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

Victor A Kovtunenko,Kohji Ohtsuka

doi:10.1515/jiip-2020-0081

Abstract

Abstract For Stokes equations under divergence-free and mixed 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 function with 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.

Highlights

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]
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 function with respect to shape variations is proved within the velocity method, and a Hadamard representation of the shape derivative by boundary integrals is derived explicitly

Summary

Introduction

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. 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. 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). 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

Lagrange formulation using adjoint state

Shape differentiability of the Lagrangian

Hadamard formula

A Proof of Trait 4