# Browsing by Subject "Low-rank approximation"

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Item A computational framework for the solution of infinite-dimensional Bayesian statistical inverse problems with application to global seismic inversion(2015-08) Martin, James Robert, Ph. D.; Ghattas, Omar N.; Biros, George; Demkowicz, Leszek; Fomel, Sergey; Marzouk, Youssef; Moser, RobertShow more Quantifying uncertainties in large-scale forward and inverse PDE simulations has emerged as a central challenge facing the field of computational science and engineering. The promise of modeling and simulation for prediction, design, and control cannot be fully realized unless uncertainties in models are rigorously quantified, since this uncertainty can potentially overwhelm the computed result. While statistical inverse problems can be solved today for smaller models with a handful of uncertain parameters, this task is computationally intractable using contemporary algorithms for complex systems characterized by large-scale simulations and high-dimensional parameter spaces. In this dissertation, I address issues regarding the theoretical formulation, numerical approximation, and algorithms for solution of infinite-dimensional Bayesian statistical inverse problems, and apply the entire framework to a problem in global seismic wave propagation. Classical (deterministic) approaches to solving inverse problems attempt to recover the “best-fit” parameters that match given observation data, as measured in a particular metric. In the statistical inverse problem, we go one step further to return not only a point estimate of the best medium properties, but also a complete statistical description of the uncertain parameters. The result is a posterior probability distribution that describes our state of knowledge after learning from the available data, and provides a complete description of parameter uncertainty. In this dissertation, a computational framework for such problems is described that wraps around the existing forward solvers, as long as they are appropriately equipped, for a given physical problem. Then a collection of tools, insights and numerical methods may be applied to solve the problem, and interrogate the resulting posterior distribution, which describes our final state of knowledge. We demonstrate the framework with numerical examples, including inference of a heterogeneous compressional wavespeed field for a problem in global seismic wave propagation with 10⁶ parameters.Show more Item Seismic modeling and imaging in complex media using low-rank approximation(2016-12) Sun, Junzhe; Fomel, Sergey B.; Biros, George; Ghattas, Omar; Sen, Mrinal K.; Zhang, YuShow more Seismic imaging in geologically complex areas, such as sub-salt or attenuating areas, has been one of the greatest challenges in hydrocarbon exploration. Increasing the fidelity and resolution of subsurface images will lead to a better understanding of geological and geomechanical properties in these areas of interest. Wavefield time extrapolation is the kernel of wave-equation-based seismic imaging algorithms, known as reverse-time migration. In exploration seismology, traditional ways for solving wave equations mainly include finite-difference and pseudo-spectral methods, which in turn involve finite-difference approximation of spatial or temporal derivatives. These approximations may lead to dispersion artifacts as well as numerical instability, therefore imposing a strict limit on the sampling intervals in space or time. This dissertation aims at developing a general framework for wave extrapolation based on fast application of Fourier integral operators (FIOs) derived from the analytical solutions to wave equations. The proposed methods are theoretically immune to dispersion artifacts and numerical instability, and are therefore desirable for applications to seismic imaging. First, I derive a one-step acoustic wave extrapolation operator based on the analytical solution to the acoustic wave equation. The proposed operator can incorporate anisotropic phase velocity, angle-dependent absorbing boundary conditions and further improvements in phase accuracy. I also investigate the numerical stability of the method using both theoretical derivations and numerical tests. Second, to model wave propagation in attenuating media, I use a visco-acoustic dispersion relation based on a constant-Q wave equation with decoupled fractional Laplacians, which allows for separable control of amplitude loss and velocity dispersion. The proposed formulation enables accurate reverse-time migration with attenuation compensation. Third, to further improve numerical stability of Q-compensation, I introduce stable Q-compensation operators based on amplitude spectrum scaling and smooth division. Next, for applications to least-squares RTM (LSRTM) and full-waveform inversion, I derive the adjoint operator of the low-rank one-step wave extrapolation method using the theory of non-stationary filtering. To improve the convergence rate of LSRTM in attenuating media, I propose Q-compensated LSRTM by replacing the adjoint operator in LSRTM with Q-compensated RTM. Finally, I extend the low-rank one-step wave extrapolation method to general elastic anisotropic media. Using the idea of eigenvalue decomposition and matrix exponential, I study the relationship between wave propagation and wave-mode decomposition. To handle the case of strong heterogeneity, I incorporate gradients of stiffnesses in wave extrapolation. Numerous synthetic examples in both 2D and 3D are used to test the practical application and accuracy of the proposed approaches.Show more Item Unsupervised learning for large-scale data(2019-09-20) Wu, Shanshan, Ph. D.; Sanghavi, Sujay Rajendra, 1979-; Dimakis, Alexandros G.; Caramanis, Constantine; Klivans, Adam R; Ward, Rachel AShow more Unsupervised learning involves inferring the inherent structures or patterns from unlabeled data. Since there is no label information, the fundamental challenge of unsupervised learning is that the objective function is not explicitly defined. The ubiquity of large-scale datasets adds another layer of complexity to the overall learning problem. When the data size or dimension is large, even algorithms with quadratic runtime may be prohibitive. This thesis presents four large-scale unsupervised learning problems. We start with two density estimation problems: given samples from a one-layer ReLU generative model or a discrete pairwise graphical model, the goal is to recover the parameters of the generative model. We then move to representation learning of high-dimensional sparse data coming from one-hot encoded categorical features. We assume that there are additional but a-priori unknown structures in their support. The goal is to learn a lossless low-dimensional embedding for the given data. Our last problem is to compute low-rank approximations of a matrix product given the individual matrices. We are interested in the setting where the matrices are too large and can only be stored in the disk. For every problem presented in this thesis, we (i) design novel and efficient algorithms to capture the inherent structure from data in an unsupervised manner; (ii) establish theoretical guarantees and compare the empirical performance with the state-of-the-art methods; and (iii) provide source code to support our experimental findingsShow more