Compressed absorbing boundary conditions via matrix probing

Absorbing layers are sometimes required to be impractically thick in order to offer an accurate approximation of an absorbing boundary condition for the Helmholtz equation in a heterogeneous medium. It is always possible to reduce an absorbing layer to an operator at the boundary by layer-stripping elimination of the exterior unknowns, but the linear algebra involved is costly. We propose to bypass the elimination procedure, and directly fit the surface-to-surface operator in compressed form from a few exterior Helmholtz solves with random Dirichlet data. The result is a concise description of the absorbing boundary condition, with a complexity that grows slowly (often, logarithmically) in the frequency parameter.Comment: 29 pages with 25 figure

Compressed Absorbing Boundary Conditions for the Helmholtz Equation

Absorbing layers are sometimes required to be impractically thick in order to offer an accurate approximation of an absorbing boundary condition for the Helmholtz equation in a heterogeneous medium. It is always possible to reduce an absorbing layer to an operator at the boundary by layer-stripping elimination of the exterior unknowns, but the linear algebra involved is costly. We propose to bypass the elimination procedure, and directly fit the surface-to-surface operator in compressed form from a few exterior Helmholtz solves with random Dirichlet data. We obtain a concise description of the absorbing boundary condition, with a complexity that grows slowly (often, logarithmically) in the frequency parameter. We then obtain a fast (nearly linear in the dimension of the matrix) algorithm for the application of the absorbing boundary condition using partitioned low rank matrices. The result, modulo a precomputation, is a fast and memory-efficient compression scheme of an absorbing boundary condition for the Helmholtz equation.Comment: PhD thesi

The method of polarized traces for the 2D Helmholtz equation

We present a solver for the 2D high-frequency Helmholtz equation in heterogeneous acoustic media, with online parallel complexity that scales optimally as O(NL), where N is the number of volume unknowns, and L is the number of processors, as long as L grows at most like a small fractional power of N. The solver decomposes the domain into layers, and uses transmission conditions in boundary integral form to explicitly define "polarized traces", i.e., up- and down-going waves sampled at interfaces. Local direct solvers are used in each layer to precompute traces of local Green's functions in an embarrassingly parallel way (the offline part), and incomplete Green's formulas are used to propagate interface data in a sweeping fashion, as a preconditioner inside a GMRES loop (the online part). Adaptive low-rank partitioning of the integral kernels is used to speed up their application to interface data. The method uses second-order finite differences. The complexity scalings are empirical but motivated by an analysis of ranks of off-diagonal blocks of oscillatory integrals. They continue to hold in the context of standard geophysical community models such as BP and Marmousi 2, where convergence occurs in 5 to 10 GMRES iterations. While the parallelism in this paper stems from decomposing the domain, we do not explore the alternative of parallelizing the systems solves with distributed linear algebra routines. Keywords: Domain decomposition; Helmholtz equation; Integral equations; High-frequency; Fast methodsUnited States. Air Force Office of Scientific Research (Grant FA9550-15-1-0078)United States. Office of Naval Research (Grant N00014-13-1-0403)National Science Foundation (U.S.) (Grant DMS-1255203

Quantification and Reconstruction in Photoacoustic Tomography

Optical absorption is closely associated with many physiological important parameters, such as the concentration and oxygen saturation of hemoglobin. Conventionally, accurate quantification in PAT requires knowledge of the optical fluence attenuation, acoustic pressure attenuation, and detection bandwidth. We circumvent this requirement by quantifying the optical absorption coefficients from the acoustic spectra of PA signals acquired at multiple optical wavelengths. We demonstrate the method using the optical-resolution photoacoustic microscopy: OR-PAM) and the acoustical-resolution photoacoustic microscopy: AR-PAM) in the optical ballistic regime and in the optical diffusive regime, respectively. The data acquisition speed in photoacoustic computed tomography: PACT) is limited by the laser repetition rate and the number of parallel ultrasound detecting channels. Reconstructing an image with fewer measurements can effectively accelerate the data acquisition and reduce the system cost. We adapted Compressed Sensing: CS) for the reconstruction in PACT. CS-based PACT was implemented as a non-linear conjugate gradient descent algorithm and tested with both phantom and in vivo experiments. Speckles have been considered ubiquitous in all scattering-based coherent imaging technologies. As a coherent imaging modality based on optical absorption, photoacoustic: PA) tomography: PAT) is generally devoid of speckles. PAT suppresses speckles by building up prominent boundary signals, via a mechanism similar to that of specular reflection. When imaging smooth boundary absorbing targets, the speckle visibility in PAT, which is defined as the ratio of the square root of the average power of speckles to that of boundaries, is inversely proportional to the square root of the absorber density. If the surfaces of the absorbing targets have uncorrelated height fluctuations, however, the boundary features may become fully developed speckles. The findings were validated by simulations and experiments. The first- and second-order statistics of PAT speckles were also studied experimentally. While the amplitude of the speckles follows a Gaussian distribution, the autocorrelation of the speckle patterns tracks that of the system point spread function