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Inverse problem of shape identification from boundary measurement for Stokes equations: Shape differentiability of Lagrangian

For Stokes equations under divergence-free andmixed boundary conditions, the inverse problem of shape identification from boundary measurement is investigated. Taking the least-square misfit as an objective function, the state-constrained optimization is treated by using an adjoint state within the Lagrange approach. The directional differentiability of a Lagrangian functionwith respect to shape variations is proved within the velocity method, and a Hadamard representation of the shape derivative by boundary integrals is derived explicitly. The application to gradient descent methods of iterative optimization is discussed.


Introduction
In the present paper, we prove the shape derivative for optimal value of a Lagrange function, which describes the inverse Stokes problem of shape identification by a least-square misfit from boundary measurements.
In a broad scope, optimization of shapes is a specific class of inverse problems; look at the survey [22]. The shape optimization is ill-posed in general because objectives have typically many local minima when varying shapes. For the theory of coefficient inverse problems, see [4], its applications in mathematical physics [36], and proper regularization in [23]. We cite [3] for the least-square method, [1,7,32] for the use of least squares in inverse scattering by obstacles, and [16] for coefficient identification in variational inequalities. Our special interest concerns variational fracture models for nonpenetrating cracks in solids [26] and their differentiability with respect to crack perturbations [27,31], which are useful for optimal control [37,38,47], overdetermined [34] and inverse problems [19,25]. The Stokes equations under consideration can be interpreted within the incompressible elasticity in solid mechanics; in this sense, the shape derivative is linked to Griffith's fracture criteria (see [2,40]).
The shape optimization in fluid mechanics was developed in [42,44]. We refer to [35,41] for the mathematical theory of incompressible flows described by Stokes and Navier-Stokes equations, to [28,29] for flow in channels and thin layers, and [5,39] for mixed variational formulations provided by boundary conditions. The overdetermined problems were studied in [11] with respect to well-posedness; the optimality condition under mixed control-state constraints was obtained in [10]. In the optimization context, Bernoullitype free boundary problems [6,24] and the coefficient identification [18] were based on least squares, while the reconstruction of obstacles immersed in a fluid by boundary measurements in [45] utilized the enclosure method.
A shape derivative is of the first importance in shape optimization because determining the first-order optimality condition with respect to perturbation of geometry. The variational approach involving shape derivatives was developed in [48] and further extended to constrained PDE models; see [12,20,21] and other works. The geometry-dependent function space formulation needs a bijective change of coordinates to transform a shape-perturbed problem to the reference geometry. The bijectivity is, however, not always the case when dealing with feasible sets due to constraints such as contact, incompressibility (divergencefree), etc. For comparison, preserving the divergence, in [43], the Piola transform was suggested to treat unsteady problems in generalized Bochner spaces on moving non-cylindrical domains. In a general case, the abstract formalism of directional derivatives of a minimax function [8] was successful to justify the shape differentiability for constrained problems within the Lagrange approach [9].
In [30], the Lagrange method was applied first to derive the shape derivative of strain energy (the Griffith formula) for curvilinear cracks constrained by the nonpenetration inequality, and for breaking-line identification under state constraints [13]. Recently, we studied Stokes and Brinkman-Stokes equations subject to the divergence-free equality with respect to the shape differentiability of its energy function [14,33]. For objective functions given in a general form, the Hadamard formula of shape derivatives by boundary integrals was formally used in [46]. In the current paper, we prove rigorously the shape differentiability of the least-square objective using the equivalent Lagrange formulation for Stokes equations and its adjoint state. The obtained analytical expression of the shape derivative and the respective Hadamard representation are advantageous for gradient descent algorithms solving the inverse problem of shape identification from boundary measurements (see Corollary 5.2).
In Section 2, we set well-posedness for a geometry-dependent forward Stokes problem under mixed boundary condition (see Proposition 2.1), existence and non-uniqueness for the inverse identification problem (see Proposition 2.2). In Section 3, an equivalent saddle-point formulation with the adjoint Stokes problem is presented in Theorem 3.1 and Corollaries 3.2 and 3.3. Based on Traits 1-4, the main theorem (Theorem 4.1) on shape derivative of the corresponding optimal value Lagrange function is proved in Section 4 and Appendix A. The Hadamard formula is established in Theorem 5.1 in Section 5, which provides an identification strategy based on the descent gradient method.

Inverse Stokes problem of shape identification
We start with the description of a family of parameter-dependent geometries where D ⊂ ℝ d is a bounded hold-all set. For every fixed time parameter t, let Ω t be a domain with the Lipschitz continuous boundary ∂Ω t and the unit normal vector n t = (n t 1 , . . . , n t d ) ⊤ which is outward to Ω t . Here and in what follows, the upper script ⊤ swaps between rows and columns. We assume that ∂Ω t consists of two nonempty, mutually disjoint sets Γ D t and Γ N t . For a given stationary force f(x) = (f 1 , . . . , f d ) ⊤ ∈ H 1 (D) d and the fluid viscosity μ > 0, we consider the forward Stokes problem under mixed Dirichlet-Neumann boundary conditions: find a flow velocity vector u t (x) = (u t 1 , . . . , u t d ) ⊤ and a pressure p t (x) satisfying the following relations (see [5,18]): (2.2c) Here the notation stands for the gradient vector ∇ := (∂/∂x 1 , . . . , ∂/∂x d ) ⊤ , the divergence div := (∇ ⋅ ), and the Laplace operator ∆. The gradient of a vector is defined as ∇u t = (∂u t i /∂x j ) d i,j=1 , the linearized strain ε(u t ) in (2.2b) is a d-by-d symmetric matrix, and ε(u t )n t in (2.2c) implies a matrix-vector multiplication.
Taking into account the no-slip boundary condition in (2.2c), we introduce the Sobolev space of admissible velocity vectors 3) The incompressibility condition in (2.2a) is determined well by the mapping where dot (⋅) in (2.5a) implies the scalar product of second order tensors.
Proof. The equilibrium equation in (2.2a) can be rewritten equivalently using by virtue of incompressibility and (2.2b). Then formulation (2.5) can be derived by the standard variational technique multiplying equations (2.2a) with the corresponding test functions w, λ and integrating them over Ω t , with the subsequent integration of the first equation by part using boundary conditions (2.2c). The quadratic term in (2.5a) determines a bounded, symmetric, bilinear quadratic form, which is strongly elliptic by the Korn-Poincaré inequality, Since Γ N t ̸ = 0, the inf-sup (LBB) condition holds for λ ∈ L 2 (Ω t ) (see [35]), Then the mapping in (2.4) is surjective, and the Ladyzhenskaya-Babuška-Brezzi-Nečas theorem follows that there exists the unique solution to (2.5).
Our long-term aim is to identify Ω t by the shape optimization approach as described in [13]. Let be an observation given at a part Γ O t ⊂ Γ N t of the boundary ∂Ω t . We introduce a least-square L 2 -misfit from the observation as the geometry-dependent objective where |w| = √ w ⊤ w denotes the Euclidean norm of vectors. In the hold-all domain D with a fixed part D D ⊂ D, admissible geometries form a set For variable shapes from (2.9) parameterized by t, the inverse Stokes problem consists in a state-constrained (MPEC) optimization problem: find Ω t such that Let Ω ⊂ ℝ d be a domain with the Lipschitz continuous boundary ∂Ω consisting of two nonempty, mutually disjoint sets Γ D and Γ N such that Ω ⊂ D. We say that Ω is a feasible geometry if there exists a pair (z, p z ) ∈ V(Ω) × L 2 (Ω) which satisfies the variational equations from Proposition 2.1 stated in Ω, Proof. If z is feasible, thus (z, p z ) satisfy (2.11), then J(u t ; Ω t ) in (2.10) attains the minimal value zero as Ω t = Ω and (u t , p t ) = (z, p z ).

Lagrange formulation using adjoint state
To express the state-constrained optimization problem (2.10) in a form suitable for analysis and numerical solution, we apply the Lagrange approach.
Based on the objective J from (2.8) and state equations (2.5), the Lagrangian function L : We formulate the corresponding saddle-point (minimax) problem: find a solution quadruple The primal state (u t , p t ) ∈ U(Ω t ) solves the forward Stokes problem (2.5). The adjoint state (v t , q t ) ∈ U(Ω t ) is the unique solution to Stokes equations (see [46]), which describes the boundary-value relations Proof. The former maximization problem in (3.2) after shortening reads for all test functions (v, q) ∈ U(Ω t ). Testing (3.6) with v = v t ± w and q = q t , we get equality (2.5a); inserting v = v t and q = q t ± λ leads to (2.5b). Conversely, from (2.5a) with w = v − v t and div(u t ) = 0 according to (2.5b), we arrive at (3.6) which hold with the equality sign. The latter minimization problem in (3.2) after shortening reads after dividing it with s ̸ = 0 and then passing the parameter s → 0, it follows equality (3.4a). The substitution of u = u t and p = p t ± λ in (3.7) follows (3.4b). Conversely, testing (3.4a) with w = u − u t such that using the incompressibility div(v t ) = 0 in (3.4b) and the convexity of J, Based on the well-posedness assertion proved in Theorem 3.1, as a corollary, we claim the equivalence below.

Corollary 3.2.
An equivalent formulation of the shape optimization problem (2.10) using adjoint state reads: find Ω t such that (3.10) Proof. For solutions (u t , p t ) to the reference problem (2.5) and (u t , p t , v t , q t ) to the minimax problem (3.2), the optimal value function ℓ : (t 0 , t 1 ) → ℝ defined as ℓ(t) := J(u t ; Ω t ) allows an equivalent representation derived straightforwardly from (2.8) and (3.1). This proves the assertion.
For a small perturbation parameter s ∈ (0, t 1 − t), we consider a perturbed according to (2.1) geometry Ω t+s ⊂ D with the Dirichlet, observation and Neumann boundaries Γ D t+s , Γ O t+s ⊂ Γ N t+s . Recalling the notation U(Ω t+s ) = V(Ω t+s ) × L 2 (Ω t+s ), the space V(Ω t+s ) is defined in accordance with (2.3) as The perturbed Stokes problem (2.5) reads: find (u t+s , p t+s ) ∈ U(Ω t+s ) such that The corresponding (v t+s , q t+s ) ∈ U(Ω t+s ) solves the perturbed adjoint equations To attain a minimum in (2.10), we look for a decreasing optimal value function ℓ(t + s) = J(u t+s ; Ω t+s ) = L(u t+s , p t+s , v t+s , q t+s ; Ω t+s ) < ℓ(t).
If the asymptotic expansion ℓ(t + s) = ℓ(t) + s∂ t J(u t ; Ω t ) + o(s) holds as s → 0 + , then we aim at the descent direction as common for gradient numerical methods, Corollary 3.3. If the one-sided limit ∂ t ℓ(t) := lim s→0 + (ℓ(t + s) − ℓ(t))/s exists, then the right derivative has two equivalent formulations Proof. Indeed, assertion (3.13) follows straightforwardly from equation (3.11) after its formal differentiation with respect to t.
Our task is to provide the limit in (3.13) called a shape derivative.

Shape differentiability of the Lagrangian
Since the optimal value function in (3.11) is shape-dependent, we utilize a coordinate transformation to fixed geometry. For t ∈ (t 0 , t 1 ) fixed, let flows The transformation determines the kinematic velocity Λ by the implicit formula Λ(t + s, y) := d ds ϕ s (ϕ −1 s (y)).
The following traits are required to prove the shape differentiability.

Trait 1. The function spaces constitute a bijective map
Proof. By the construction, coordinate transformation (4.1) builds a diffeomorphism, thus preserving integrable functions and first derivatives that form L 2 and H 1 spaces entering (4.5).
Based on Trait 1, after transformation to the reference geometry Ω t , perturbed objective functioñ for all (u, p, v, q) ∈ U(Ω t+s ) 2 . At s = 0, relations (4.6) imply that We formulate corresponding to (4.6b) a perturbed saddle-point problem: find a solution quadruple By virtue of (4.6b) applied to (4.9), we derive the transformed saddle point (ũ t+s ,p t+s ,ṽ t+s ,q t+s ) := (u t+s ∘ ϕ s , p t+s ∘ ϕ s , v t+s ∘ ϕ s , q t+s ∘ ϕ s ), which satisfies perturbed minimax problem (4.7) in the reference space Ω t .

Trait 4. For the saddle points in (3.3) and (4.8)
, there exists a subsequence s k for k → ∞ such that, as s k → 0 + , (ũ t+s ,p t+s ,ṽ t+s ,q t+s ) → (u t , p t , v t , q t ) strongly in U(Ω t ) 2 . (4.15) The proof of Trait 4 is rather technical and given in Appendix A. Based on Traits 1-4, we establish the following theorem on differentiability.

Hadamard formula
Provided by a smooth solution to the Stokes problem (see [15]), a Hadamard representation of the shape derivative ∂ t ℓ(t) by boundary integrals is presented next.
Theorem 5.1. Assume that the primal and adjoint solutions of (2.5) and (3.4) are smooth such that If Λ is constant outside some domain O t ⊂ O with C 2,0 -smooth boundary ∂O t and outward normal vector n t , then an equivalent expression of the shape derivative (4.16) holds, where τ t is a tangential vector at the boundary positive oriented to n t in 2d, and b t = τ t × n t is a binomial vector within the moving frame at the respective boundary in 3d. The notation m(u), the scalar D 1 , D 3 and vector-valued ) are defined as follows:

2)
and the curvature is t := div τ t n t .
Proof. Since ∇Λ = 0 in Ω t \ O t and the solution (u t , p t , v t , q t ) is smooth in O t , the shape derivative from Theorem 4.1 in accordance with formula (4.13) reads Integrating the terms entering (4.13) by parts in O t , we have and using the incompressibility div(u t ) = div(v t ) = 0, whereas the integral After summation, employing the equilibrium equations (2.2a), (3.5a) and the identity gathering the like terms yields Applying the divergence theorem to the first integral over O t in the right-hand side, we arrive at formulas for D 3 and D 4 in (5.1) and (5.2). Integration along the boundary Γ O t is given by the following formula (see [48, equation (2.125)]): Together with the identity ∇m(u) = ∇u ⊤ (u − z) − ∇z ⊤ (u − z), this leads to the expression of J ∂O t (Λ) involving D 1 , D 2 and m(u) in (5.1) and (5.2).
The important corollary deals with the inverse problem of shape identification (3.10) and guarantees the descent direction for optimization as suggested in (3.12). In the following consideration, we decompose the vectors into orthogonal, normal and tangential components at the boundary,
Corollary 5.2 gives practical formulas for numerical simulation of the inverse Stokes problem by gradient methods.

A Proof of Trait 4
We split the proof in three blocks: uniform estimate, weak convergence and strong convergence.
The application of limit as s k → 0 + due to weak convergences (A.8) and the Korn-Poincaré inequality (2.6) provide the strong convergence by means of lim sup On the other hand, subtracting (2.5a) with w = u from (A.2), we get the inequality Dividing it with the norm of v such that