PDE Constrained Shape Optimization as Optimization on Shape Manifolds

28/10/2015
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The novel Riemannian view on shape optimization introduced in [14] is extended to a Lagrange–Newton as well as a quasi–Newton approach for PDE constrained shape optimization problems.

PDE Constrained Shape Optimization as Optimization on Shape Manifolds

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application/pdf PDE Constrained Shape Optimization as Optimization on Shape Manifolds Kathrin Welker, Volker Schulz, Martin Siebenborn

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            <title>PDE Constrained Shape Optimization as Optimization on Shape Manifolds</title></titles>
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        <publicationYear>2015</publicationYear>
        <resourceType resourceTypeGeneral="Text">Text</resourceType><subjects><subject>Shape optimization</subject><subject>Riemannian manifold</subject><subject>Newton method</subject><subject>Quasi–Newton method</subject><subject>Limited memory BFGS</subject></subjects><dates>
	    <date dateType="Created">Sun 8 Nov 2015</date>
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            <date dateType="Submitted">Mon 15 Oct 2018</date>
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The novel Riemannian view on shape optimization introduced in [14] is extended to a Lagrange–Newton as well as a quasi–Newton approach for PDE constrained shape optimization problems.

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PDE Constrained Shape Optimization as Optimization on Shape Manifolds Volker H. Schulz, Martin Siebenborn and Kathrin Welker Trier University GSI 2015 2nd Conference on Geometric Science of Information Ecole Polytechnique October 30th, 2015 Outline 1 Introduction 2 Lagrange–Newton approach 3 Quasi–Newton approach 4 Conclusion and outlook K. Welker (Trier University) Shape Optimization on Shape Manifolds October 30th, 2015 1 / 27 Constrained shape optimization problem min Ω J(Ω) s.t. PDE constraints • J real–valued shape differentiable objective function • Ω shape, i.e., simply connected and compact subset of R2 with Ω /= ∅ and C∞ boundary ∂Ω K. Welker (Trier University) Shape Optimization on Shape Manifolds October 30th, 2015 2 / 27 Constrained shape optimization problem min Ω J(Ω) s.t. PDE constraints • J real–valued shape differentiable objective function • Ω shape, i.e., simply connected and compact subset of R2 with Ω /= ∅ and C∞ boundary ∂Ω How does the set of all these shapes look like? K. Welker (Trier University) Shape Optimization on Shape Manifolds October 30th, 2015 2 / 27 Shape space Definition (cf. [2]) Let M be compact manifold and N a Riemannian manifold with dim(M) < dim(N). The space of all submanifolds of M in N is defined as Be(M, N) = Emb(M, N)/Diff(N). b1 ∈ Be(S2 , R3 ) b2 /∈ Be(S2 , R3 ) c1 ∈ Be(S1 , R2 ) c2 /∈ Be(S1 , R2 ) [2] P. Michor and D. Mumford. Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms. Documenta Mathematica, 10:217–245, 2005. K. Welker (Trier University) Shape Optimization on Shape Manifolds October 30th, 2015 3 / 27 Shape manifold The shape space Be = Be(S1 , R2 ) with the Sobolev metric g1 Tc Be × Tc Be → R, (h, k) ↦ ∫ c=∂Ω αβ + Aα′ β′ ds = ((I − A c )α, β)L2(c) is a Riemannian manifold for A > 0 (cf. [2]). h = αn, k = βn denote two elements from the tangent space Tc Be ≅ {h h = αn, α ∈ C∞ (S1 , R)}. [2] P. Michor and D. Mumford. Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms. Documenta Mathematica, 10:217–245, 2005. K. Welker (Trier University) Shape Optimization on Shape Manifolds October 30th, 2015 4 / 27 Shape manifold The shape space Be = Be(S1 , R2 ) with the Sobolev metric g1 Tc Be × Tc Be → R, (h, k) ↦ ∫ c=∂Ω αβ + Aα′ β′ ds = ((I − A c )α, β)L2(c) is a Riemannian manifold for A > 0 (cf. [2]). h = αn, k = βn denote two elements from the tangent space Tc Be ≅ {h h = αn, α ∈ C∞ (S1 , R)}. Riemannian shape gradient Let (M, g) denote a Riemannian manifold. The Riemannian shape gradient is a representation of DJ(Ω) such that DJ(Ω)[V ] = g(grad J(Ω), V ), ∀V ∈ TΩM. If the shape derivative is given in the form DJ(Ω)[V ] = ∫c γ ⟨V , n⟩ ds and (Be, g1 ) is considered, then the Riemannian shape gradient grad J is given as the normal vector field grad J = qn with (I − A c )q = γ. [2] P. Michor and D. Mumford. Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms. Documenta Mathematica, 10:217–245, 2005. K. Welker (Trier University) Shape Optimization on Shape Manifolds October 30th, 2015 4 / 27 Solution techniques min Ω J(Ω) s.t. PDE constraints Find a critical point of the optimization problem, i.e., solve the Karush–Kuhn–Tucker (KKT) or first order necessary optimality conditions gradL(ξ) = 0 (∗) where L denotes the Lagrangian to the optimization problem. Newton or quasi–Newton method to equation (∗) • Newton method – k. iteration: compute increment ∆ξ as solution of HessL(ξk )∆ξ = −gradL(ξk ) • quasi–Newton method – k. iteration: compute update formula Hk for HessL(ξk ) or the increment ∆ξ = −H−1 k gradL(ξk ) K. Welker (Trier University) Shape Optimization on Shape Manifolds October 30th, 2015 5 / 27 Outline 1 Introduction 2 Lagrange–Newton approach 3 Quasi–Newton approach 4 Conclusion and outlook K. Welker (Trier University) Shape Optimization on Shape Manifolds October 30th, 2015 6 / 27 Problem formulation • interface problem • Ω = (0,1)2 ⊂ R2 • Ω1,Ω2 ⊂ Ω with ∂Ω1 ⋂∂Ω2 = Γ, Ω1 ⊍Γ⊍Ω2 = Ω Ω1 Ω2 Γint The interface Γint is an element of B0 e([0,1],R2 ) = Emb0 ([0,1],R2 )/Diff0 ([0,1]) where Emb0 ([0,1],R2 ) = {φ ∈ C∞ ([0,1],R2 ) φ(0) = (0.5,0),φ(1) = (0.5,1),φ injective immersion}, Diff0 ([0,1]) = {φ [0,1] → [0,1] φ(0) = (0.5,0),φ(1) = (0.5,1),φ diffeomorphism}. K. Welker (Trier University) Shape Optimization on Shape Manifolds October 30th, 2015 7 / 27 Poisson–type model problem min Γint J(Ω) ≡ 1 2 ∫ Ω (y − ¯y)2 dx + µ∫ Γint 1ds s.t. − y = f in Ω y = 0 on ∂Ω jumping coefficient: f ≡ ⎧⎪⎪ ⎨ ⎪⎪⎩ f1 = const. in Ω1 f2 = const. in Ω2 interface conditions (continuity of the state and the flux at Γint): y = 0, ∂y ∂n = 0 on Γint • n unit outer normal to Ω1 at Γint • jump symbol ⋅ denotes the discontinuity across the interface Γint and is defined by v = v1 − v2 where v1 = v Ω1 and v2 = v Ω2 K. Welker (Trier University) Shape Optimization on Shape Manifolds October 30th, 2015 8 / 27 Lagrangian, adjoint problem and shape derivative Lagrangian L(y, Γint, p) = J(Ω) + a(y, p) − b(p) = 1 2 ∫ Ω (y − ¯y)2 dx + µ ∫ Γint 1ds + ∫ Ω ∇yT ∇pdx − ∫ Γint ∂y ∂n p ds − ∫ Ω fpdx Adjoint problem − p = − (y − y) in Ω p = 0 on ∂Ω interface conditions: p = 0, ∂z ∂n = 0 on Γint Shape derivative DJ(Ω)[V ] = ∫ Γint ⟨V , n⟩ (− f p + µκ) ds (κ curvature corresponding to n) K. Welker (Trier University) Shape Optimization on Shape Manifolds October 30th, 2015 9 / 27 Lagrange–Newton approach gradL(ξ) = 0, ξ = (y, Γint, p) ∈ F = {(H1 0 (Ω), Γint, H1 0 (Ω)) Γint ∈ B0 e([0, 1], R2 )} Newton method to this equation – k. iteration: compute increment ∆ξ as solution of HessL(ξk )∆ξ = −gradL(ξk ) (∗) Solve equation (∗) in a weak formulation, i.e., for all h = (z, w, q)⊺ ∈ T(y,Γint,p)F: H11(z, z) + H12(w, z) + H13(q, z) = −a(z, p) − ∂ ∂y J(Ω)z H21(z, w) + H22(w, w) + H23(q, w) = −DL(y, Γint, p)[w] H31(z, q) + H32(w, q) + H33(q, q) = −a(y, q) + b(q) Key–observation in [3]: H22 symmetric in the solution of the optimization problem This observation motivates a Riemannian SQP method where away from the solution only expressions in H22 are used which are nonzero at the solution. [3] V. Schulz. A Riemannian View on Shape Optimization. Foundations of Comput. Math., 14:483–501, 2014. K. Welker (Trier University) Shape Optimization on Shape Manifolds October 30th, 2015 10 / 27 Equivalent linear–quadratic problem If the term H22(w,w) = g (HessL(y,Γint,p)w,w) is replaced by an approximation ˆH22(w,w) such that • ˆH22(w,w) omits all terms in H22(w,w), which are zero at the solution • the reduced Hessian of HessL built with the approximation ˆH22 is coercive then equation (∗) is for h = (z,w,q)⊺ equivalent to the linear–quadratic problem min (z,w) 1 2 (H11(z,z) + 2H12(w,z) + ˆH22(w,w)) + ∂ ∂y J(Ω)z + DJ(Ω)[w] s.t. a(z, ¯q) + D (a(y,q) − b(q))[w] = −a(y, ¯q) + b(¯q), ∀¯q ∈ H(u) K. Welker (Trier University) Shape Optimization on Shape Manifolds October 30th, 2015 11 / 27 Poisson–type model problem – QP min (z,w) 1 2 (H11(z, z) + 2H12(w, z) + ˆH22(w, w)) + ∂ ∂y J(Ω)z + DJ(Ω)[w] s.t. a(z, ¯q) + D (a(y, q) − b(q)) [w] = −a(y, ¯q) + b(¯q) , ∀¯q ∈ H(u) Quadratic problem – optimal control problem min (z,w) ∫ Ω z2 2 + (y − ¯y)z dx + ∫ Γint µκw − f pw ds + 1 2∫ Γint µ ( ∂w ∂τ ) 2 − f κpw2 ds s.t. − z = y + f in Ω ∂z ∂n = f1w, − ∂z ∂n = f2w on Γint z = 0 on ∂Ω ( ∂ ∂τ derivative tangential to Γint, κ curvature) adjoint problem: − q = −z − (y − ¯y) in Ω q = 0 on ∂Ω design equation: 0 = − f (p + κpw + q) + µκ − µ∂2 w ∂τ2 on Γint K. Welker (Trier University) Shape Optimization on Shape Manifolds October 30th, 2015 12 / 27 Optimization algorithm evaluate measurements solve adjoint PDE solve optimal control problem (QP) solve linear elasticity equa- tions and deform mesh QP is solved by a CG–iteration for the reduced problem (design equation), i.e., iterate over w ⇒ each time the CG–iteration needs a residual of the design equation from wk : 1. compute the state variable zk from the optimal control problem (QP) 2. compute the adjoint variable qk from the adjoint equation 3. compute the residual rk = − f (p + κpwk + qk ) + µκ − µ∂2 wk ∂τ2 K. Welker (Trier University) Shape Optimization on Shape Manifolds October 30th, 2015 13 / 27 Implementation details • f ≡ ⎧⎪⎪ ⎨ ⎪⎪⎩ f1 = 1000 in Ω1 f2 = 1 in Ω2 • µ = 10 K. Welker (Trier University) Shape Optimization on Shape Manifolds October 30th, 2015 14 / 27 Implementation details • f ≡ ⎧⎪⎪ ⎨ ⎪⎪⎩ f1 = 1000 in Ω1 f2 = 1 in Ω2 • µ = 10 • data y are generated from a solution of the state equation with Γint being the straight line connection of the points (0.5,0) and (0.5,1) K. Welker (Trier University) Shape Optimization on Shape Manifolds October 30th, 2015 14 / 27 Implementation details • f ≡ ⎧⎪⎪ ⎨ ⎪⎪⎩ f1 = 1000 in Ω1 f2 = 1 in Ω2 • µ = 10 • data y are generated from a solution of the state equation with Γint being the straight line connection of the points (0.5,0) and (0.5,1) • starting point of the iterations is described by a B–spline defined by the two control points (0.6,0.7) and (0.4,0.3) K. Welker (Trier University) Shape Optimization on Shape Manifolds October 30th, 2015 14 / 27 Implementation details • f ≡ ⎧⎪⎪ ⎨ ⎪⎪⎩ f1 = 1000 in Ω1 f2 = 1 in Ω2 • µ = 10 • data y are generated from a solution of the state equation with Γint being the straight line connection of the points (0.5,0) and (0.5,1) • starting point of the iterations is described by a B–spline defined by the two control points (0.6,0.7) and (0.4,0.3) • distance of each shape uk to the solution u∗ is approximated by dist(uk ,u∗ ) = ∫ u∗ ⟨uk ,(1,0)T ⟩ − 1 2 ds K. Welker (Trier University) Shape Optimization on Shape Manifolds October 30th, 2015 14 / 27 Numerical results ([5]) It.-No. Ω1 h ≈ 6000 triangles Ω2 h ≈ 24000 triangles Ω3 h ≈ 98000 triangles 0 0.0705945 0.070637 0.0706476 1 0.0043115 0.004104 0.0040465 2 0.0003941 0.000104 0.0000645 ⇒ quadratic convergence on the finest grid [5] V. Schulz, M. Siebenborn, and K. W. Towards a Lagrange–Newton approach for PDE constrained shape optimization. In G. Leugering et al., Trends in PDE Constrained Optimization, volume 165 of International Series of Numerical Mathematics. Springer, 2014. K. Welker (Trier University) Shape Optimization on Shape Manifolds October 30th, 2015 15 / 27 Outline 1 Introduction 2 Lagrange–Newton approach 3 Quasi–Newton approach 4 Conclusion and outlook K. Welker (Trier University) Shape Optimization on Shape Manifolds October 30th, 2015 16 / 27 Problem formulation • interface identification problem • diffusion process • homogeneous concentration in Ω for t = 0 • higher concentration on the top in the beginning • two materials with different permeability • variable boundary Γint ∈ Be(S1 ,R2 ) ⇒ fit to measured concentration Ω2 Ω1 Γtop Γint Γleft Γright Γbottom n • Ω ⊂ R2 open with Ω /= ∅ • Ω1,Ω2 ⊂ Ω with ∂Ω1 ⋂∂Ω2 = Γint, Ω1 ⊍Γint ⊍Ω2 = Ω K. Welker (Trier University) Shape Optimization on Shape Manifolds October 30th, 2015 17 / 27 Parabolic diffusion model problem min Γint J(Ω) = j(Ω) + jreg (Ω) ≡ 1 2 ∫ T 0 ∫ Ω (y − ¯y)2 dx ds + µ ∫ Γint 1 ds s.t. ∂y ∂t − div(k∇y) = f in Ω × (0, T] y = 1 on Γtop × (0, T] ∂y ∂n = 0 on (Γbottom ∪ Γleft ∪ Γright) × (0, T] y = y0 in Ω × {0} k ≡ ⎧⎪⎪ ⎨ ⎪⎪⎩ k1 = const. in Ω1 × (0, T] k2 = const. in Ω2 × (0, T] f ≡ const. in Ω × (0, T] interface conditions (continuity of the state and the flux at Γint): y = 0, k ∂y ∂n = 0 on Γint × (0, T] K. Welker (Trier University) Shape Optimization on Shape Manifolds October 30th, 2015 18 / 27 Adjoint equation − ∂p ∂t − div(k∇p) = −(y − y) in Ω × [0, T) p = 0 on Γtop × [0, T) ∂p ∂n = 0 on (Γbottom ∪ Γleft ∪ Γright) × [0, T) p = 0 in Ω × {T} p = 0 on Γint × [0, T) k ∂p ∂n = 0 on Γint × [0, T) p1 = −k1p on (Γbottom ∪ Γleft ∪ Γright) × [0, T) p2 = k1 ∂p ∂n on Γtop × [0, T) K. Welker (Trier University) Shape Optimization on Shape Manifolds October 30th, 2015 19 / 27 Shape derivative In many cases, the shape derivative arises in two equivalent national forms: DJΩ(Ω)[V ] = ∫ Ω F(x)V (x) dx (domain formulation) DJΓ(Ω)[V ] = ∫ Γ l(s) ⟨V (s), n(s)⟩ ds (boundary formulation) where F(x) is a (differential) operator acting linearly on the perturbation vector field V and l Γ → R with DJΩ(Ω)[V ] = DJ(Ω)[V ] = DJΓ(Ω)[V ]. K. Welker (Trier University) Shape Optimization on Shape Manifolds October 30th, 2015 20 / 27 Shape derivative In many cases, the shape derivative arises in two equivalent national forms: DJΩ(Ω)[V ] = ∫ Ω F(x)V (x) dx (domain formulation) DJΓ(Ω)[V ] = ∫ Γ l(s) ⟨V (s), n(s)⟩ ds (boundary formulation) where F(x) is a (differential) operator acting linearly on the perturbation vector field V and l Γ → R with DJΩ(Ω)[V ] = DJ(Ω)[V ] = DJΓ(Ω)[V ]. DjΩ(Ω)[V ] + ∫ Γint ⟨V , n⟩ µκ ds = dJ(Ω)[V ] = DjΓint (Ω)[V ] + ∫ Γint ⟨V , n⟩ µκ ds DjΩ(Ω)[V ] = ∫ T 0 ∫ Ω −k∇y⊺ (∇V + ∇V ⊺ ) ∇p − p∇f ⊺ V + div(V ) ( 1 2 (y − y)2 + ∂y ∂t p + k∇y⊺ ∇p − fp) dx dt DjΓ(Ω)[V ] = ∫ T 0 ∫ Γint k ∇y⊺ 1 ∇p2 ⟨V , n⟩ ds dt K. Welker (Trier University) Shape Optimization on Shape Manifolds October 30th, 2015 20 / 27 Riemannian limited BFGS update limited memory BFGS ρk ← 1 g1(wk ,sk ) q ← gradJ(ck ) for i = k − 1,...,k − m do si ← Tsi wi ← Twi αi ← ρi g1 (si ,q) q ← q − αi di end for z ← gradJ(ck ) q ← g1 (wk−1,sk−1) g1(wk−1,wk−1) gradJ(ck ) for i = k − m,...,k − 1 do βi ← ρi g1 (wi ,z) q ← q + (αi − βi )si end for return q = H−1 k gradJ(ck ) • implements quasi–Newton update formula for H−1 k • computes the BFGS update H−1 k gradJ(ck) • memory contains only m shape gradients • sk distance between iterated shapes • wk difference of iterated shape gradients • T vector transport of elements in tangential space to updated shape K. Welker (Trier University) Shape Optimization on Shape Manifolds October 30th, 2015 21 / 27 Implementation details • µ = 0.0001 , T = 20 , A = 0.001, k ≡ ⎧⎪⎪ ⎨ ⎪⎪⎩ k1 = 1 in Ω1 k2 = 0.001 in Ω2 K. Welker (Trier University) Shape Optimization on Shape Manifolds October 30th, 2015 22 / 27 Implementation details • µ = 0.0001 , T = 20 , A = 0.001, k ≡ ⎧⎪⎪ ⎨ ⎪⎪⎩ k1 = 1 in Ω1 k2 = 0.001 in Ω2 • data y are generated from a solution of the state equation for the setting Ω2= {x x 2 ≤ 0.5} K. Welker (Trier University) Shape Optimization on Shape Manifolds October 30th, 2015 22 / 27 Implementation details • µ = 0.0001 , T = 20 , A = 0.001, k ≡ ⎧⎪⎪ ⎨ ⎪⎪⎩ k1 = 1 in Ω1 k2 = 0.001 in Ω2 • data y are generated from a solution of the state equation for the setting Ω2= {x x 2 ≤ 0.5} K. Welker (Trier University) Shape Optimization on Shape Manifolds October 30th, 2015 22 / 27 Numerical results ([4]) 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 1E-05 0.0001 0.001 0.01 0.1 full BFGS l5-BFGS l2-BFGS gradient • BFGS methods are superior to the gradient method • superlinear convergence of limited BFGS methods • different number of gradients stored – successful with very few gradients in memory [4] V. Schulz, M. Siebenborn, K. W. Structured inverse modeling in parabolic diffusion problems. submitted to SICON, 2014. K. Welker (Trier University) Shape Optimization on Shape Manifolds October 30th, 2015 23 / 27 Outline 1 Introduction 2 Lagrange–Newton approach 3 Quasi–Newton approach 4 Conclusion and outlook K. Welker (Trier University) Shape Optimization on Shape Manifolds October 30th, 2015 24 / 27 Conclusion Lagrange–Newton approach • shape gradient for Poisson–type interface problem • Riemannian SQP method • quadratic convergence rates on the finest grid K. Welker (Trier University) Shape Optimization on Shape Manifolds October 30th, 2015 25 / 27 Conclusion Lagrange–Newton approach • shape gradient for Poisson–type interface problem • Riemannian SQP method • quadratic convergence rates on the finest grid Quasi–Newton approach • shape gradient for parabolic interface problem • Riemannian limited BFGS update • superlinear convergence rates – successful with very few gradients in memory K. Welker (Trier University) Shape Optimization on Shape Manifolds October 30th, 2015 25 / 27 Outlook • optimize only pieces of a shape • consider domains with Lipschitz–boundaries or Ck –boundaries, k < ∞ • use the domain expression of the shape derivative to deform the mesh ⇒ done in [6] Volker H. Schulz, Martin Siebenborn, and K. W. A novel Steklov–Poincaré type metric for efficient PDE constrained optimiziation in shape space. 2015. submitted to SIOPT. Outlook Analyze the new shape manifold introduced in [6] and which is different from the shape space Be. K. Welker (Trier University) Shape Optimization on Shape Manifolds October 30th, 2015 26 / 27 References [1] P. A. Absil, R. Mahony, and R. Sepulchre. Optimization algorithms on matrix manifolds. Princeton University Press, 2008. [2] P. M. Michor and D. Mumford. An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach. 23:74–113, 2007. [3] V. H. Schulz. A Riemannian view on shape optimization. Foundations of Computational Mathematics, 14:483–501, 2014. [4] V. H. Schulz, M. Siebenborn, and K. W. Structured inverse modeling in parabolic diffusion problems. 2014. submitted to SICON. [5] V. H. Schulz, M. Siebenborn, and K. W. Towards a Lagrange–Newton approach for PDE constrained shape optimization. In G. Leugering et al., Trends in PDE Constrained Optimization, volume 165 of International Series of Numerical Mathematics. Springer, 2014. [6] V. H. Schulz, M. Siebenborn, and K. W. A novel Steklov–Poincaré type metric for efficient PDE constrained optimiziation in shape space. 2015. submitted to SIOPT. Thank you for your attention! K. Welker (Trier University) Shape Optimization on Shape Manifolds October 30th, 2015 27 / 27