146 research outputs found
Scalable iterative methods for sampling from massive Gaussian random vectors
Sampling from Gaussian Markov random fields (GMRFs), that is multivariate
Gaussian ran- dom vectors that are parameterised by the inverse of their
covariance matrix, is a fundamental problem in computational statistics. In
this paper, we show how we can exploit arbitrarily accu- rate approximations to
a GMRF to speed up Krylov subspace sampling methods. We also show that these
methods can be used when computing the normalising constant of a large
multivariate Gaussian distribution, which is needed for both any
likelihood-based inference method. The method we derive is also applicable to
other structured Gaussian random vectors and, in particu- lar, we show that
when the precision matrix is a perturbation of a (block) circulant matrix, it
is still possible to derive O(n log n) sampling schemes.Comment: 17 Pages, 4 Figure
Structural Variability from Noisy Tomographic Projections
In cryo-electron microscopy, the 3D electric potentials of an ensemble of
molecules are projected along arbitrary viewing directions to yield noisy 2D
images. The volume maps representing these potentials typically exhibit a great
deal of structural variability, which is described by their 3D covariance
matrix. Typically, this covariance matrix is approximately low-rank and can be
used to cluster the volumes or estimate the intrinsic geometry of the
conformation space. We formulate the estimation of this covariance matrix as a
linear inverse problem, yielding a consistent least-squares estimator. For
images of size -by- pixels, we propose an algorithm for calculating this
covariance estimator with computational complexity
, where the condition number
is empirically in the range --. Its efficiency relies on the
observation that the normal equations are equivalent to a deconvolution problem
in 6D. This is then solved by the conjugate gradient method with an appropriate
circulant preconditioner. The result is the first computationally efficient
algorithm for consistent estimation of 3D covariance from noisy projections. It
also compares favorably in runtime with respect to previously proposed
non-consistent estimators. Motivated by the recent success of eigenvalue
shrinkage procedures for high-dimensional covariance matrices, we introduce a
shrinkage procedure that improves accuracy at lower signal-to-noise ratios. We
evaluate our methods on simulated datasets and achieve classification results
comparable to state-of-the-art methods in shorter running time. We also present
results on clustering volumes in an experimental dataset, illustrating the
power of the proposed algorithm for practical determination of structural
variability.Comment: 52 pages, 11 figure
Some fast algorithms in signal and image processing.
Kwok-po Ng.Thesis (Ph.D.)--Chinese University of Hong Kong, 1995.Includes bibliographical references (leaves 138-139).AbstractsSummaryIntroduction --- p.1Summary of the papers A-F --- p.2Paper A --- p.15Paper B --- p.36Paper C --- p.63Paper D --- p.87Paper E --- p.109Paper F --- p.12
A block -circulant based preconditioned MINRES method for wave equations
In this work, we propose an absolute value block -circulant
preconditioner for the minimal residual (MINRES) method to solve an all-at-once
system arising from the discretization of wave equations. Since the original
block -circulant preconditioner shown successful by many recently is
non-Hermitian in general, it cannot be directly used as a preconditioner for
MINRES. Motivated by the absolute value block circulant preconditioner proposed
in [E. McDonald, J. Pestana, and A. Wathen. SIAM J. Sci. Comput.,
40(2):A1012-A1033, 2018], we propose an absolute value version of the block
-circulant preconditioner. Our proposed preconditioner is the first
Hermitian positive definite variant of the block -circulant
preconditioner, which fills the gap between block -circulant
preconditioning and the field of preconditioned MINRES solver. The
matrix-vector multiplication of the preconditioner can be fast implemented via
fast Fourier transforms. Theoretically, we show that for properly chosen
the MINRES solver with the proposed preconditioner has a linear
convergence rate independent of the matrix size. To the best of our knowledge,
this is the first attempt to generalize the original absolute value block
circulant preconditioner in the aspects of both theory and performance.
Numerical experiments are given to support the effectiveness of our
preconditioner, showing that the expected optimal convergence can be achieved
X-ray CT Image Reconstruction on Highly-Parallel Architectures.
Model-based image reconstruction (MBIR) methods for X-ray CT use accurate
models of the CT acquisition process, the statistics of the noisy measurements,
and noise-reducing regularization to produce potentially higher quality images
than conventional methods even at reduced X-ray doses. They do this by
minimizing a statistically motivated high-dimensional cost function; the high
computational cost of numerically minimizing this function has prevented MBIR
methods from reaching ubiquity in the clinic. Modern highly-parallel hardware
like graphics processing units (GPUs) may offer the computational resources to
solve these reconstruction problems quickly, but simply "translating" existing
algorithms designed for conventional processors to the GPU may not fully
exploit the hardware's capabilities.
This thesis proposes GPU-specialized image denoising and image reconstruction
algorithms. The proposed image denoising algorithm uses group coordinate
descent with carefully structured groups. The algorithm converges very
rapidly: in one experiment, it denoises a 65 megapixel image in about 1.5
seconds, while the popular Chambolle-Pock primal-dual algorithm running on the
same hardware takes over a minute to reach the same level of accuracy.
For X-ray CT reconstruction, this thesis uses duality and group coordinate
ascent to propose an alternative to the popular ordered subsets (OS) method.
Similar to OS, the proposed method can use a subset of the data to update the
image. Unlike OS, the proposed method is convergent. In one helical CT
reconstruction experiment, an implementation of the proposed algorithm using
one GPU converges more quickly than a state-of-the-art algorithm converges
using four GPUs. Using four GPUs, the proposed algorithm reaches near
convergence of a wide-cone axial reconstruction problem with over 220 million
voxels in only 11 minutes.PhDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/113551/1/mcgaffin_1.pd
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