Second order directional shape derivatives of integrals on submanifolds

Anton Schiela, Julian Ortiz
2021 Mathematical Control and Related Fields
We compute first and second order shape sensitivities of integrals on smooth submanifolds using a variant of shape differentiation. The result is a quadratic form in terms of one perturbation vector field that yields a second order quadratic model of the perturbed functional. We discuss the structure of this derivative, derive domain expressions and Hadamard forms in a general geometric framework, and give a detailed geometric interpretation of the arising terms. 2020 Mathematics Subject
more » ... tics Subject Classification. Primary: 53A07, 49Q10, 49Q12. While the first shape derivatives coincide in all approaches, the second shape derivatives differ among the approaches. The reason is that for given vector fields v the corresponding transformations φ(t, ·) differ up to second order, depending on the chosen approach. Moreover, in order to obtain a bilinear form, classical definitions of shape hessians employ two vector fields v i and two temporal parameters t i , the combination of which defines φ. For example in the perturbation of identity method the definition φ(t 1 , t 2 , x) = x + t 1 v 1 + t 2 v 2 has been considered in [11, 9, 4] . For the velocity method φ(t 1 , t 2 , x) has been defined as the composition of two mappings [2, Sect. 9.6]. Consequently φ depends on v 1 and v 2 in a non-commutative way, which leads to a non-symmetric shape hessian. A connection to the second Lie derivative has been drawn in [1, 6] , applications in image segmentation can be found in  . Relations between these variants and application of Newton's method have been discussed in  . In this paper we start with a single family of transformations φ(t, ·) : S → R d , use only a single vector field v = φ t (0, ·) on S and look for a quadratic approximation of the perturbed integral. We end up with a quadratic form q(v) in terms of a single vector field. This contrasts with the approaches mentioned above which all yield bilinear forms in two vector fields. In addition, we observe that a linear term arises that depends on an acceleration field v t = φ tt (0, ·). A symmetric bilinear form can be derived, if needed, by differentiating q at 0 with respect to v twice. Concerning the geometry of S we choose the setting of a k-dimensional submanifold S ⊂ R d with (possibly empty) boundary. This includes the well known special cases S = Ω, where Ω is an open domain in R d and S = ∂Ω (as e.g. discussed in the text books  and ) but also a couple of others, such as hypersurfaces with boundaries and curves. Except for  , where a structure theorem is derived for first order shape derivatives, little work on shape calculus has been done in this general setting. Apart from the higher generality, a benefit is a unified view on the different cases of shape derivatives, which are traditionally treated by separate computations. Finally, we perform a derivation and geometrical interpretation of the Hadamard form of the second derivative which includes a splitting into normal and tangential components of the vector fields. This allows to give each term of the Hadamard form a specific geometric interpretation, which we try to illuminate, also with the help of geometric examples. Notation. Let us fix the following standard notation and recall basic definitions from differential geometry. Throughout this paper we consider R d equipped with the canonical basis {e i } i=1...d of unit vectors and the standard scalar product a·b := d i=1 a i b i . The spaces of linear and bilinear mappings between vector spaces X and Y are denoted by L(X, Y ) and L (2) (X, Y ), respectively. We define a smooth k-dimensional submanifold S of R d with boundary ∂S as usual in differential geometry via atlases of (boundary adapted) local charts of submanifolds. Its tangent space at x ∈ S is denoted by T x S, its orthogonal complement, the normal space, by N x S = (T x S) ⊥ and its dual, the cotangent space T x S * . Correspondingly, we have the tangent bundle T S, the normal bundle T N and the co-tangent bundle T S * . By general vector fields on S we denote mappings v : S → R d (identifying T x R d with R d itself). If v(x) ∈ T x S for all x ∈ S, then we call v a tangential vector field on S, if v(x) ∈ N x S for all x ∈ S, then v is called a normal vector field on S.