Recovering added mass in nanoresonator sensors from finite axial eigenfrequency data

In this paper we present a method for solving a finite inverse eigenvalue problem arising in the determination of added distributed mass in nanoresonator sensors by measurements of the first N natural frequencies of the free axial vibration under clamped end conditions. The method is based on an iterative procedure that produces an approximation of the unknown mass density as a generalized Fourier partial sum of order N, whose coefficients are calculated from the first N eigenvalues. To avoid trivial non-uniqueness due to the symmetry of the initial configuration of the nanorod, it is assumed that the mass variation has support contained in half of the axis interval. Moreover, the mass variation is supposed to be small with respect to the total mass of the initial nanorod. An extended series of numerical examples shows that the method is efficient and gives excellent results in case of continuous mass variations. The determination of discontinuous coefficients exhibits no negligible oscillations near the discontinuity points, and requires more spectral data to obtain good reconstruction. A proof of local convergence of the iteration algorithm is provided for a family of finite dimensional mass coefficients. Surprisingly enough, in spite of its local character, the identification method performs well even for not necessarily small mass changes. To the authors' knowledge, this is the first quantitative study on the identification of distributed mass attached on nanostructures modelled within generalized continuum mechanics theories by using finite eigenvalue data

Effective Holographic Theories for low-temperature condensed matter systems

The IR dynamics of effective holographic theories capturing the interplay between charge density and the leading relevant scalar operator at strong coupling are analyzed. Such theories are parameterized by two real exponents $(\gamma,\delta)$ that control the IR dynamics. By studying the thermodynamics, spectra and conductivities of several classes of charged dilatonic black hole solutions that include the charge density back reaction fully, the landscape of such theories in view of condensed matter applications is characterized. Several regions of the $(\gamma,\delta)$ plane can be excluded as the extremal solutions have unacceptable singularities. The classical solutions have generically zero entropy at zero temperature, except when $\gamma=\delta$ where the entropy at extremality is finite. The general scaling of DC resistivity with temperature at low temperature, and AC conductivity at low frequency and temperature across the whole $(\gamma,\delta)$ plane, is found. There is a codimension-one region where the DC resistivity is linear in the temperature. For massive carriers, it is shown that when the scalar operator is not the dilaton, the DC resistivity scales as the heat capacity (and entropy) for planar (3d) systems. Regions are identified where the theory at finite density is a Mott-like insulator at T=0. We also find that at low enough temperatures the entropy due to the charge carriers is generically larger than at zero charge density.Comment: (v3): Added discussion on the UV completion of the solutions, and on extremal spectra in the charged case. Expanded discusion on insulating extremal solutions. Many other refinements and corrections. 126 pages. 48 figure

Inversion for subbottom sound velocity profiles in the deep and shallow ocean

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution February 2005This thesis investigates the application of acoustic measurements in the deep and shallow ocean to infer the sound velocity profile (svp) in the seabed. For the deep water ocean, an exact method based on the Gelfand-Levitan integral equation is evaluated. The input data is the complex plane-wave reflection coefficient estimated from measurements of acoustic pressure in water. We apply the method to experimental data and estimate both the reflection coefficient and the seabed svp. A rigorous inversion scheme is hence applied in a realistic problem. For the shallow ocean, an inverse eigenvalue technique is developed. The input data are the eigenvalues associated with propagating modes, measured as a function of source-receiver range. We investigate the estimation of eigenvalues from acoustic fields measured in laterally varying environments. We also investigate the errors associated with estimating varying modal eigenvalues, analogous to the estimation of time-varying frequencies in multicomponent signals, using time-varying autoregressive (TVAR) methods. We propose and analyze two AR sequential estimators, one for model coefficients, another for the zeros of the AR characteristic polynomial. The decimation of the pressure field defined in a discrete range grid is analyzed as a tool to improve AR estimation. The nonlinear eigenvalue inverse problem of estimating the svp from a sequence of eigenvalues is solved by iterating linearized approximations. The solution to the inverse problem is proposed in the form of a Kalman filter. The resolution and variance of the eigenvalue inverse problem are analyzed in terms of the Cramer-Rao lower bound and the Backus-Gilbert (BG) resolution theory. BG theory is applied to the design of shallow-water experiments. A method is developed to compensate for the Doppler deviation observed in experiments with moving sources.I am grateful for the support of my work provided by the WHOI Academic Programs Office and the Office of Naval Research

Recovery of a distributed order fractional derivative in an unknown medium

In this work, we study an inverse problem of recovering information about the weight in distributed-order time-fractional diffusion from the observation at one single point on the domain boundary. In the absence of an explicit knowledge of the medium, we prove that the one-point observation can uniquely determine the support bound of the weight. The proof is based on asymptotics of the data, analytic continuation and Titchmarch convolution theorem. When the medium is known, we give an alternative proof of an existing result, i.e., the one-point boundary observation uniquely determines the weight. Several numerical experiments are also presented to complement the analysis

Recovery of multiple parameters in subdiffusion from one lateral boundary measurement

This work is concerned with numerically recovering multiple parameters simultaneously in the subdiffusion model from one single lateral measurement on a part of the boundary, while in an incompletely known medium. We prove that the boundary measurement corresponding to a fairly general boundary excitation uniquely determines the order of the fractional derivative and the polygonal support of the diffusion coefficient, without knowing either the initial condition or the source. The uniqueness analysis further inspires the development of a robust numerical algorithm for recovering the fractional order and diffusion coefficient. The proposed algorithm combines small-time asymptotic expansion, analytic continuation of the solution and the level set method. We present extensive numerical experiments to illustrate the feasibility of the simultaneous recovery. In addition, we discuss the uniqueness of recovering general diffusion and potential coefficients from one single partial boundary measurement, when the boundary excitation is more specialized