Ru Yu Lai

Increasing Stability for the Conductivity and Attenuation Coefficients

In this work we consider stability of recovery of the conductivity and attenuation coefficients of the stationary Maxwell and Schrodinger equations from a complete set of (Cauchy) boundary data. By using complex geometrical optics solutions we derive some bounds which can be viewed as an evidence of increasing stability in these inverse problems when the frequency is growing.

Uniqueness and stability of Lamé parameters in elastography

This paper concerns an hybrid inverse problem involving elastic measurements called Transient Elastography (TE) which enables detection and characterization of tissue abnormalities. In this paper we assume that the displacements are modeled by linear isotropic elasticity system and the tissue displacement has been obtained by the first step in hybrid methods. We reconstruct Lame parameters of this system from knowledge of the tissue displacement. More precisely, we show that for a sufficiently large number of solutions of the elasticity system and for an open set of the well-chosen boundary conditions, Lame parameters can be uniquely and stably reconstructed. The set of well-chosen boundary conditions is characterized in terms of appropriate complex geometrical optics solutions.

Inverse Boundary Value Problem for the Stokes and the Navier–Stokes Equations in the Plane

In this paper, we prove in two dimensions the global identifiability of the viscosity in an incompressible fluid by making boundary measurements. The main contribution of this work is to use more natural boundary measurements, the Cauchy forces, than the Dirichlet-to-Neumann map previously considered in Imanuvilov and Yamamoto (Global uniqueness in inverse boundary value problems for Navier–Stokes equations and Lamé ststem in two dimensions. arXiv:1309.1694, 2013) to prove the uniqueness of the viscosity for the Stokes equations and for the Navier–Stokes equations.

Increasing stability for the diffusion equation

We study the phenomenon of increasing stability of the diffusion and absorption coefficients in the diffusion equation. We derive some bounds which can be viewed as evidence of increasing stability when the frequency is growing. These bounds hold under a priori assumptions on the diffusion and absorption coefficients.

An inverse problem from condensed matter physics

We consider the problem of reconstructing the features of a weak anisotropic background potential by the trajectories of vortex dipoles in a nonlinear Gross-Pitaevskii equation. At leading order, the dynamics of vortex dipoles are given by a Hamiltonian system. If the background potential is sufficiently smooth and flat, the background can be reconstructed using ideas from the boundary and the lens rigidity problems. We prove that reconstructions are unique, derive an approximate reconstruction formula, and present numerical examples.

Nonparaxial Near-Nondiffracting Accelerating Optical Beams

We show that new families of accelerating and almost nondiffracting beams (solutions) for Maxwell’s equations can be constructed. These are complex geometrical optics (CGO) solutions to Maxwell’s equations with nonlinear limiting Carleman weights. They have the form of wave packets that propagate along circular trajectories while almost preserving a transverse intensity profile. We also show similar waves constructed using the approach combining CGO solutions and the Kelvin transform.