arXiv subjects

Alden Waters

Publications and source records attributed to Alden Waters.

24 records · Page 2Linked to original sources

arXiv2015

Stability Analysis in Magnetic Resonance Elastography II

We consider the inverse problem of finding unknown elastic parameters from internal measurements of displacement fields for tissues. In the sequel to Ammari, Waters, Zhang (2015), we use pseudodifferential methods for the problem of recovering the shear modulus for Stokes systems from internal data. We prove stability estimates in $d=2,3$ with reduced regularity on the estimates and show that the presence of a finite dimensional kernel can be removed. This implies the convergence of the Landweber numerical iteration scheme. We also show that these hypotheses are natural for experimental use in constructing shear modulus distributions.

math.AP↗

arXiv2014

Stability Analysis for Magnetic Resonance Elastography

We consider the inverse problem of finding unknown elastic parameters from internal measurements of displacement fields for tissues. The measurements are made on the entirety of a smooth domain. Since tissues can be modeled as quasi-incompressible fluids, we examine the Stokes system and consider only the recovery of shear modulus distributions. Our main result is to establish Lipschitz stable estimates on the shear modulus distributions from internal measurements of displacement fields. These estimates imply convergence of a numerical scheme known as the Landweber iteration scheme for reconstructing the shear modulus distributions.

math.AP↗

arXiv2014

Spectral Rigidity for Periodic Schrödinger Operators in Dimension 2

We consider two dimensional real-valued analytic potentials for the Schrödinger equation which are periodic over a lattice $L$. Under certain assumptions on the form of the potential and the lattice $L$, we can show there is a large class of analytic potentials which are Floquet rigid and dense in the set of $C^\infty(R^2/L)$ potentials. The result extends the work of Eskin et. al, in "On isospectral periodic potentials in $R^n$, II."

math.AP↗

arXiv2014

Stable determination of X-ray transforms of time dependent potentials from the dynamical Dirichlet-to-Neumann map

We consider compact smooth Riemmanian manifolds with boundary of dimension greater than or equal to two. For the initial-boundary value problem for the wave equation with a lower order term $q(t,x)$, we can recover the X-ray transform of time dependent potentials $q(t,x)$ from the dynamical Dirichlet-to-Neumann map in a stable way. We derive Hölder stability estimates for the X-ray transform of $q(t,x)$ which do not require measurements on the entirety of the boundary. The essential technique involved is the Gaussian beam Ansatz, and the proofs are done with the minimal assumptions on the geometry for the Ansatz to be well-defined.

math.AP↗

arXiv2013

Stability estimate and the Aharonov-Bohm effect for the Klein-Gordon equation with time-dependent vector potentials

We consider the relativistic Schrödinger (Gordon-Klein) equation with a time dependent vector and a scalar potential on a bounded cylindrical domain. Using a standard Geometric Optics Ansatz, we establish a logarithmic stability estimate for the recovery of the vector potentials. We also study the recovery of the potentials in the presence of obstacles. It is shown that under some geometric restrictions, the recovery of the potentials is possible up to a gauge transform.

math.AP↗

arXiv2010

A Parametrix Construction for the Wave Equation with Low Regularity Coefficients Using a Frame of Gaussians

We construct a frame of complex Gaussians for the space of $L^2(\mathbb{R}^n)$ functions. When propagated along bicharacteristics for the wave equation, the frame can be used to build a parametrix with suitable error terms. When the coefficients of the wave equation have more regularity, propagated frame functions become Gaussian beams.

math.AP↗