166 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
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&
Spectral behavior of preconditioned non-Hermitian multilevel block Toeplitz matrices with matrix-valued symbol
This note is devoted to preconditioning strategies for non-Hermitian
multilevel block Toeplitz linear systems associated with a multivariate
Lebesgue integrable matrix-valued symbol. In particular, we consider special
preconditioned matrices, where the preconditioner has a band multilevel block
Toeplitz structure, and we complement known results on the localization of the
spectrum with global distribution results for the eigenvalues of the
preconditioned matrices. In this respect, our main result is as follows. Let
, let be the linear space of complex matrices, and let be functions whose components
belong to .
Consider the matrices , where varies
in and are the multilevel block Toeplitz matrices
of size generated by . Then
, i.e. the family
of matrices has a global (asymptotic)
spectral distribution described by the function , provided
possesses certain properties (which ensure in particular the invertibility of
for all ) and the following topological conditions are met:
the essential range of , defined as the union of the essential ranges
of the eigenvalue functions , does not
disconnect the complex plane and has empty interior. This result generalizes
the one obtained by Donatelli, Neytcheva, Serra-Capizzano in a previous work,
concerning the non-preconditioned case . The last part of this note is
devoted to numerical experiments, which confirm the theoretical analysis and
suggest the choice of optimal GMRES preconditioning techniques to be used for
the considered linear systems.Comment: 18 pages, 26 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
FAST SOLUTION METHODS FOR CONVEX QUADRATIC OPTIMIZATION OF FRACTIONAL DIFFERENTIAL EQUATIONS
In this paper, we present numerical methods suitable for solving convex
quadratic Fractional Differential Equation (FDE) constrained optimization
problems, with box constraints on the state and/or control variables. We
develop an Alternating Direction Method of Multipliers (ADMM) framework, which
uses preconditioned Krylov subspace solvers for the resulting sub-problems. The
latter allows us to tackle a range of Partial Differential Equation (PDE)
optimization problems with box constraints, posed on space-time domains, that
were previously out of the reach of state-of-the-art preconditioners. In
particular, by making use of the powerful Generalized Locally Toeplitz (GLT)
sequences theory, we show that any existing GLT structure present in the
problem matrices is preserved by ADMM, and we propose some preconditioning
methodologies that could be used within the solver, to demonstrate the
generality of the approach. Focusing on convex quadratic programs with
time-dependent 2-dimensional FDE constraints, we derive multilevel circulant
preconditioners, which may be embedded within Krylov subspace methods, for
solving the ADMM sub-problems. Discretized versions of FDEs involve large dense
linear systems. In order to overcome this difficulty, we design a recursive
linear algebra, which is based on the Fast Fourier Transform (FFT). We manage
to keep the storage requirements linear, with respect to the grid size ,
while ensuring an order computational complexity per iteration of
the Krylov solver. We implement the proposed method, and demonstrate its
scalability, generality, and efficiency, through a series of experiments over
different setups of the FDE optimization problem
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