243 research outputs found
Cauchy problem for Ultrasound Modulated EIT
Ultrasound modulation of electrical or optical properties of materials offers
the possibility to devise hybrid imaging techniques that combine the high
electrical or optical contrast observed in many settings of interest with the
high resolution of ultrasound. Mathematically, these modalities require that we
reconstruct a diffusion coefficient for , a bounded domain
in \Rm^n, from knowledge of for , where
is the solution to the elliptic equation in
with on .
This inverse problem may be recast as a nonlinear equation, which formally
takes the form of a 0-Laplacian. Whereas Laplacians with are
well-studied variational elliptic non-linear equations, is a limiting
case with a convex but not strictly convex functional, and the case
admits a variational formulation with a functional that is not convex. In this
paper, we augment the equation for the 0-Laplacian with full Cauchy data at the
domain's boundary, which results in a, formally overdetermined, nonlinear
hyperbolic equation.
The paper presents existence, uniqueness, and stability results for the
Cauchy problem of the 0-Laplacian. In general, the diffusion coefficient
can be stably reconstructed only on a subset of described as
the domain of influence of the space-like part of the boundary for
an appropriate Lorentzian metric. Global reconstructions for specific
geometries or based on the construction of appropriate complex geometric optics
solutions are also analyzed.Comment: 26 pages, 6 figure
An invariance principle for Brownian motion in random scenery
We prove an invariance principle for Brownian motion in Gaussian or
Poissonian random scenery by the method of characteristic functions. Annealed
asymptotic limits are derived in all dimensions, with a focus on the case of
dimension , which is the main new contribution of the paper.Comment: 22 pages, to appear in EJ
- β¦