Photoacoustic Tomography in a Rectangular Reflecting Cavity

Almost all known image reconstruction algorithms for photoacoustic and thermoacoustic tomography assume that the acoustic waves leave the region of interest after a finite time. This assumption is reasonable if the reflections from the detectors and surrounding surfaces can be neglected or filtered out (for example, by time-gating). However, when the object is surrounded by acoustically hard detector arrays, and/or by additional acoustic mirrors, the acoustic waves will undergo multiple reflections. (In the absence of absorption they would bounce around in such a reverberant cavity forever). This disallows the use of the existing free-space reconstruction techniques. This paper proposes a fast iterative reconstruction algorithm for measurements made at the walls of a rectangular reverberant cavity. We prove the convergence of the iterations under a certain sufficient condition, and demonstrate the effectiveness and efficiency of the algorithm in numerical simulations.Comment: 21 pages, 6 figure

Theory and modeling of molecular modes in the NMR relaxation of fluids

Traditional theories of the NMR autocorrelation function for intramolecular dipole pairs assume single-exponential decay, yet the calculated autocorrelation of realistic systems display a rich, multi-exponential behavior resulting in anomalous NMR relaxation dispersion (i.e., frequency dependence). We develop an approach to model and interpret the multi-exponential autocorrelation using simple, physical models within a rigorous statistical mechanical development that encompasses both rotational and translational diffusion in the same framework. We recast the problem of evaluating the autocorrelation in terms of averaging over a diffusion propagator whose evolution is described by a Fokker-Planck equation. The time-independent part admits an eigenfunction expansion, allowing us to write the propagator as a sum over modes. Each mode has a spatial part that depends on the specified eigenfunction, and a temporal part that depends on the corresponding eigenvalue (i.e., correlation time) with a simple, exponential decay. The spatial part is a probability distribution of the dipole-pair, analogous to the stationary states of a quantum harmonic oscillator. Drawing inspiration from the idea of inherent structures in liquids, we interpret each of the spatial contributions as a specific molecular mode. These modes can be used to model and predict NMR dipole-dipole relaxation dispersion of fluids by incorporating phenomena on the molecular level. We validate our statistical mechanical description of the distribution in molecular modes with molecular dynamics simulations interpreted without any relaxation models or adjustable parameters: the most important poles in the Pad{\'e}-Laplace transform of the simulated autocorrelation agree with the eigenvalues predicted by the theory

Filling in CMB map missing data using constrained Gaussian realizations

For analyzing maps of the cosmic microwave background sky, it is necessary to mask out the region around the galactic equator where the parasitic foreground emission is strongest as well as the brightest compact sources. Since many of the analyses of the data, particularly those searching for non-Gaussianity of a primordial origin, are most straightforwardly carried out on full-sky maps, it is of great interest to develop efficient algorithms for filling in the missing information in a plausible way. We explore practical algorithms for filling in based on constrained Gaussian realizations. Although carrying out such realizations is in principle straightforward, for finely pixelized maps as will be required for the Planck analysis a direct brute force method is not numerically tractable. We present some concrete solutions to this problem, both on a spatially flat sky with periodic boundary conditions and on the pixelized sphere. One approach is to solve the linear system with an appropriately preconditioned conjugate gradient method. While this approach was successfully implemented on a rectangular domain with periodic boundary conditions and worked even for very wide masked regions, we found that the method failed on the pixelized sphere for reasons that we explain here. We present an approach that works for full-sky pixelized maps on the sphere involving a kernel-based multi-resolution Laplace solver followed by a series of conjugate gradient corrections near the boundary of the mask.Comment: 22 pages, 14 figures, minor changes, a few missing references adde

Hypersonic Boundary-Layer Stability Across a Compression Corner

Stability of a hypersonic boundary-layer over a compression corner was investigated numerically. The three-dimensional compressible Navier-Stokes equations were solved using a fifth-order weighted essentially non-oscillating (ArENO) shock capturing scheme to study the shock wave and boundary-layer interactions. The boundary-layer stability was studied in three distinct regions: upstream of the separation region, inside the separation region and downstream of the separation region. After the mean flow field was computed, linear stability theory was employed to predict the unstable disturbance modes in different flow regions and also to find the most amplified disturbance frequency across the compression corner. Gortler instability computations were performed to study the influence of the streamline curvatures on boundary-layer stability, and PSE(parabolized stability equation) method was employed to obtain the initial disturbances for direct numerical simulation. To study the boundary-layer stability by direct numerical simulation, two- or three-dimensional initial disturbances were introduced at the initial streamwise location of the computational domain. Two-dimensional disturbance evolution simulation shows that two-dimensional high frequency linear disturbances grow exponentially upstream and downstream of the separation region and remain neutral in the separation region, but two-dimensional low frequency linear disturbances only grow in a narrow area inside the separation region and remain neutral upstream and downstream of the separation region. Two-dimensional nonlinear disturbances will saturate downstream of the separation region when their amplitudes reach quit large amplitude. The three-dimensional disturbance evolution simulations show that three-dimensional linear mono-frequency disturbances are less amplified than its two-dimensional counterpart across the compression corner. The three-dimensional nonlinear mono-frequency disturbance evolution indicates that mode (0,2) is responsible for the oblique breakdown. Three-dimensional disturbances are much more amplified with the presence of two-dimensional primary disturbance due to the secondary instability. Finally, the simulations of three-dimensional random frequency disturbance evolution with the presence of a two-dimensional primary disturbance show that the secondary instability first occurs downstream of the separation region and a fundamental or K-type breakdown will be triggered by this secondary instability