Stitching Data: Recovering a Manifold’s Geometry from Geodesic Intersections

Let (M, g) be a Riemannian manifold with boundary. We show that knowledge of the length of each geodesic, and where pairwise intersections occur along the corresponding geodesics allows for recovery of the geometry of (M, g) (assuming (M, g) admits a Riemannian collar of a uniform radius). We call this knowledge the ‘stitching data’. We then pose a boundary measurement problem called the ‘delayed collision data problem’ and apply our result about the stitching data to recover the geometry from the collision data (with some reasonable geometric restrictions on the manifold).

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

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.

Identification of a harmonically varying external source in wave equation from Neumann-type boundary measurement

In this paper, the identification problem of recovering the spatial source {F\in L^{2}(0,l)} in the wave equation {u_{tt}=u_{xx}+F(x)\cos(\omega t)}, with harmonically varying external source {F(x)\cos(\omega t)} and with the homogeneous boundary {u(0,t)=u(l,t)=0}, {t\in(0,T)}, and initial {u(x,0)=u_{t}(x,0)=0}, {x\in(0,l)}, conditions, is studied. As a measurement output {g(t)}, the Neumann-type boundary measurement {g(t):=u_{x}(0,t)}, {t\in(0,T)}, at the left boundary {x=0} is used. It is assumed that the observation {g\in L^{2}(0,T)} may has a random noise. We propose combination of the boundary control for PDEs, adjoint method and Tikhonov regularization, for identification of the unknown source {F\in L^{2}(0,l)}. Our approach based on weak solution theory of PDEs and, as a result, allows use of nonsmooth input/output data. Introducing the input-output operator {\Phi F:=u_{x}(0,t;F)}, {\Phi:L^{2}(0,l)\mapsto L^{2}(0,T)}, where {u(x,t;F)} is the solution of the wave equation with above homogeneous boundary and initial conditions, we first prove the compactness of this operator. This allows to obtain the uniqueness of regularized solution of the identification problem, i.e. the minimum of the regularized cost functional {J_{\alpha}(F):=J(F)+\frac{1}{2}\alpha\|F\|_{L^{2}(0,l)}^{2}}, where {J(F)=\frac{1}{2}\|u_{x}(0,\cdot\,;F)-g\|_{L^{2}(0,T)}^{2}}. Then the adjoint problem approach is used to derive a formula for the Fréchet gradient of the cost functional {J(F)}. Use of the gradient formula in the conjugate gradient algorithm (CGA) allows to construct a fast algorithm for recovering the unknown source {F(x)}. A comprehensive set of benchmark numerical examples, with up to 10 noise level random noisy data, illustrate the usefulness and effectiveness of the proposed approach.