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COMPUTING f(A)b VIA LEAST SQUARES POLYNOMIAL APPROXIMATIONS
, 2009
"... Given a certain function f, various methods have been proposed in the past for addressing the important problem of computing the the matrixvector product f(A)b without explicitly computing the matrix f(A). Such methods were typically used to compute a specific function f, a common case being that ..."
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Cited by 15 (7 self)
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Given a certain function f, various methods have been proposed in the past for addressing the important problem of computing the the matrixvector product f(A)b without explicitly computing the matrix f(A). Such methods were typically used to compute a specific function f, a common case being that of the exponential. This paper discusses a procedure based on least squares polynomials that can, in principle, be applied to any (continuous) function f. The idea is to start by approximating the function by a spline of a desired accuracy. Then, a particular definition of the function inner product is invoked that facilitates the computation of the least squares polynomial to this spline function. Since the function is approximated by a polynomial, the matrix A is referenced only through a matrixvector multiplication. In addition, the choice of the inner product makes it possible to avoid numerical integration. As an important application, we consider the case when f(t) = √ t and A is a sparse, symmetric positivedefinite matrix, which arises in sampling from a Gaussian process distribution. The covariance matrix of the distribution is defined by using a covariance function that has a compact support, at a very large number of sites that are on a regular or irregular grid. We derive error bounds and show extensive numerical results to illustrate the effectiveness of the proposed technique.
Parameter Estimation in High Dimensional Gaussian Distributions
, 2012
"... In order to compute the loglikelihood for high dimensional Gaussian models, it is necessary to compute the determinant of the large, sparse, symmetric positive definite precision matrix. Traditional methods for evaluating the loglikelihood, which are typically based on Choleksy factorisations, are ..."
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Cited by 9 (5 self)
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In order to compute the loglikelihood for high dimensional Gaussian models, it is necessary to compute the determinant of the large, sparse, symmetric positive definite precision matrix. Traditional methods for evaluating the loglikelihood, which are typically based on Choleksy factorisations, are not feasible for very large models due to the massive memory requirements. We present a novel approach for evaluating such likelihoods that only requires the computation of matrixvector products. In this approach we utilise matrix functions, Krylov subspaces, and probing vectors to construct an iterative numerical method for computing the loglikelihood.
Iterative Numerical Methods for Sampling from High Dimensional Gaussian Distributions
, 2011
"... Many applications require efficient sampling from Gaussian distributions. The method of choice depends on the dimension of the problem as well as the structure of the covariance (Σ) or precision matrix (Q). The most common blackbox routine for computing a sample is based on Cholesky factorisation. ..."
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Cited by 4 (0 self)
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Many applications require efficient sampling from Gaussian distributions. The method of choice depends on the dimension of the problem as well as the structure of the covariance (Σ) or precision matrix (Q). The most common blackbox routine for computing a sample is based on Cholesky factorisation. In high dimensions, computingtheCholesky factor of Σor Q may beprohibitive duetomassive fillin. We comparedifferent methods for computing the samples iteratively adapting ideas from numerical linear algebra. These methods assume that matrixvector products, Qv, are fast to compute. We show that some of the methods are competitive and faster than Cholesky sampling and that a parallel version of one method on a Graphical Processing Unit (GPU) using CUDA can introduce a speedup of up to 30x. Moreover, one method is used to sample from the posterior distribution of petroleum reservoir parameters in a North Sea field, given seismic reflection data on a large 3D grid.
Think continuous: Markovian Gaussian models in spatial statistics
, 2011
"... Gaussian Markov random fields (GMRFs) are frequently used as computationally efficient models in spatial statistics. Unfortunately, it has traditionally been difficult to link GMRFs with the more traditional Gaussian random field models as the Markov property is difficult to deploy in continuous spa ..."
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Gaussian Markov random fields (GMRFs) are frequently used as computationally efficient models in spatial statistics. Unfortunately, it has traditionally been difficult to link GMRFs with the more traditional Gaussian random field models as the Markov property is difficult to deploy in continuous space. Following the pioneering work of Lindgren et al. (2011), we expound on the link between Markovian Gaussian random fields and GMRFs. In particular, we discuss the theoretical and practical aspects of fast computation with continuously specified Markovian Gaussian random fields, as well as the clear advantages they offer in terms of clear, parsimonious and interpretable models of anisotropy and nonstationarity. 1
Specifying gaussian markov random fields with incomplete orthogonalfactorizationusinggivensrotations
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Scalable iterative methods for sampling from massive Gaussian random vectors
, 2013
"... Sampling from Gaussian Markov random fields (GMRFs), that is multivariate Gaussian random 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 accurate approximations to ..."
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Sampling from Gaussian Markov random fields (GMRFs), that is multivariate Gaussian random 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 accurate 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 likelihoodbased inference method. The method we derive is also applicable to other structured Gaussian random vectors and, in particular, 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.
models with discrete structures
, 2012
"... The use of systems of stochastic PDEs as priors for multivariate models with discrete structures by ..."
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The use of systems of stochastic PDEs as priors for multivariate models with discrete structures by
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"... Abstract. Fractional differential equations are becoming increasingly used as a modelling tool for processes associated with anomalous diffusion or spatial heterogeneity. However, the presence of a fractional differential operator causes memory (time fractional) or nonlocality (space fractional) iss ..."
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Abstract. Fractional differential equations are becoming increasingly used as a modelling tool for processes associated with anomalous diffusion or spatial heterogeneity. However, the presence of a fractional differential operator causes memory (time fractional) or nonlocality (space fractional) issues that impose a number of computational constraints. In this paper we develop efficient, scalable techniques for solving fractionalinspace reaction diffusion equations using the finite element method on both structured and unstructured grids via robust techniques for computing the fractional power of a matrix times a vector. Our approach is showcased by solving the fractional Fisher and fractional Allen–Cahn reactiondiffusion equations in two and three spatial dimensions, and analyzing the speed of the traveling wave and size of the interface in terms of the fractional power of the underlying Laplacian operator.