Nonsmooth Schur Newton Methods and Applications

Nonsmooth Schur Newton Methods and Applications C. Gräser, R. Kornhuber, and U. Sack IMA Annual Program Year Workshop Numerical Solutions of Partial Differential Equations: Fast Solution Techniques November 29 December 3, 2010 Matheon

Outline Motivation: Phase separation in alloys (Matheon C17: Kh., Sprekels, Sack) - Cahn-Larché equations (Dreyer & Müller 1999,..., Garcke 01,05,..., Weikard 01,...) Nonsmooth PDE constrained optimal control - nonsmooth saddle point problem - globally convergent nonsmooth Schur Newton methods (Gräser & Kh. 06, 09, Gräser 10) - numerical experiments: robust convergence Phase separation in alloys: Some numerical results

The Lead-Free Movement classical solder joints: SnPb alloy distribution of lead by electronic waste: 20.000.000 kg/year environmentally friendly manufacturing: lead-free solders (EU: July 1, 2006) SnAg, CuSn, SnAgCu, SnAgCuSb,...

Assessing Solder Joint Reliability in Lead-Free Assemblies cracks: and failure: main reasons: thermomechanical stress demixing of alloys (phase separation)

Experimental results: Aging of a AgCu Alloy 2 hours 5 hours 20 hours 40 hours T. Bo hme & W.H. Mu ller 2006

Cahn Larché Equations u t = ( M(u) w ) w = (γ(u) u) + Ψ (u) + W,u (u,ε) W,ε (u,ε) = 0 concentration of component α: u [ 1,1] mobility: M(u) = ω(u)m α + (1 ω(u))m β, ω(u) [0, 1] surface tension: γ(u) = ω(u)γ α + (1 ω(u))γ β nonsmooth (logarithmic) free energy: Ψ(u) = 1 2ε T( (1 + u)log(1 + u) + (1 u)log(1 u) ) 1 2ε T c(1 u 2 ) strain energy: W(u,ε) = 1 2C(u)(ε ε(u)) : (ε ε(u)) elasticity tensor: C(u), linearized strain: ε, eigenstrain ε(u)

Some References Cahn Hilliard equations: Elliott et al. (1991, 1991, 1992, 1993,...,1996, 1997...), Barrett & Blowey (1995,..., 2001),... Cahn Larché equations: modelling: Dreyer & Müller (1999, 2001,...) logarithmic free energy analysis: Garcke (2003, 2005) existence, logarithmic free energy Sprekels et al. (2002) viscous case, existence + uniqueness, logarithmic free energy numerical analysis: Garcke, Rumpf & Weikard (2001) qualitative study, quartic free energy, Newton solver Garcke & Weikard (2005) convergence of the discretization, quartic free energy Merkle (2005) quantitative numerical study, piecewise polynomial free energy, Newton solver not covered: logarithmic free energy widely open: multicomponent alloys

Discretization in time: semi implicit Euler scheme (Blowey & Elliott 92) fully implicit: second order terms semi implicit: free energy Ψ = Φ + Σ unconditionally stable Φ convex implicit Σ concave explicit frozen mobilities, surface tension, and coefficients in elasticity decouples Cahn-Hilliard equation and elasticity adaptive multigrid solvers for linear elasticity available (..., Sander,...) fast adaptive solver for the time-discrete Cahn-Hilliard equation needed! in space: piecewise finite elements S j = span {λ (j) p p N j } missing: convergence analysis ongoing work: systematic numerical assessment

Discrete Spatial Problem u k j, v + τ(m wk j (γ u k j, v) + Φ (u k j ),v wk j, v, v) = uk 1 j, v = (W,u(u k 1 j, ε k ),v) Σ (u k 1 j ), v singularly perturbed nonlinear system: variational inequality for T 0 no classical Newton linearization 500 400 300 200 100 (multicomponent) Cahn-Hilliard: high numerical complexity efficient and reliable algebraic solvers strongly varying solution adaptivity in space and time 0 100 200 300 400 500 1.5 1 0.5 0 0.5 1 1.5 Φ for shallow quench, deep quench and deep quench limit

Nonsmooth PDE-Constrained Optimal Control (Blowey&Elliott 92, Gräser&Kh. 06) minimize J (y, q) = 1 2 a(y, y) + 1 2b(q,q) + φ(y, q) f(y) g(q) pde constraints: c(y,v) + q, v = h(v) v S j a(, ), b(, ) s.p.d., c(, ) s.p.s.d. φ convex, l.s.c., proper L 2 control problem with control constraints: a(y, v) = y, v, b(q, v) = ε q, v, c(y, v) = ( y, v), φ(y,q) = χ [0,1] (q) spatial Cahn-Hilliard problem: u = q, w = λ (Lagrange multiplier) c(y,v) = τ(m y, z), a(y, z) = c(y, z) + y, 1 z,1, b(q,v) = (γ q, v) + q,1 v, 1, φ(y,q) = p N j Φ(q(p)) α p

Karush-Kuhn-Tucker: NonSmooth Saddle-Point Problem ( F B T ) ( u ) ( f ) B C λ g u = (y, q) T, F = A + Φ A R n,n s.p.d., B R m,n, C R m,m symmetric, positive semi-definite sufficient assumptions on nonlinearity: F = ϕ, ϕ : R n R strictly convex, l.s.c., proper, coercive: ( ϕ) 1 single valued, Lipschitz

Recent Approaches to Algebraic Solution splitting algorithm of Lions & Mercier 79: (Blowey & Elliott 92,... ) box constraints, global convergence proof, expensive and slow block Gauß Seidel relaxation: (Barrett 98, Barrett, Nürnberg & Styles 05,...) box constraints, global convergence proof, slow multilevel block Gauß Seidel: (Baňas & Nürnberg 09) box constraints, no convergence proof nonsmooth Schur Newton methods: (Gräser & Kornhuber 07, 06-09, Gräser 09) piecewise smooth Φ (box constraints, log. potential), global convergence proof active set approach to box constraints: (Blank, Butz & Garcke 09) box constraints, locally exact Moreau-Yosida-type regularization of box constraints: (Hintermüller, Hinze &Tber 09) box constraints, convergence analysis in function space

Recent Approaches to Algebraic Solution splitting algorithm of Lions & Mercier 79: (Blowey & Elliott 92,... ) box constraints, global convergence proof, expensive and slow block Gauß Seidel relaxation: (Barrett 98, Barrett, Nürnberg & Styles 05,...) box constraints, global convergence proof, slow multilevel block Gauß Seidel: (Baňas & Nürnberg 09) box constraints, no convergence proof nonsmooth Schur Newton methods: (Gräser & Kornhuber 07, 06-09, Gräser 09) box constraints, log. potential and other Φ, global convergence proof active set approach to box constraints: (Blank, Butz & Garcke 09) box constraints, locally exact Moreau-Yosida-type regularization of box constraints: (Hintermüller, Hinze &Tber 09) box constraints, convergence analysis in function space

Nonlinear Schur Complement F B T!!! u f B C λ g H(λ) = 0, H(λ) = BF 1 (f B T λ) + Cλ + g Proposition Let ϕ : R n R denote the polar functional of ϕ with F = ϕ J (λ) = ϕ (f B T λ) + 1 2 (Cλ, λ) + (g, λ) is Fréchet differentiable H = J H(λ) = 0 is equivalent to unconstrained (!) convex minimization λ R n : J (λ) J (v) v R n

Gradient-Related Descent Methods λ ν+1 = λ ν + ρ ν d ν, d ν = S 1 ν J (λ ν ), S ν s.p.d. Assumption on d ν : c v 2 (S ν v, v) C v 2 ν N Assumption on ρ ν : J (λ ν + ρ ν d ν ) J (λ ν ) c( J (λ ν ),d ν ) 2 / d ν 2 Theorem: (..., Ortega & Rheinboldt 70, Nocedal 92, Powell 98, Deuflhard 04,...) The iteration is globally convergent.

Damping Strategies classical Armijo damping: two parameters, expensive evaluation of J inexact damping: approximate the solution of (H(λ ν + ρd ν ), d ν ) = 0 by bisection computational cost: evaluation of ( ϕ) 1 in each bisection step numerical experience: usually 1 step, but up to 8 steps in exceptional cases monotonicity test: (Gräser & Kh. 09, Gräser 10) no damping necessary, if d ν σ d ν 1, σ < 1

Inexact Evaluation of S 1 ν Proposition: Let d ν d ν = S 1 ν J (λ ν ). Then the accuracy conditions d ν d ν 1 ν d ν, (H(λ ν ), d ν ) < 0 ν N preserve convergence.

Selection of S ν : Nonsmooth Schur Newton Methods gradient-related descent method: λ ν+1 = λ ν ρ ν S 1 ν H(λ ν ) H(λ ν ) = Bu ν+1 + Cλ ν + g, u ν+1 = F 1 (f B T λ ν ) smooth nonlinearity: Newton s method: S ν = H (λ ν ) = B ( F (u ν+1 ) ) 1 B T + C nonsmooth nonlinearity: nonsmooth Newton: S ν = H (λ ν ) = B ( δf(u ν+1 ) ) + B T + C

Selection of S ν : Nonsmooth Schur Newton Methods gradient-related descent method: λ ν+1 = λ ν ρ ν S 1 ν H(λ ν ) H(λ ν ) = Bu ν+1 + Cλ ν + g, u ν+1 = F 1 (f B T λ ν ) smooth nonlinearity: Newton s method: S ν = H (λ ν ) = B ( (F 1 ) (f B T λ ν ) ) B T + C nonsmooth nonlinearity: nonsmooth Newton: S ν = H (λ ν ) = B ( δ(f 1 )(f B T λ ν ) ) B T + C

Selection of S ν : Nonsmooth Schur Newton Methods gradient-related descent method: λ ν+1 = λ ν ρ ν S 1 ν H(λ ν ) H(λ ν ) = Bu ν+1 + Cλ ν + g, u ν+1 = F 1 (f B T λ ν ) smooth nonlinearity: Newton s method: S ν = H (λ ν ) = B ( (F 1 ) (f B T λ ν ) ) B T + C nonsmooth nonlinearity: nonsmooth Newton: S ν = H (λ ν ) = B ( δ(f 1 )(f B T λ ν )) ) B T + C

Selection of S ν : Nonsmooth Schur Newton Methods gradient-related descent method: λ ν+1 = λ ν ρ ν S 1 ν H(λ ν ) H(λ ν ) = Bu ν+1 + Cλ ν + g, u ν+1 = F 1 (f B T λ ν ) smooth nonlinearity: Newton s method: S ν = H (λ ν ) = B ( (F 1 ) (f B T λ ν ) ) B T + C nonsmooth nonlinearity: nonsmooth Newton: S ν = H (λ ν ) = B ( δ(f 1 )(F(u ν+1 )) ) B T + C

Selection of δ(f 1 )(F(u ν+1 )): Truncated Derivatives ( ) + postulating the chain rule: δ(f 1 )(F(u ν+1 )) := ˆF (u ν+1 ) Φ piecewise smooth: (logarithmic potential) F = A+ Φ: δ(f 1 )(F(u ν+1 )) = ( A + Φ (u ν+1 )) + truncation at large Φ uniformly bounded Φ with finitely many jumps truncation at jumps characteristic function Φ = χ [ 1,1] : (box constraints) F = A + χ [ 1,1] : δ(f 1 )(F(u ν+1 )) = (Â(u ν+1 )) + truncation at active nodes

Selection of δ(f 1 )(F(u ν+1 )): Truncated Derivatives ( ) + postulating the chain rule: δ(f 1 )(F(u ν+1 )) := ˆF (u ν+1 ) Φ piecewise smooth: (logarithmic potential) F = A+Φ : δ(f 1 )(F(u ν+1 )) = ( A + Φ (u ν+1 ) ) 1 truncation at large Φ uniformly bounded Φ with finitely many jumps truncation at jumps characteristic function Φ = χ [ 1,1] : (box constraints) F = A + χ [ 1,1] : δ(f 1 )(F(u ν+1 )) = (Â(u ν+1 )) + truncation at active nodes

Convergence Results for Nonsmooth Schur Newton Methods Theorem: (Gräser & Kh. 09, Gräser 09) global convergence (exact and inexact version) independent of any parameters piecewise smooth and uniformly bounded Φ : S ν = B ( A + Φ (u ν+1 ) ) + B T + C H(λ ν ) (B-derivative) for rank B = n locally quadratic convergence for non-degenerate problems inexact version: asymptotic linear convergence for C s.p.d. characteristic function Φ = χ [ 1,1] finite termination

nonsmooth Schur Newton iteration: Interpretations u ν+1 = F 1 (f B T λ ν ), λ ν+1 = λ ν ρ ν S 1 ν H(λ ν ), H(λ ν ) = Bu ν+1 B T + Cλ ν + g S ν = B ( δ(f 1 )(F(u ν+1 )) ) B T + C preconditioned Uzawa iteration: evaluation of F 1 : semilinear elliptic problem (obstacle potential: active set) nonlinear Gauß-Seidel, multigrid (Kh. 94,02,...), damped Jacobi ( Blank et al.), (inexact) evaluation of Sν 1 : linear saddle point problem multigrid (..., Vanka 86, Zulehner & Schöberl 03,...), exact extension and globalization of primal-dual active set strategies: (Gräser 07) L 2 -control: preconditioned Uzawa primal-dual active set (Hintermüller, Ito, Kunisch 03)

nonsmooth Schur Newton iteration: Interpretations u ν+1 = F 1 (f B T λ ν ), λ ν+1 = λ ν ρ ν S 1 ν H(λ ν ), H(λ ν ) = Bu ν+1 B T + Cλ ν + g S ν = B ( δ(f 1 )(F(u ν+1 )) ) B T + C preconditioned Uzawa iteration: evaluation of F 1 : semilinear elliptic problem (obstacle potential: active set) nonlinear Gauß-Seidel, multigrid (Kh. 94,02,...), damped Jacobi ( Blank et al.), (inexact) evaluation of Sν 1 : linear saddle point problem multigrid (..., Vanka 86, Zulehner & Schöberl 03,...), exact extension and globalization of primal-dual active set strategies: (Gräser 07) nonsmooth Schur-Newton for L 2 -control primal-dual active set (Hintermüller, Ito, Kunisch 03)

Numerical Properties convergence properties: superlinear convergence no damping and fast convergence for good initial iterates nested iteration provides good initial iterates nested iteration provides mesh-independent convergence Cahn-Hilliard: obstacle potential: linear saddle-point problem dominates complexity logarithmic potential: robust convergence for varying temperature T [0,T c ]

Numerical Properties convergence properties: superlinear convergence no damping and fast convergence for good initial iterates nested iteration provides good initial iterates mesh-independent convergence by nested iteration Cahn-Hilliard: obstacle potential: linear saddle-point problem dominates complexity logarithmic potential: robust convergence for varying temperature T [0,T c ]

Anisotropic Cahn-Hilliard Equation with Logarithmic Potential interfacial energy: u u g 2 ( u), g(y) = d i=0 y 2 i + δ y 2 parameters: γ = 10 3, δ = 10 3, temperature T = 10 3, T c = 1 discretization parameters: τ = 10 3, h min = 2 7 γ 1/2 /4 t = 1τ t = 10τ t = 100τ t = 250τ

temperature: T = 10 1, 10 3, 10 10 Iteration Histories 1 work unit: CPU time for 1 multigrid step for u ν+1 = F 1 (f B T λ ν ) 10 0 10 2 10 4 error 10 6 10 8 10 10 10 12 10 14 T=1e 01 T=1e 03 T=1e 10 0 100 200 300 400 500 600 700 multigrid steps bad initial iterate superlinear convergence

Iteration Histories temperature: T = 10 1, 10 3, 10 10 1 work unit: CPU time for 1 multigrid step for u ν+1 = F 1 (f B T λ ν ) nested iteration: 10 0 10 2 10 4 10 0 10 2 10 4 T=1e 01 T=1e 03 T=1e 10 error 10 6 10 8 error 10 6 10 8 10 10 10 12 10 14 T=1e 01 T=1e 03 T=1e 10 0 100 200 300 400 500 600 700 multigrid steps bad initial iterate 10 10 10 12 10 14 0 100 200 300 400 500 600 700 multigrid steps good initial iterate robust superlinear convergence, error reduction / work unit 0.75 (nested iteration)

Asymptotic Mesh-Dependence of Nonsmooth Schur Newton number of iteration steps to roundoff error over dofs: 25 20 Newton damped Newton iteration steps 15 10 5 0 10 2 10 3 10 4 10 5 degrees of freedom good initial iterate: nested iteration

Small Scale Computations for an AgCu-Alloy realistic material data (Böhme et al 08), small length scale L = 0.1µm γ 6 10 4 discretization: mesh size h = 3 10 3 γ 1/2 /10, time step τ = 10 4 γ initial condition t = 0.25s t = 2.5s t = 12.5s equilibrium concentrations and convergence rates:

Large Scale Computations for an AgCu-Alloy experimental length scale: L = 10µm γ 6 10 8 discretization: mesh size h min = 1.5 10 5 γ 1/2 /10, time step τ = 10 4 experimental data initial condition t = 50 min (12 time steps) Schur-Newton solver: fast, mesh independent convergence (5.6 10 6 unknowns) severe complexity issues: improved implementation: code cleaning, linear solver, parallelization,... averaged quantities: coarsening rates

Conclusion Nonsmooth Schur Newton methods: preconditioned nonlinear Uzawa iterations globally convergent, no additional (regularization) parameters numerical properties: mesh independence and robustness straightforward generalization to the vector-valued case Hierarchical a posteriori error erstimates: numerical properties: mesh independence and robustness Phase separation in alloys: satisfying small-scale computations towards comparisons with experiments...