Numerical Solution of Nonsmooth Problems and Application to Damage Evolution and Optimal Insulation

Transcription

1 Numerical Solution of Nonsmooth Problems and Application to Damage Evolution and Optimal Insulation Sören Bartels University of Freiburg, Germany Joint work with G. Buttazzo (U Pisa), L. Diening (U Bielefeld), M. Milicevic (U Freiburg), R. Nochetto (U Maryland), A. Salgado (U Tennessee), M. Thomas (WIAS Berlin) FoCM 2017, Barcelona, 17 July / 32

2 Nonsmooth Models Obstacle problem Minimize 1 2 Ω subject to u χ Placticity (friction) models t p I K (σ), div C ( ε(u) p ) = f Sparse reconstruction u 2 dx fu dx Ω Minimize x l 1 subject to Ax = b Image denoising Minimize Du + α u g 2 2 Ω 2 / 32

3 Outline 1 Introduction 2 Thin Insulating Films 3 Total Variation Minimization 4 BV Regularized Damage Evolution 5 Computing Insulating Films 6 Summary 3 / 32

4 Thin Insulating Films 4 / 32

5 Thin Insulation Ω Model of insulation: ε \ Ω Ω R d conducting body Ω εl(s) 0 insulating layer of width εl(s) conductivity of layer is ε u = f ε u = 0 u = 0 in Ω c ε PDE system: u C(Ω ε ) with u = f ε u = 0 u = 0 in Ω in Ω ε \ Ω on Ω ε Limit ε 0 leads to Robin BC: u = f u + l n u = 0 in Ω on Ω See: Brézis, Caffarelli & Friedman 80, Acerbi & Buttazzo 86 Operator form of limit model A l u = f with A l : H 1 (Ω) H 1 (Ω) A l u, v = u v dx Ω + l 1 uv ds Ω A l coercive & symmetric 5 / 32

6 Loss of Heat Associated heat equation t u + A l u = 0, u(0) = u 0 Prinicipal eigenvalue determines long-time behaviour λ m,l = inf A lu, u λ m,l u 2 A l u, u u =1 Decay of variance of enthalpy 1 d 2 dt u 2 = A l u, u λ m,l u 2 = u(t) u 0 e λ m,lt λ m,l smallest eigenfrequency Dependence on available mass m = Ω l ds? Optimal distribution l? Class of admissible mass distributions { I m = l L 1 ( Ω) : l 0, Ω } l ds = m 6 / 32

7 Optimal Distribution For fixed m optimize λ m,l over l I m and note interchange λ m = inf l I m λ m,l = inf u =1 Given u H 1 (Ω) optimal thickness l I m is Problem reduces to λ m = l(s) = m inf J m(u), J m (u) = u =1 u(s) u L 1 ( Ω) Ω inf A l u, u l I m u 2 dx + 1 m ( Ω ) 2 u ds Existence of eigenfunctions u m Nonlocal, nondifferentiable, constrained problem Thickness l proportional to trace of u 7 / 32

8 Symmetry Break Recall λ m = min u =1 J m(u), J m (u) = u m u 2 L 1 ( Ω) m λ m continuous, strictly decreasing, λ m 0 as m Dirichlet and Neumann eigenvalues for λ D = min u =1, u Ω =0 u 2 > λ N = min u =1, Ω u dx=0 u 2 λ m λ D for m 0 and λ m0 = λ N for some m 0 > 0 λ D λ N m λ m Theorem (Bucur, Buttazzo & Nitsch 16). Nonradial eigenfunctions u m for Ω = B R (0) iff m < m 0, particularly, optimal mass distributions l leave gaps. Gap symmetric and/or connected? m 0 8 / 32

9 Proof (sketched) radial u m solves 1D Sturm Liouville problem and has no roots u m satisfies PDE ( u m, v) + 1 m u m L ( Ω)( 1 σ(um ), v ) Ω = λ m(u m, v) with σ(u m ) u m, i.e., σ(u m ) = 1 since u m > 0 testing u N with v(x) = c 1 x 2 c 2 gives (u N, 1) Ω = ( u N, v) = λ N (u N, v) = 0 u m L 2,H u 1 N since λ m (u m, u N ) = ( u m, u N ) = λ N (u N, u m ) for ε small u m + εu N L1 ( Ω) = u m L1 ( Ω) implies ( um + εu ) N λ m J m = J m(u m ) + ε 2 u N 2 u m + εu N 1 + ε 2 = λ m + ε 2 λ N 1 + ε 2 contradicts λ m > λ N hence u m not radial 9 / 32

10 Total Variation Minimization 10 / 32

11 BV Model Problem For noisy image g : Ω R find u BV (Ω) minimal for I(u) = Du + α u g 2 2 Proposed by Rudin, Osher & Fatemi 92 Convex, nondifferentiable minimization problem Captures discontiniuties, staircasing effect Ω = Contributions: Elliott & Mikelić 91, Chambolle & P.-L. Lions 97, Feng & Prohl 03, Hintermüller & Kunisch 04, Tsai & Osher 05, Osher, Burger, Goldfarb, Xu & Yin 05, Wu & Tai 09, Chambolle & Pock 11, Wang & Lucier 11, Stamm & Wihler 15, / 32

12 Properties of BV Function v BV (Ω) if v L 1 (Ω) and Dv = Dv (Ω) = sup Ω ξ C 0 (Ω;Rd ), ξ 1 Examples: (i) v(x) = sign(x), Dv = 2δ 0, Dv (Ω) = 2 (ii) v(x) = χ A (x), Dv (Ω) = Per Ω (A) v div ξ dx < Ω Existence of solutions by convexity and compactness W 1,1 (Ω) BV (Ω) strictly and v L 1 (Ω) = Dv (Ω) C (Ω) BV (Ω) weakly but not strongly dense Intermediate convergence in BV (Ω): v j v in L 1 (Ω) & Dv j (Ω) Dv (Ω) 12 / 32

13 Aspects and Tools Choice of finite element space V h BV (Ω) Convergence rates unter appropriate regularity conditions A posteriori error estimates and local mesh refinement Effective iterative numerical solution Strong convexity: For arbitrary v BV (Ω) L 2 (Ω) α 2 u v 2 I(v) I(u) σ I (v, w) Strong duality: Sharp relation I(v) D(q) with q H N (div; Ω) and D(q) = 1 2α div q + αg 2 + α 2 g 2 I K1 (0)(q) Combination: For minimizer u and arbitrary v and q α 2 u v 2 I(v) I(u) I(v) D(q) v w 13 / 32

14 Convergence Rates P0 FE fails as perimeter approximated incorrectly: h u h L 1 χ {x<0} = Du h (Ω) 2 P1 FE quasi-interpolant via regularization and interpolation Lemma (B., Nochetto & Salgado 12). There exists ũ h,ε S 1 (T h ) with ũ h,ε L 1 (1 + cε + chε 1 ) Du (Ω), u ũ h,ε L 1 c(h 2 ε 1 + ε) Du (Ω). Implies rate O(h 1/4 ) with ε = h 1/2 : α 2 u u h 2 I(u h ) I(u) inf I(v h ) I(u) ch 1/2 v h S 1 (T h ) 1.5 Optimal rate is O(h 1/2 ) for generic u BV (Ω) TVD property ũ h,ε Du (Ω) yields optimality u P h u h h = 1/10 h = 1/20 h = 1/ / 32

15 Adaptivity Computable localized error control via approximation of dual (B. 15): α 2 u u h 2 I(u h ) D(p h ) = u h L 1 = Ω T T h η T (u h, p h ), p h u h dx + 1 2α div p h α(u h g) 2 provided p h H N (div; Ω) with p h 1; note η T (u h, p h ) 0 Nearly optimal rate h 1/2 N 1/4 Dual solution in P1 vector fields Better: H(div; Ω) FE spaces 15 / 32

16 Solution via ADMM Decouple nondifferentiability from gradient via p = u L τ (u, p; λ) = p dx + α 2 u g 2 + (λ; u p) H + τ 2 u p 2 H Ω Formally, e.g., on discrete level, for arbitrary τ 0 inf sup u,p λ L τ (u, p; λ) = inf u I(u) Choice of Hilbert space H: weighted L 2 norm on FE spaces Iterative solution via decoupled minimization: (Glowinski & Morocco 76, Gabay & Mercier 76, He & Yuan 12, Deng & Yin 16) Algorithm. Choose (u 0, λ 0 ) and τ > 0; set j = 1. (1) Compute minimizer p j for p L τ ( u j 1, p; λ j 1). (2) Compute minimizer u j for u L τ ( u, p j ; λ j 1 ). (3) Update λ j 1 via λ j = λ j 1 + τ( u j p j ). 16 / 32

17 Properties Unconditional convergence if τ > 0 Proof holds for sequence of decreasing step sizes τ j+1 τ j Monotonicity of residuals Error control via residual R j = λ j λ j 1 2 H + τ 2 j (u j u j 1 ) 2 H u u j 2 + u j u j 1 2 C 0 τ 1 j R 1/2 j Linear convergence of residuals R j+1 γr j, 0 < γ < 1 Motivates adaptive step size adjustment (B. & Milicevic 17): Start with τ 1 and γ (0, 1) Accept τ if γ-reduction satisfied, decrease τ otherwise If τ = 1 increase γ and re-start 17 / 32

18 Experiment Comparison to ADMM and accelerated version (Goldfarb et al. 13) Hyphens indicate more than 10 4 steps Variable step size variant best for large initial step sizes Restricted to convex problems 18 / 32

19 Subdifferential Flow Gradient flows most general and robust tool to minimize Formal PDE for TV flow and backward Euler method t u = div u u, d tu k = div uk u k Unconditional energy stability by convexity d t u k 2 + d t Ω u k dx d t u k 2 u k + Ω u k d tu k dx = 0 Practical variant semiimplicit and regularized d t u k u k = div u k 1 ε Violates monotonicity property and might depend critically on ε 19 / 32

20 Unconditional Stability Continuous differentiation formula d dt a(t) = a(t)a (t) a(t) Discrete product and quotient rule = 1 2 d dt a(t) 2 a(t) d t a k b k = (d t a k )b k + a k 1 (d t b k ), d t 1 c k = d tc k c k 1 c k Binomial formula Crucial formula d t a k = d t a k 2 a k 2a k d t a k = d t a k 2 + τ d t a k 2 = d t a k 2 a k 1 + ak 2 d t 1 a k = d t a k 2 a k 1 ak d t a k a k 1 = 1 d t a k 2 2 a k 1 τ d ta k 2 2 a k 1 1 d t a k 2 2 a k 1 Implies unconditional convergence of semi-implicit scheme 20 / 32

21 Experiments Change of height proportional to mean curvature of level set Evolution of characteristic function of disk u 0 = χ B1 Exact solution given by u(t, x) = (1 2t) + χ B1 (x) Error u u h L (L 2 ) depends on ε External Movie implicit ε = h 1/2 ε = h ε = h / 32

22 BV Regularized Damage Evolution 22 / 32

23 BV Regularized Damage Model Energy and dissipation for displacement ϕ and damage variable z: E(ϕ, z) = f (z)ε(ϕ) : Cε(φ) dx + Dz (Ω) + I [0,1] (z) dx Ω Ω { ϱ v if v (, 0], R(v) = R(v) dx, R(v) = + if v > 0. Ω Evolution of q = (ϕ, z) via Biot s equation 0 q E(q) + q R( q) Energetic solutions are globally stable and fulfill energy balance E ( q(t) ) E ( q ) + R ( z z(t) ) E ( q(t) ) ( ) ( ) t + Diss R z, [s, t] = E q(s) + r E ( q(r) ) dr with Diss R (z, [s, t]) = sup { N j=1 R( z(t j ) z(t j 1 ) ), t 0 <... < t N }. derivative-free formulation existence theories: Mielke 05; partial damage: Thomas 13 s 23 / 32

24 Experiment Time-stepping via decoupled minimization External Link 24 / 32

25 Computing Insulating Films 25 / 32

26 Gradient Flow Minimization problem J m (u) = u m u 2 L 1 ( Ω) subject to u = 1 Regularized gradient flow with u ε = (u 2 + ε 2 ) 1/2 ( t u, v) + ( u, v) + 1 m u L 1 ε ( u ), v = µ(u, v) u ε Ω u(t) = 1 implies t u L 2 u; RHS disappears if v L 2 u Algorithm (B. 17). Given u k 1 compute u k with d t u k L 2 u k 1 and (d t u k, v) + ( u k, v) + 1 ( u k ) m uk 1 L 1 ε u k 1, v ε for all v with v L 2 u k 1 with quotient d t u k = (u k u k 1 )/τ. Iteration (essentially) unconditionally stable Ω = 0 26 / 32

27 Spatial Discretization For triangulation T h of Ω use P1 FE-space S 1 (T h ) = { } v h C(Ω) : v h T P 1 (T ) for all T T h. and discretized functional J m,ε,h (u h ) = u h ( ) 2 β z u h (z) ε m z N h Ω Stability estimate carries over to J m,ε,h Lemma (B. 17). If eigenfunction u = u m satisfies u H 2 then 0 min u h S 1 (T h ) J m (u h ) J m (u) ch u 2 H 2 (Ω), Jm,ε,h (u h ) J m (u h ) c( uh H1 (Ω) + 1) 2 (ε + h 1/2 ). Consistency estimates imply Γ convergence via density 27 / 32

28 Experiments Dirichlet and Neumann eigenfunctions on unit disk Optimal and constant l with m = 0.4 on unit disk Optimal and constant l with m = 0.1 on unit square (λ D = 2π 2, λ N = π 2 ) 28 / 32

29 Iteration Numbers Ω = B 1 (0) R 2, m = 1/2, h = 2 l, ε = h/10, τ = 1 l #N h #T h K stop λ m,h Ω = (0, 1) 2 R 2, m = 1/8, h = 2 l, ε = h/10, τ = 1 l #N h #T h K stop λ m,h Fixed stopping criterion: d t u k h ε stop = / 32

30 Rotational Shapes Symmetrical ellipsoids E aa with radii (a, a, r a ), a [1, 1.5] Optimal assembled ellipsoid E ab with radii (a, b, r ab ) Fixed volume c 0 = B 1 and mass m = 6.0 λ m(e aa) λ m(e opt ab ) (i) (ii) (i) a (ii) High resolution via two-dimensional reduction Egg shape optimal among convex shapes? 30 / 32

31 Summary 31 / 32

32 Summary Reduced convergence rates for nondifferentiable problems Iterative solution via regularized gradient flows Semi-implicit discretization unconditionally stable Future topics Convergence of adaptive schemes Optimal isolating bodies Error estimates for damage evolution Further information 32 / 32