254 research outputs found
Accelerating Cosmic Microwave Background map-making procedure through preconditioning
Estimation of the sky signal from sequences of time ordered data is one of
the key steps in Cosmic Microwave Background (CMB) data analysis, commonly
referred to as the map-making problem. Some of the most popular and general
methods proposed for this problem involve solving generalised least squares
(GLS) equations with non-diagonal noise weights given by a block-diagonal
matrix with Toeplitz blocks. In this work we study new map-making solvers
potentially suitable for applications to the largest anticipated data sets.
They are based on iterative conjugate gradient (CG) approaches enhanced with
novel, parallel, two-level preconditioners. We apply the proposed solvers to
examples of simulated non-polarised and polarised CMB observations, and a set
of idealised scanning strategies with sky coverage ranging from nearly a full
sky down to small sky patches. We discuss in detail their implementation for
massively parallel computational platforms and their performance for a broad
range of parameters characterising the simulated data sets. We find that our
best new solver can outperform carefully-optimised standard solvers used today
by a factor of as much as 5 in terms of the convergence rate and a factor of up
to in terms of the time to solution, and to do so without significantly
increasing the memory consumption and the volume of inter-processor
communication. The performance of the new algorithms is also found to be more
stable and robust, and less dependent on specific characteristics of the
analysed data set. We therefore conclude that the proposed approaches are well
suited to address successfully challenges posed by new and forthcoming CMB data
sets.Comment: 19 pages // Final version submitted to A&
Accelerating Cosmic Microwave Background map-making procedure through preconditioning
Estimation of the sky signal from sequences of time ordered data is one of
the key steps in Cosmic Microwave Background (CMB) data analysis, commonly
referred to as the map-making problem. Some of the most popular and general
methods proposed for this problem involve solving generalised least squares
(GLS) equations with non-diagonal noise weights given by a block-diagonal
matrix with Toeplitz blocks. In this work we study new map-making solvers
potentially suitable for applications to the largest anticipated data sets.
They are based on iterative conjugate gradient (CG) approaches enhanced with
novel, parallel, two-level preconditioners. We apply the proposed solvers to
examples of simulated non-polarised and polarised CMB observations, and a set
of idealised scanning strategies with sky coverage ranging from nearly a full
sky down to small sky patches. We discuss in detail their implementation for
massively parallel computational platforms and their performance for a broad
range of parameters characterising the simulated data sets. We find that our
best new solver can outperform carefully-optimised standard solvers used today
by a factor of as much as 5 in terms of the convergence rate and a factor of up
to in terms of the time to solution, and to do so without significantly
increasing the memory consumption and the volume of inter-processor
communication. The performance of the new algorithms is also found to be more
stable and robust, and less dependent on specific characteristics of the
analysed data set. We therefore conclude that the proposed approaches are well
suited to address successfully challenges posed by new and forthcoming CMB data
sets.Comment: 19 pages // Final version submitted to A&
Conjugate-Gradient Preconditioning Methods for Shift-Variant PET Image Reconstruction
Gradient-based iterative methods often converge slowly for tomographic image reconstruction and image restoration problems, but can be accelerated by suitable preconditioners. Diagonal preconditioners offer some improvement in convergence rate, but do not incorporate the structure of the Hessian matrices in imaging problems. Circulant preconditioners can provide remarkable acceleration for inverse problems that are approximately shift-invariant, i.e., for those with approximately block-Toeplitz or block-circulant Hessians. However, in applications with nonuniform noise variance, such as arises from Poisson statistics in emission tomography and in quantum-limited optical imaging, the Hessian of the weighted least-squares objective function is quite shift-variant, and circulant preconditioners perform poorly. Additional shift-variance is caused by edge-preserving regularization methods based on nonquadratic penalty functions. This paper describes new preconditioners that approximate more accurately the Hessian matrices of shift-variant imaging problems. Compared to diagonal or circulant preconditioning, the new preconditioners lead to significantly faster convergence rates for the unconstrained conjugate-gradient (CG) iteration. We also propose a new efficient method for the line-search step required by CG methods. Applications to positron emission tomography (PET) illustrate the method.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85979/1/Fessler85.pd
A fast semi-direct least squares algorithm for hierarchically block separable matrices
We present a fast algorithm for linear least squares problems governed by
hierarchically block separable (HBS) matrices. Such matrices are generally
dense but data-sparse and can describe many important operators including those
derived from asymptotically smooth radial kernels that are not too oscillatory.
The algorithm is based on a recursive skeletonization procedure that exposes
this sparsity and solves the dense least squares problem as a larger,
equality-constrained, sparse one. It relies on a sparse QR factorization
coupled with iterative weighted least squares methods. In essence, our scheme
consists of a direct component, comprised of matrix compression and
factorization, followed by an iterative component to enforce certain equality
constraints. At most two iterations are typically required for problems that
are not too ill-conditioned. For an HBS matrix with
having bounded off-diagonal block rank, the algorithm has optimal complexity. If the rank increases with the spatial dimension as is
common for operators that are singular at the origin, then this becomes
in 1D, in 2D, and
in 3D. We illustrate the performance of the method on
both over- and underdetermined systems in a variety of settings, with an
emphasis on radial basis function approximation and efficient updating and
downdating.Comment: 24 pages, 8 figures, 6 tables; to appear in SIAM J. Matrix Anal. App
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
MINRES-QLP: a Krylov subspace method for indefinite or singular symmetric systems
CG, SYMMLQ, and MINRES are Krylov subspace methods for solving symmetric
systems of linear equations. When these methods are applied to an incompatible
system (that is, a singular symmetric least-squares problem), CG could break
down and SYMMLQ's solution could explode, while MINRES would give a
least-squares solution but not necessarily the minimum-length (pseudoinverse)
solution. This understanding motivates us to design a MINRES-like algorithm to
compute minimum-length solutions to singular symmetric systems.
MINRES uses QR factors of the tridiagonal matrix from the Lanczos process
(where R is upper-tridiagonal). MINRES-QLP uses a QLP decomposition (where
rotations on the right reduce R to lower-tridiagonal form). On ill-conditioned
systems (singular or not), MINRES-QLP can give more accurate solutions than
MINRES. We derive preconditioned MINRES-QLP, new stopping rules, and better
estimates of the solution and residual norms, the matrix norm, and the
condition number.Comment: 26 pages, 6 figure
Preconditioners for Krylov subspace methods: An overview
When simulating a mechanism from science or engineering, or an industrial process, one is frequently required to construct a mathematical model, and then resolve this model numerically. If accurate numerical solutions are necessary or desirable, this can involve solving large-scale systems of equations. One major class of solution methods is that of preconditioned iterative methods, involving preconditioners which are computationally cheap to apply while also capturing information contained in the linear system. In this article, we give a short survey of the field of preconditioning. We introduce a range of preconditioners for partial differential equations, followed by optimization problems, before discussing preconditioners constructed with less standard objectives in mind
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