalization Strategies 19

22 Additive and Multiplicative Trust-Region Methods Nonlinear Domain Decomposition Scheme R. Krause (Università della Svizzera italiana) Nonlinear Domain Decomposition Methods 18

23 Non-linear problems Non-linear Multigrid 1: procedure FAS (two level)(f h, F H, uh, 0 IH, h Ph H, V h, V H,TOL) Non-linearity, start value, and tolerance 2: k = 0 3: while F h (uh k ) > TOL or u k+1 h u h > TOL do 4: Solve (approximately) F h (uh k + c h ) = 0 in V h Fine Grid Solve 5: ū h = uh k + c h 6: uh 0 = Ph H (ū h ) 7: Solve (approximately) F H (uh 0 + c H ) = F (uh) 0 (IH) h T F h (ū h ) Coarse solve 8: u k+1 h = ūh k + IHc h H Update 9: k k : end while 11: return u k solution found 12: end procedure local convergence [Reusken 87] and investigation [Brabazona, Hubbard, Jimack 14] global convergence - gradient operators (damping) [Hackbusch, Reusken 89] global convergence - non-convex minimization (TR) [Gratton et al. 08; Groß, K 09] Linearize in function space [Deuflhard, Weiser 97, 98;], nested iteration [Bank, Rose 82] F = J: MG/OPT- multilevel optimization (MG/OPT)[Nash 99,... ] F = J+convex: monotone multigrid [Kornhuber 94; Kornhuber, K 01,... ] R. Krause (Università della Svizzera italiana) Nonlinear Domain Decomposition Methods 7

24 On the Projection Operator Projection vs. Restriction [G K 09] Note For our examples from nonlinear mechanics: restriction almost never yields initial coarse level iterates for which the objective function satisfies the standard Linesearch assumptions. R. Krause (ICS Lugano) On the Application of Nonlinear Preconditioning Strategies on Parallel Systems 15

25 On the Projection Operator Projection vs. Restriction [G K 09] The influence of the coarse level correction in nonlinear multigrids or domain decomposition methods R. Krause (ICS Lugano) On the Application of Nonlinear Preconditioning Strategies on Parallel Systems 14

26 Additive and Multiplicative Trust-Region Methods RMTR strategy [Gratton et al. 2008; Gratton et al. 2009; Groß,K 2009] The RMTR method 1 compute m 1 pre smoothing trust region steps to approximately solve H k (u k ) < H k (P k+1 u k+1 ) w.r.t u k 2B k, ku k kapple k 2 if (k is not coarsest level) Compute B k 1,andH k 1, u k 1,0 = P k u k,m1 call RMTR on level k 1andreceiveacorrections k 1 8 < u k,m1 + I k 1 s k 1 if M = H k (u k,m1 ) H k (u k,m1 +I k 1 s k 1 ) u k,m1 +1 = : u k,m1 otherwise Update trust-region k,m 1 +1 H k 1 (P k u k,m1 ) H k 1 (P k u k,m1 +s k 1 ) 3 compute m 2 post smoothing trust region steps to approximately solve H k (u k ) < H(u k,m1 +1) w.r.t u k 2B k, ku k kapple k 4 return final iterate R. Krause (Università della Svizzera italiana) Nonlinear Domain Decomposition Methods 19

27 Examples Ogden Materials Propagation of correction, Standard Approach Z J (u) = dtr(e)+ (tr(e)) 2 +(µ d)tr(e 2 )+d (det(i + ru))dx 2 E = 1 2 (ru + rut + ru T ru), (v) = ln(v), d > 0

28 Examples Ogden Materials Propagation of Correction, Multi Level Approach stresses and deformations after each V-Zyklus Multi level trust region Methode

29 Additive and Multiplicative Trust-Region Methods MPTS [Groß, K 2009] MPTS: a generalization of RMTR Almost arbitrary domain decomposition methods possible: Multigrid methods Alternating domain decomposition methods and nonlinear Jacobi methods Convergence to first-order critical points Theorem: If the search directions/corrections are chosen su ciently well, the norm of the gradients and of B are either bounded on a compact set, then MPTS is globally convergent. Even more: global convergence can be guaranteed without global smoothing, ifan (overlapping) domain decomposition is employed. R. Krause (Università della Svizzera italiana) Nonlinear Domain Decomposition Methods 20

30 Examples Convergence and Control Efficiency: Multilevel ansatz significantly increases convergence speed Highly non linear boundary value problem Norm of gradient vs. # iterations, different approaches Solution strategy: Projected cg + non linear Gauß-Seidel for quadratic problems ( fixed number of iterations for each ) Reassembling of the Hessian after every 4 steps adaptive strategies possible

31 Numerical Examples - APTS/MPTS Cylinder Contact Problem - Performance of Trust-Region Methods Energy optimal displacements First-order su ciency conditions krj(u)k 2 after each Trust-Region step; Comparison between seq. Trust-Region, APTS, MPTS, combinedapts/mpts = AMPTS (F b= 4localTrust-RegionstepsoneachD k,4globaltrust-regionstepsinorderto compute s) R. Krause (Università della Svizzera italiana) Nonlinear Domain Decomposition Methods 23

32 Numerical Examples Unconstrained Minimization Problem- Linesearch method energy optimal displacements First-order su ciency conditions krj(u)k 2 after each cycle; Comparison between seq. Linesearch, APLS, MPLS, combined APLS/MPLS = AMPLS (F b= 4 local Linesearch steps on each D k, 4 global Linesearch steps in order to compute s) 330,999 unknowns 8 processors C. Groß, R. Krause (University of Lugano) On the Application of Nonlinear Preconditioning Strategies 29

33 Numerical Results - GASPIN Comparisons Evolution of the objective function J(u i )andthenormofthegradientkg i k for globalized Aspin employing di erent numbers of processors 240 cores 480 cores 960 cores 1920 cores Overall Time Solver global TR problem Solver local QP Problem Assembling Nonlinear Iterations Computation time in seconds. R. Krause (Università della Svizzera italiana) Nonlinear Domain Decomposition Methods 34