On Some Variational Optimization Problems in Classical Fluids and Superfluids

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1 On Some Variational Optimization Problems in Classical Fluids and Superfluids Bartosz Protas Department of Mathematics & Statistics McMaster University, Hamilton, Ontario, Canada URL: Funded by Early Researcher Award (ERA) and NSERC Computational Time Provided by SHARCNET New Challenges in Mathematical Modelling and Numerical Simulation of Superfluids, CIRM 216

2 Agenda Formulation of the Problem Gradient Minimization Sobolev Gradients First-Order Geometry Second-Order Geometry Riemannian Conjugate Gradients

3 Collaborators Ionut Danaila (Université de Rouen) Diego Ayala (former Ph.D. student, now at University of Michigan) Charles Doering (University of Michigan)

4 PROBLEM I for Computation Ground States in Bose-Einstein Condensates (joint work with I. Danaila)

5 Formulation of the Problem Gradient Minimization Sobolev Gradients Gross-Pitaevskii Free Energy Functional (non-dimensional form) [ 1 E(u) = D 2 u 2 + C trap u ] 2 C g u 4 ic Ω u A t u dx, u 2 2 = u(x) 2 dx = 1, D R d where D ψ u =, ψ wavefunction, ψ : D C d/2 N x s N number of atoms in the condensate A t = [y, x, ], x s characteristic length scale C trap (x, y, z) trapping potential C g, C Ω constants C Ω characterizes the effect of rotation

6 Formulation of the Problem Gradient Minimization Sobolev Gradients Dirichlet boundary conditions: u = on D Variational optimization, E : H 1 (D) R min E(u) u H 1(D) subject to u L2 (D) = 1 Minimizers constrained to a nonlinear manifold M in H 1 (D) M := { u H 1 (D) : u L2 (D) = 1 } Computational approaches: Euler-Lagrange equation for E(u) = nonlinear eigenvalue problem Direct minimization of E(u) via a gradient method

7 Formulation of the Problem Gradient Minimization Sobolev Gradients Steepest-gradient approach where: ũ = lim n u(n) E ( u (u)) u (n+1) = u (n) τ n E ( u (u)), n =, 1,..., u () = u, (initial guess), the minimizer ( ground state ) gradient of E(u) at u (n) τ n = argmin τ> E ( u (n) τ E ( u (u))) optimal step size Key issues: Regularity of the minimizers ũ H 1 (D) = Sobolev gradients Enforcement of the constraint ũ M = Riemannian optimization

8 Formulation of the Problem Gradient Minimization Sobolev Gradients Gâteaux differential of the Gross-Pitaevskii Energy Functional E (u; v) = lim ɛ ɛ 1 [E(u + ɛv) E(u)], u, v X X some function space Riesz Representation Theorem: E (u; ) bounded linear functional on X = v X E (u; v) = X E(u), v X Relevant inner products (Danaila & Kazemi 21) u, v L2 = u, v H 1 = u, v HA = D D D u, v dx, u, v + u, v dx u, v + A u, A v dx, where u, v = uv A = + ic Ω A t

9 Formulation of the Problem Gradient Minimization Sobolev Gradients Different Sobolev gradients (X = L 2, H 1, H A ) E L2 H1 (u; v) = R L2 E(u), v = R H1 E(u), v = R H A E(u), v H A The L 2 gradient L2 E(u) = 2 ( 1 ) 2 2 u + C trap u + C g u 2 u ic Ω A t u, The Sobolev gradient G = HA E(u) obtained from the L 2 gradient via an elliptic boundary-value problem v H 1 (D) D [( 1 + C 2 Ω (x 2 + y 2 ) ) Gv + G v 2iC Ω A t Gv ] dx = D 1 2 u v + [ C trap u + C g u 2 u ic Ω A t u ] v dx

10 First-Order Geometry Second-Order Geometry Riemannian Conjugate Gradients is general approach based on differential geometry here made simple by the constraint u L2(D) = 1 Intrinsic approach with optimization performed directly on the manifold M without reference to the space H 1 (D) optimization problem becomes unconstrained can apply more efficient optimization algorithms (conjugate gradients, Newton s method) Reference: P.-A. Absil, R. Mahony and R. Sepulchre, Optimization Algorithms on Matrix Manifolds, Princeton University Press, (28).

11 First-Order Geometry Second-Order Geometry Riemannian Conjugate Gradients Projection of the gradient G on the tangent subspace T u M P un,h A G = G R ( u n, G L 2) R ( u n, v HA L 2) v H A, v HA, v HA = u n, v L 2, v H A where There is some freedom in choosing the subtracted field (v HA ) Approach equivalent to constraint enforcement via Lagrange multipliers Error in constraint satisfaction O(τ n )

12 First-Order Geometry Second-Order Geometry Riemannian Conjugate Gradients Retraction R u : T u M M maps a tangent vector ξ T u M back to the manifold M

13 First-Order Geometry Second-Order Geometry Riemannian Conjugate Gradients For our constraint manifold M R u (ξ) = u + ξ u + ξ L2 (D) retraction = normalization Riemannian steepest descent approach u n+1 = R un (τ n P un,ha G(u n )), n =, 1, 2,... u = u where τ n = argmin τ> E (R un (τp un,h A G(u n )))

14 First-Order Geometry Second-Order Geometry Riemannian Conjugate Gradients Results: Abrikosov Lattice (C g = 1, C Ω =.9) Finite-element approximation of energy functional and PDE operators implementation in FreeFEM++ (with 1 5 P1 elements) u, initial guess (Thomas-Fermi approximation) u, final state (Thomas-Fermi approximation)

15 First-Order Geometry Second-Order Geometry Riemannian Conjugate Gradients n Riemannian Grad n steepest descent for banana function step size τ n

16 First-Order Geometry Second-Order Geometry Riemannian Conjugate Gradients Consider min x R N f (x), where f : R N R Nonlinear Conjugate Gradients Method x n+1 = x n + τ n d n, n =, 1,... x = x descent direction dn is defined as d n = g n + β n d n 1, n = 1, 2,... d = g, g n = f (x n ) momentum coefficients β n ensure conjugacy of decent directions β n = β FR n := g n, g n X g n 1, g n 1 X (Fletcher-Reeves), β n = β PR n := g n, (g n g n 1 ) X g n 1, g n 1 X (Polak-Ribiére)

17 In the Riemannian setting First-Order Geometry Second-Order Geometry Riemannian Conjugate Gradients g n 1, d n 1 T xn 1 and g n, d n T xn, hence cannot be added or multiplied... Need a mapping between the tangent spaces T un 1 M and T un M Vector transport T η (ξ) : T M T M T M, ξ, η T M describing how the vector field ξ is transported along the manifold M by the field η

18 First-Order Geometry Second-Order Geometry Riemannian Conjugate Gradients Conclusions (I) Riemannian optimization accelerates the computation of BEC ground states Key enablers for Riemannian Conjugate Gradients: projections onto Tun M retractions from T un M onto M, vector transport between Tun 1 and T un Riemannian Newton s method?

19 First-Order Geometry Second-Order Geometry Riemannian Conjugate Gradients PROBLEM II Probing Fundamental Bounds in Hydrodynamics (joint work with D. Ayala and Ch. Doering)

20 Navier-Stokes equation (Ω = [, L] d, d = 2, 3) v + (v )v + p ν v =, in Ω (, T ] t v =, in Ω (, T ] v = v in Ω at t = Boundary Condition on Γ (, T ] 2D Case Existence Theory Complete smooth and unique solutions exist for arbitrary times and arbitrarily large data 3D Case Weak solutions (possibly nonsmooth) exist for arbitrary times Classical (smooth) solutions (possibly nonsmooth) exist for finite times only Possibility of blow-up (finite-time singularity formation) One of the Clay Institute Millennium Problems ($ 1M!) Equations

21 What is known? Available Estimates A Key Quantity Enstrophy E(t) v 2 dω (= v 2 2) Ω Smoothness of Solutions Bounded Enstrophy E(t) (Foias & Temam, 1989) NaN? max E(t) <??? t [,T ] E() Can estimate de(t) using the momentum equation, Sobolev s embeddings, Young and Cauchy-Schwartz inequalities,... Remark: incompressibility not used in these estimates... t t

22 2D Case: de(t) C 2 ν E(t)2 3D Case: Grönwall s lemma and energy equation yield t E(t) < smooth solutions exist for all times de(t) 27C 2 128ν 3 E(t)3 corresponding estimate not available... upper bound on E(t) blows up in finite time E() E(t) 1 4 CE()2 ν t 3 singularity in finite time cannot be ruled out!

23 Problem of Lu & Doering (28), I Can we actually find solutions which saturate a given estimate? Estimate de(t) where de(t) ce(t) 3 at a fixed instant of time t de(t) max v H 1 (Ω), v= subject to E(t) = E = ν v v v v dω Ω E is a parameter Solution using a gradient-based descent method

24 Problem of Lu & Doering (28), II [ ] de(t) = 8.97 max 1 4 E vorticity field (top branch)

25 How about solutions which saturate de(t) finite time window [, T ]? ce(t) 3 over a where E(t) = t de(τ) dτ max E(T ) v H 1 (Ω), v= subject to E() = E dτ + E E and T are parameters In principle doable, but will try something simpler first...

26 Relevant Estimates 1D Burgers instantaneous 1D Burgers finite time 2D Navier Stokes instantaneous 2D Navier Stokes finite time 3D Navier Stokes instantaneous 3D Navier Stokes finite time Best Estimate Sharp? de(t) ( 3 1 ) 1/3 E(t) 5/3 Yes 2 π 2 ν Lu & Doering (28) [ max t [,T ] E(t) E 1/3 + ( ) L 2 ( 1 ) ] 4/3 3 No E 4 π 2 ν Ayala & P. (211) dp(t) ( ν E ) P 2 + C 1 ( E ν ) P dp(t) C 2 max t> P(t) ν K 1/2 P 3/2 [ ] 2 P 1/2 + C 2 K 1/2 4ν 2 E de(t) 27C 2 128ν 3 E(t) 3 E(t) E() 1 4 CE()2 ν 3 t Yes Lu & Doering (28)

27 Question #1 ( small ) Sharpness of instantaneous estimates (at some fixed t) max u max u de dp (1D, 3D) (2D) Question #2 ( big ) Sharpness of finite time estimates (at some time window [, T ], T > ) [ max max u [ max u t [,T ] E(t) ] max P(t) t [,T ] ] (1D, 3D) (2D) E(t) E() E(t) E() max T max t t

28 Relevant Estimates 1D Burgers instantaneous 1D Burgers finite time 2D Navier Stokes instantaneous 2D Navier Stokes finite time 3D Navier Stokes instantaneous 3D Navier Stokes finite time Best Estimate Sharp? de(t) ( 3 1 ) 1/3 E(t) 5/3 Yes 2 π 2 ν Lu & Doering (28) [ max t [,T ] E(t) E 1/3 + ( ) L 2 ( 1 ) ] 4/3 3 No E 4 π 2 ν Ayala & P. (211) dp(t) ( ν E ) P 2 + C 1 ( E ν ) P dp(t) C 2 ν K 1/2 P 3/2 [ ] 2 max t> P(t) P 1/2 + C 2 K 1/2 4ν 2 E de(t) 27C 2 128ν 3 E(t) 3 E(t) E() 1 4 CE()2 ν 3 t Yes Lu & Doering (28)

29 2D vorticity equation in a periodic box (ω = e z ω) ω + J(ω, ψ) = ν ω t where ψ = ω J(f, g) = f xg y f y g x Enstrophy uninteresting in 2D flows (w/o boundaries) 1 d ω 2 dω = ν ( ω) 2 dω < 2 Ω Ω Evolution equation for the vorticity gradient ω ω t + (u ) ω = ν ω + ω u }{{} stretching term

30 Palinstrophy P(t) Ω ( ω(t, x)) 2 dω = Ω ( ψ(t, x)) 2 dω Also of interest Kinetic Energy K(t) u(t, x) 2 dω = Ω Ω ( ψ(t, x)) 2 dω Poincaré s inequality K (2π) 2 E (2π) 2 P

31 Estimates for the Rate of Growth of Palinstrophy dp(t) = J( ψ, ψ) 2 ψ dω ν ( 2 ψ) 2 dω R P (ψ) Ω dp(t) ( ν ) ( ) E P 2 + C 1 P E ν (Doering & Lunasin, 211) dp(t) C 2 ν K1/2 P 3/2 (Ayala, 212) Using Poincaré s inequality (may not be sharp) dp(t) Bound on growth in finite time max t> P(t) Ω C ν P2, [ P 1/2 + C 2 4ν 2 K1/2 E ] 2 (Ayala, 212)

32 Maximum Growth of dp(t) for fixed E >, P > (2π) 2 E max R P (ψ) where ψ S P,E S P,E = ψ H4 (Ω) : 1 2 ( ψ) 2 dω = P Ω ( ψ) 2 dω = E Ω 1 2 Maximum Growth of dp(t) for fixed K >, P > (2π) 4 K max R P (ψ) where ψ S P,K S P,K = ψ H4 (Ω) : ( ψ) 2 dω = P Ω ( ψ) 2 dω = K Ω

33 y y Small Palinstrophy Limit: P (2π) 2 E ϕ = arg max R (ϕ), R (ϕ) = ν ( 2 ϕ) 2 dω, ϕ S Ω { S = ϕ H 4 (Ω) : 1 } ( ψ) 2 dω = (2π)2 ( ψ) 2 dω 2 2 Ω Ω Optimizers: Eigenfunctions of the Laplacian ( ϕ Ker( )) x ϕ(x, y) = sin(π(y x)) sin(π(y +x)) x ϕ(x, y) = sin(2πx) sin(2πy)

34 Numerical Solution of Maximization Problem Discretization of Gradient Flow dψ dτ = H4 R ν (ψ), ψ() = ψ ψ (n+1) = ψ (n) τ (n) H4 R ν (ψ (n) ), ψ () = ψ Gradient in H 4 (Ω) (via variational techniques) [ Id L 8 4] H4 R ν = L 2 R ν (Periodic BCs) L 2 R ν (ψ) = 2 J( ψ, ψ) + J(ψ, 2 ψ) + J( 2 ψ, ψ) 2ν 4 ψ Constraint satisfaction via arc minimization

35 y y y y Maximizers with Fixed (K, P ) Estimate: dp(t) C 2 ν K 1/2 P 3/ RP ( ψ) P max dp vs. P, K = x (a) P 1P c x (b) P 1P c RP ( ψ) P max dp P 3/2 as P x (c) P 1 4 P c (d) P 1 4 P c x 3

36 y y y y Maximizers with Fixed (E, P ) Estimate: dp(t) ( ν E ) P 2 + C 1 ( E ν ) P RP ( ψ) x x P max dp vs. P, E = 1 3 (a) P P c (b) P 1P c RP ( ψ) P max dp vs. P x (c) P 1 2 P c x (d) P 1 7/2 P c

37 y y x x y y x x Maximizers with Fixed (K, P ) Finite-Time Estimate: max t> P(t) [ P 1/2 + C 2 4ν 2 K 1/2 E ] max P(t) t> 1 7 E P -constraint {K, P }-constraint E (a) t =.213 (b) t =.458 (c) t =.633 (d) t =.1265

38 Relevant Estimates 1D Burgers instantaneous 1D Burgers finite time 2D Navier Stokes instantaneous 2D Navier Stokes finite time 3D Navier Stokes instantaneous 3D Navier Stokes finite time Best Estimate Sharp? de(t) ( 3 1 ) 1/3 E(t) 5/3 Yes 2 π 2 ν Lu & Doering (28) [ max t [,T ] E(t) E 1/3 + ( ) L 2 ( 1 ) ] 4/3 3 No E 4 π 2 ν Ayala & P. (211) dp(t) max t> P(t) ( ν E ) P 2 + C 1 ( E ν ) P dp(t) C 2 ν K 1/2 P 3/2 [Yes] Ayala & P. (213) [ P 1/2 + C 2 4ν 2 K 1/2 E ] 2 [Yes] Ayala & P. (213) de(t) 27C 2 128ν 3 E(t) 3 E(t) E() 1 4 CE()2 ν 3 t Yes Lu & Doering (28)???

39 Conclusions (II) Variational analysis and numerical optimization as tools for assessing sharpness of PDEs analysis motivated by big open questions in mathematical fluid mechanics (including one of Clay Millennium Problems ) 2D extreme vortex states saturate the worst-case bounds analysis is sharp Ongoing work on the 3D case (the real problem) so far, no evidence of blow-up, but results still from conclusive...

40 References L. Lu and C. R. Doering, Limits on Enstrophy Growth for Solutions of the Three-dimensional Navier-Stokes Equations Indiana University Mathematics Journal 57, , 28. D. Ayala and B. Protas, On Maximum Enstrophy Growth in a Hydrodynamic System, Physica D 24, , 211. D. Ayala and B. Protas, Maximum Palinstrophy Growth in 2D Incompressible Flows: Instantaneous Case, Journal of Fluid Mechanics , 214. D. Ayala and B. Protas, Vortices, Maximum Growth and the Problem of Finite-Time Singularity Formation, Fluid Dynamics Research (Special Issue for IUTAM Symposium on Vortex Dynamics), 46, 3144, 214. D. Ayala and B. Protas, Extreme Vortex States and the Growth of Enstrophy in 3D Incompressible Flows, (submitted; see arxiv: ), 216.