LECTURE 3: DISCRETE GRADIENT FLOWS

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1 LECTURE 3: DISCRETE GRADIENT FLOWS Department of Mathematics and Institute for Physical Science and Technology University of Maryland, USA Tutorial: Numerical Methods for FBPs Free Boundary Problems and Related Topics Isaac Newton Institute, January 2014

2 Outline Subdifferentials and Coercivity Subgradient Operators Angle-Bounded Operators Space Discretization Numerical Experiments for TV Flow The Allen-Cahn Equation

3 OUTLINE Subdifferentials and Coercivity Subgradient Operators Angle-Bounded Operators Space Discretization Numerical Experiments for TV Flow The Allen-Cahn Equation

4 Subdifferentials Space: H separable Hilbert space. Functional: φ : H R proper lower semicontinuous (l.s.c) convex function with (nonempty, convex) domain D(φ). Subdifferential: F = φ is the subdifferential of φ: w F(w) w, v w φ(v) φ(w) v D(φ) with D(F) densily contained in D(φ). The operator F is in general multivalued. Monotonocity: F(v) F(w), v w 0 v, w D(F).

5 Energy Solutions Strong Solution: Given u 0 D(F) and f L 1 (0, T ; H), a function u C 0 ([0, T ]; H) is a strong solution of u (t) + F(u(t)) = f(t) a.e. t (0, T ) if it is absolutely continuous in (0, T ) and satisfies the equation. Energy Solutions: Multiply the equation by u(t) v to obtain the variational inequality u (t) f(t), u(t) v + φ(u(t)) φ(v) 0 v D(φ). Energy Identity: If f L 2 (0, T ; H) then t φ(u(t)) is absolutely continuous in (0, T ) and u (t) 2 + d dt φ(u(t)) = f(t), u (t) a.e. t (0, T ). Dissipation Inequality: If u is a strong solution and f is absolutely continuous 1 d 2 dt u (t) 2 f (t), u (t).

6 Example 1: Parabolic Obstacle Problem Space: H = L 2 (Ω) K = {v H 1 0 (Ω) : v(x) ψ(x) a.e. x Ω}. Functional: If I K is the indicator function of K, then φ(w) = 1 Z w 2 + I K(w), D(φ) = K 2 Ω Variational Inequality: u solves Z Z tu(u v) + u (u v) f(u v) v K. Ω Ω

7 Example 2: Degenerate Parabolic PDE Space: H = H 1 (Ω) with w = Gw L 2 (Ω), G = ( ) 1 with homogeneous Dirichlet data. Functional: Let β : R R be continuous and monotone, and Z Z w(x) φ(w) = β(r)dr dx, D(φ) = {v L 2 (Ω) : j(v) L 1 (Ω)}. Ω 0 {z } =j(w(x)) Subdifferential: F(w) = β(w), D(F) = {v D(φ) : β(v) H 1 0 (Ω)} Equation: u solves t β(u) = f.

8 Example 3: Total Variation Flow φ(v) = R v D(φ) := Ω L2 (Ω) BV (Ω). u t div = 0 F(u) = φ(u) = div u u u u Characterization of the subdifferential φ [Andreu et al. 2004]: Let X = z L (Ω) : divz L 2 (Ω), z n Ω = 0 B 1(X) unit ball of X Z divz φ(u) z B 1(X) : z u = φ(u) Ω We exploit this characterization to restate the TV flow: given u D(φ) find u H 1 (0, T ; L 2 (Ω) L (0, T ; D(φ)) and z L (0, T ; X) Initial condition: u t=0 = u 0 TV flow: u t, w + R z w = 0, w D(φ) Ω R divz φ(u): z B 1(X) (q z) u 0 Ω q B1(X).

9 Coercivity σ(w; v) := φ(v) φ(w) F(w), v w 0 ρ(w, v) := σ(w, v) + σ(v; w) = F(w) F(v), w v Variational Inequality: u f, u v + φ(u) φ(v) + σ(u; v) 0 v D(φ). If ρ is p-coercive, p 2, ρ(w; v) [w v] p w, v D(F), then σ(w, v) 1 [w v]p p w D(F), v D(φ). We will ignore coercivity in the rest of the discussion.

10 OUTLINE Subdifferentials and Coercivity Subgradient Operators Angle-Bounded Operators Space Discretization Numerical Experiments for TV Flow The Allen-Cahn Equation

11 Hilbert Theory: Subgradient Operators Energy Solutions: (with vanishing forcing term) u t + F(u) = 0 in H (Hilbert space) F = φ (φ convex) = u (t) 2 + d φ(u(t)) = 0. dt u, u v + φ(u) φ(v) v D(φ) Backward Euler Method: 1 U n U n 1, U n V + φ(u n) φ(v ) V D(φ) Un = E n := Un φ(un) φ(un 1) 0 (Numerical Dissipation)

12 A Posteriori Error Analysis (Steps 1 and 2) We follow [N, Savaré, Verdi 00]: 1. ODE for U(t) = t t n 1 U n + tn t U n 1 (linear interpolant of U n 1 and U n) U, U v + φ(u) φ(v ) U, U U n + φ(u) φ(un) {z } R(t)=residual 2. Estimate of R(t) U U n = t tn `Un U n 1 φ(u) tn t φ(u n 1) + t tn 1 φ(u n) (φ convex) = R(t) = (t n t)e n

13 A Posteriori Error Analysis (Steps 1 and 2) We follow [N, Savaré, Verdi 00]: 1. ODE for U(t) = t t n 1 U n + tn t U n 1 (linear interpolant of U n 1 and U n) U, U v + φ(u) φ(v ) U, U U n + φ(u) φ(un) {z } R(t)=residual 2. Estimate of R(t) U U n = t tn `Un U n 1 φ(u) tn t φ(u n 1) + t tn 1 φ(u n) (φ convex) = R(t) = (t n t)e n

14 A Posteriori Error Analysis (Steps 1 and 2) We follow [N, Savaré, Verdi 00]: 1. ODE for U(t) = t t n 1 U n + tn t U n 1 (linear interpolant of U n 1 and U n) U, U v + φ(u) φ(v ) U, U U n + φ(u) φ(un) {z } R(t)=residual 2. Estimate of R(t) U U n = t tn `Un U n 1 φ(u) tn t φ(u n 1) + t tn 1 φ(u n) (φ convex) = R(t) = (t n t)e n

15 A Posteriori Error Analysis (Steps 3 and 4) 3. Error Equation v = U V = u u, u U + φ(u) φ(u) 0 U, U u + φ(u) φ(u) R(t) = 1 d 2 dt u U 2 R(t) = (t n t)e n 4. Integration in Time max u 0 t T U 2 u 0 U NX τne 2 n n=1

16 A Posteriori Error Analysis (Steps 3 and 4) 3. Error Equation v = U V = u u, u U + φ(u) φ(u) 0 U, U u + φ(u) φ(u) R(t) = 1 d 2 dt u U 2 R(t) = (t n t)e n 4. Integration in Time max u 0 t T U 2 u 0 U NX τne 2 n n=1

17 A Posteriori Error Analysis (Steps 3 and 4) 3. Error Equation v = U V = u u, u U + φ(u) φ(u) 0 U, U u + φ(u) φ(u) R(t) = 1 d 2 dt u U 2 R(t) = (t n t)e n 4. Integration in Time max u 0 t T U 2 u 0 U NX τne 2 n n=1

18 A Posteriori Error Analysis (Steps 3 and 4) 3. Error Equation v = U V = u u, u U + φ(u) φ(u) 0 U, U u + φ(u) φ(u) R(t) = 1 d 2 dt u U 2 R(t) = (t n t)e n 4. Integration in Time max u 0 t T U 2 u 0 U NX τne 2 n n=1

19 A Posteriori Error Analysis via Residual Monotone Terms Continuous Dissipation Inequality: If u is a strong solution and f = 0, then Discrete Dissipation Inequality: 1 τ 2 n Residual Monotone Term: because implies E n = τ 1 n d dt u (t) 2 0. U n U n U τn 1 2 n 1 U n n N E n τn 1 F(U n) F(U n 1), U n U n 1 = D n τ 1 n φ(u n) φ(u n 1) F(U n 1), U n U n 1 φ(u n 1) φ(u n) τn U 2 n U n 1 2 F(U n 1), U n U n 1 + τ 1 F(U n), U n U n 1. n

20 A Priori Rate of Convergence We follow [N, Savaré, Verdi 00]. Similar results in [Baiocchi 89, Rulla 96]. Weak Regularity: 0 φ(u 0) C NX n=1 τ 2 ne n φ(u 0) max 1 n N τn {z } =τ U 0 = u 0 D(φ) max 0 t T u U p τφ(u 0) Strong Regularity: F(U 0) C NX τne 2 n 1 2 F(U0) 2 max 2 1 n N n=1 {z } =τ 2 U 0 = u 0 D(F) max 0 t T u U τ 2 F(U 0)

21 A Priori Rate of Convergence: Proofs Weak Regularity: Since 0 E n τn `φ(un 1) 1 φ(u n), and φ can be assumed to be nonnegative after an affine modification, we deduce U 0 = u 0 NX NX τne 2 n τ φ(u n 1) φ(u n) τφ(u 0) n=1 with τ := max 1 n N, whence max u 0 t T U 2 n=1 NX τne 2 n φ(u 0) τ. n=1 Strong Regularity: Since τn 1 `Un U n 1 = F(Un) we have 0 D n = F(U n) F(U n 1), F(U n) 1 F(U n 1) 2 F(U n) 2 2 whence for U 0 = u 0 max u 0 t T U 2 NX τnd 2 n 1 2 F(u0) 2 τ 2. n=1

22 Examples of Subgradient Operators 1. Example 1: Parabolic Obstacle Problems (H = L 2 (Ω), D(φ) = H 1 0 (Ω), D(F) = H 2 (Ω)) u ψ : u t, u v + u, (u v) 0 v ψ Z n := U n Un Un 1 E n 1 (U n U n 1) Z n Z n 1, U n U n 1 2. Example 2: Degenerate Parabolic Problems (H = H 1 (Ω), D(φ) = L 2 (Ω), D(F) = H 1 0 (Ω)) u t β(u) = 0 (β non-decreasing) E n 1 β(u n) β(u n 1), U n U n 1

23 Example 3: Time-Discrete TV Flow Continuous Problem: Z u t, w + z w = 0, w D(φ) Ω Z z B 1(X) (q z) u 0 q B 1(X) Ω Semidiscrete Variational Scheme: [Bartels, N, Salgado 13] Given τ > 0 find {U n } h 1 (L 2 (Ω)) and {Z n } l (X) such that Z 1 τ U n+1 U n, w + Z n+1 w = 0, w D(φ) Ω Z `q Z n+1 U n+1 0, q B 1(X). Ω A Priori Error Estimates (H = L 2 (Ω)) u 0 D(φ) u U 2 L (H) u 0 U 0 2 H + τφ(u 0 ). u 0 D( φ) u U 2 L (H) u 0 U 0 2 H + τ 2 φ(u 0 ) 0 2 H.

24 OUTLINE Subdifferentials and Coercivity Subgradient Operators Angle-Bounded Operators Space Discretization Numerical Experiments for TV Flow The Allen-Cahn Equation

25 Hilbert Theory: Angle-Bounded Operators (N, Savaré, Verdi 00) u + F(u) = 0, U + F(U n) = 0 1 d 2 dt u U 2 = F(u) F(U n), u U = tn 1 t F(u) F(U n), u U n + tn t F(U n) F(u), u U n 1 γ > 0 : F(v) F(w), w z γ F(v) F(z), v z

26 Examples of Angle-Bounded Operator Example 1: Linear Advection-Diffusion PDE F(v) = ν 2 v + b v + cv, H = L 2(Ω) ( b(x) b 0 d(x) = 1 2 divb(x) + c(x) d2 0 γ 2 = 1 4 b ν 2 (d ν2 p 2 0 ) u U 2 L (L 2 ), ν 2 (u U) 2 L 2 (L 2 ) u 0 U 0 2 L 2 X N Z + 2γ 2 ν 2 (U n U n d(x) U n U n 1 2 n=1 τ 2 n Ω Example 2: Obstacle Problem for Advection-Diffusion PDE F(v) = ν 2 v + b v + cv, v ψ, H = L 2(Ω)

27 OUTLINE Subdifferentials and Coercivity Subgradient Operators Angle-Bounded Operators Space Discretization Numerical Experiments for TV Flow The Allen-Cahn Equation

28 Space Discretization: Abstract Theory (Bartels, N, Salgado 13) Let {H h } h>0 be a family of finite dimensional subspaces of H and φ h : H h R convex and lower semicontinuous functionals such that: Monotonicity: For every v h H h φ h (v h ) φ(v h ). Approximation: For every h > 0, there is an operator C h : W H h with W D(φ) dense such that C h w w H ε 1(h) w W, w W D(φ). Energy diminishing: For all w W φ h (C h w) φ(w) ε 2(h) w W.

29 A Priori Error Estimate Semidiscrete problem Fully discrete problem 1 τ U n+1 U n, U n+1 w + φ(u n+1 ) φ(w) 0. 1 n+1 Uh Uh n, U n+1 h W h + φ h (U k+1 h ) φ h (W h ) ρ n+1, U n+1 h W h, τ with ρ n+1 a convenient perturbation (inexact solution). Theorem (abstract a priori error estimate) If U 0 h W uniformly in h and ρ l (H) Cτ, then U U h 2 l (H) U 0 U 0 h 2 H + C`ε 1(h) + ε 2(h) U l (W) + Cτ.

30 Example: Fully Discrete TV Flow Direct discretization: find {U n h } H h H = L 2 (Ω) such that 1 n+1 Uh Uh n, U n+1 h W h + φ h (U n+1 h ) φ h (W h ) ρ n+1, u n+1 h W h. τ This is equivalent to: find {Uh n } H h and {Z n h} X such that Z 1 n+1 Uh Uh n, W h + Z n+1 h W h = ρ k+1, W h W h V h, τ Ω Z `q Z n+1 h U n+1 h 0 q B 1(X). Ω Questions: The dual variable Z n h is not discrete. How to discretize it? If we discretize it, how do we solve the variational inequality?

31 Fully Discrete TV Flow: Quadrature Finite element space: Let H h be constructed using P 1 or Q 1 continuous finite elements. Scheme: The TV flow with quadrature is to find {U n h } H h such that 1 n+1 Uh Uh n, U n+1 h W h +φ h (U n+1 h ) φ h (W h ) ρ k+1, U k+1 h W h, τ Quadrature: the quadrature rule is φ h (v h ) = Q h ( v h ) = X QX ω q T v h (ξ q) φ(v h ). T T h q=1

32 Fully Discrete TV Flow: Approximation Properties Monotonicity: If Q h is supported at the vertices of the cells, then Z φ h (w h ) = Q h ( w h ) w h = φ(w h ), Approximation: If C h w = Π h w ε is the Clement interpolant of w ε, a C approximation of function w L (Ω) BV (Ω), we get h 2 C h w w L 2 (Ω) C Ω 1 ε + ε 2 {z } =ε 1 (h) w 1 2 L (Ω) Dw (Ω) 1 2. Energy diminishing: If Q h is exact for linears, then φ h (w) φ(w) = Q h ( C h w ) Dw (Ω) h 1 C ε + ε 2 Dw (Ω) {z } =ε 2 (h) w BV (Ω).

33 Fully Discrete TV Flow: A Priori Error Analysis Theorem (a priori error estimates) Let u 0 L (Ω) BV (Ω) and U 0 h = Π h u 0. Then U U h l (L 2 (Ω)) Ch 1 6. Assume we can construct a TV-diminishing interpolation operator C h, i.e. Z Z C h w w, Ω Ω The error becomes O(h 1 4 ). This suboptimal rate is related to lack of time regularity; compare with heat equation. The error for total variation minimization (image processing) becomes O(h 1 2 ). This is optimal given the regularity L (Ω) BV (Ω) B 1 2 (L 2 (Ω)).

34 Construction of the TV Diminishing Operator (Bartels, N, Salgado 13) Let Ω = S 1 S 1 be a periodic domain. The construction extends to non-periodic domains and dimension d > 2. V h p.w. Q 1, continuous. For a vertex z N h W z := 1 h xh y Z Q w v Let Π h : L 1 (Ω) V h be Π h w(z) := W z hy hx

35 TV Diminishing Operator: Stability Properties Theorem (stability) The operator Π h is TV diminishing Z Π h w Dw (Ω) Ω and is L p stable for 1 p, i.e. Πw L p (Ω) w L p (Ω), w BV (Ω) w L p (Ω) The key is to exploit the symmetry of the construction Notice that the constant is also = 1 in the L p estimate

36 TV Diminishing Operator: Approximation Properties Theorem (error estimates) Let h = max{h x, h y}. If w BV (Ω) w Πw L 1 (Ω) Ch Dw (Ω) If w BV (Ω) L p (Ω) w Πw L q (Ω) Ch 1 s Dw (Ω) 1 s w s L p (Ω), 1 q = 1 s + s 1 p If w W 2 1 (Ω) w Πw L 1 (Ω) Ch 2 w W 2 1 (Ω) The operator preserves linears locally = second order

37 OUTLINE Subdifferentials and Coercivity Subgradient Operators Angle-Bounded Operators Space Discretization Numerical Experiments for TV Flow The Allen-Cahn Equation

38 Experiment 1: Characteristic of a Ball If u 0 = χ BR, then [Andreu et. al. 2004] u(x, t) = (1 λt) + χ BR (x), λ = 2 R. The computational convergence rate is O(h 1 2 ) with τ = τ = h 10. No oscillations. No diffusion of the support B R (no regularization) Excellent agreement with the extinction time (not covered by theory).

39 Experiment 2: Effect of Global Forcing (Condensation) We consider f = constant, Neumann BC, u 0 = χ BR and u t div = f. u u characteristic of ball B R Forcing f prevents χ BR to decrease: no diffusion; Characteristic χ Ω/BR grows with constant speed until it merges with χ BR, and next χ Ω grows with the same speed.

40 Experiment 3: Characteristic of a Disconnected Set Let u 0 = P 3 i=1 χ B(x i,r i ), where the centers lie on an equilateral triangle of length 1 and R i 0.2. Then [Andreu et.al. 2004] 3X u(x, t) = λ it) i=1(1 + χ B(xi,R i )(x), λ i = 2 R i

41 Experiment 4: Characteristic of an Annulus Let u 0 = kχ E with E = B R \ B r and k R. Then [Andreu et al. 04] u(x, t) = ( sgn(k) ( k λ Et) + χ E(x) + λ rtχ Br, t < T 1, sgn(κ) ( κ λ r(t T 1)) + χ BR, t > T 1 with λ E = 2t R r, λr = 2 r T 1 = k λ E + λ r, κ = T 1λ r

42 Experiment 4: Characteristic of an Annulus No diffusion of the support. Although not covered by the theory, there is good agreement for the meeting time. Although not covered by the theory, there is good agreement for the extinction time.

43 Experiment 5: Comparison with Regularized TV Flow [Feng-Prohl 03]: If u ε is the solution of the regularized TV flow u ε tu ε div p = 0, ε2 + u ε 2 then u u ε L (L 2 ) Cε 1 2. If u ε L 2 (H 2 ) ε 1, our analysis yields provided ε h dt. u U ε,h L (L 2 ) Ch 1 2

44 Experiment 5: Comparison with Regularized TV Flow 0.5 ε=0 ε=h ε=h u x The mesh size is h = 2 5, the time-step t = 2 10, and T=5. Notice that the requirement t = O(h 2 ) needed for the L 2 -convergence of the regularized flow (Feng-Prohl) is valid.

45 OUTLINE Subdifferentials and Coercivity Subgradient Operators Angle-Bounded Operators Space Discretization Numerical Experiments for TV Flow The Allen-Cahn Equation

46 The Allen-Cahn Equation as a Gradient Flow Space: H = L 2 (Ω) with norm u, v = ε R Ω uv Functional: Z φ(v) = Ω ε 2 u ` u ε Gradient Flow in L 2 : If f(u) = u` u 2 1, then ε tu ε u + 1 ε f(u) = 0 Error Analysis: Since φ is a Lipschitz perturbation of a convex functional, the a priori error analysis applies but with a stability constant C = e T /ε2. This exponential deterioration in the small parameter ε is in general unavoidable as corresponds to the logistic ODE ε t + 1 f(u) = 0. ε Developed Interfaces: Once diffuse interfaces form, the exponential behavior can be replaced a priori by a polynomial dependence on ε 1 [Feng-Prohl 02].

47 The Allen-Cahn Equation: A Posteriori Error Analysis (Kessler-N, Bartels) Singularly Perturbed Parabolic PDE: ε tu, v + ε u, v + 1 ε f(u), v = 0, f(u) = u`u 2 1 Finite Element Discretization: U n V n : for all V V n ε U n I nu n 1, V + ε U n, V + 1 ε f(inun 1), V = 0 Piecewise Linear Interpolant: Residual: U(t) = t tn 1 U n + tn t U n 1 R(t), v := ε tu, v + ε U, v + 1 f(u), v ε

48 A Posteriori Error Estimate Without Exponential Spectral Estimate [DeMottoni-Schatzman, X. Chen] ε v 2 L 2 (Ω) + 1 ε f (u)v, v λ 0ε v 2 L 2 (Ω) v H 1 (Ω) E 0, E 1 computable estimators in terms of the residual R(t) Tolerance: δ Cε 4 Conditions on E 0, E 1: λ T, λ 0 T, λ 1 T e 2λ 0T u 0 U 0 L2 (Ω) λ T δ, E 0 λ 0 T εδ, E 1 λ 1 T ε 2 δ A posteriori error estimate (with polynomial dependence on ε): u U L2 (0,T ;L 2 (Ω)) δ