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25
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 8 (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.
Reducing the influence of tiny normwise relative errors on performance profiles
 Manchester Institute for Mathematical Sciences, The University of Manchester
, 2011
"... Reports available from: And by contacting: ..."
Shaping Social Activity by Incentivizing Users
"... Events in an online social network can be categorized roughly into endogenous events, where users just respond to the actions of their neighbors within the network, or exogenous events, where users take actions due to drives external to the network. How much external drive should be provided to each ..."
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Cited by 5 (4 self)
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Events in an online social network can be categorized roughly into endogenous events, where users just respond to the actions of their neighbors within the network, or exogenous events, where users take actions due to drives external to the network. How much external drive should be provided to each user, such that the network activity can be steered towards a target state? In this paper, we model social events using multivariate Hawkes processes, which can capture both endogenous and exogenous event intensities, and derive a time dependent linear relation between the intensity of exogenous events and the overall network activity. Exploiting this connection, we develop a convex optimization framework for determining the required level of external drive in order for the network to reach a desired activity level. We experimented with event data gathered from Twitter, and show that our method can steer the activity of the network more accurately than alternatives. 1
A Catalogue of Software for Matrix Functions. Version 1.0
, 2014
"... A catalogue of software for computing matrix functions and their Fréchet derivatives is presented. For a wide variety of languages and for software ranging from commercial products to open source packages we describe what matrix function codes are available and which algorithms they implement. ..."
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Cited by 4 (4 self)
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A catalogue of software for computing matrix functions and their Fréchet derivatives is presented. For a wide variety of languages and for software ranging from commercial products to open source packages we describe what matrix function codes are available and which algorithms they implement.
Exponential Taylor methods: analysis and implementation
 Comput. Math. Appl
"... For the time integration of semilinear systems of dierential equations, a class of multiderivative exponential integrators is considered. The methods are based on a Taylor series expansion of the semilinearity about the numerical solution, the required derivatives are computed by automatic dierentia ..."
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Cited by 1 (1 self)
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For the time integration of semilinear systems of dierential equations, a class of multiderivative exponential integrators is considered. The methods are based on a Taylor series expansion of the semilinearity about the numerical solution, the required derivatives are computed by automatic dierentiation. Inserting these derivatives into the variationofconstants formula results in an exponential integrator which requires the action of the exponential of an augmented Jacobian only. The convergence properties of such exponential integrators are analyzed, and potential sources of numerical instabilities are identied. In particular, it is shown that local linearization gives rise to better stability for sti problems. A number of numerical experiments illustrate the theoretical results.
MARKOV CHAIN APPROXIMATIONS FOR TRANSITION DENSITIES OF LÉVY PROCESSES
, 1211
"... Abstract. We consider the convergence of a continuoustime Markov chain approximation X h, h> 0, to an Rvalued Lévy process X. The state space of X h is an equidistant lattice and its Qmatrix is chosen to approximate the generator of X. Under a general sufficient condition for the existence of ..."
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Abstract. We consider the convergence of a continuoustime Markov chain approximation X h, h> 0, to an Rvalued Lévy process X. The state space of X h is an equidistant lattice and its Qmatrix is chosen to approximate the generator of X. Under a general sufficient condition for the existence of transition densities of X, we establish sharp convergence rates of the normalised probability mass function of X h to the probability density function of X. 1.
The Leja method in Python supervised by
"... The goal of this bachelor thesis is to write a stateoftheart implementation of the Leja method for computing the action of matrix exponential in Python based, amongst others, on the NumPy and SciPy packages. The developed code is going to be published on the homepage of the numerical analysis gro ..."
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The goal of this bachelor thesis is to write a stateoftheart implementation of the Leja method for computing the action of matrix exponential in Python based, amongst others, on the NumPy and SciPy packages. The developed code is going to be published on the homepage of the numerical analysis group. Description: The Leja method, described in Caliari et al. [2004], Bergamaschi et al. [2006], Caliari and Ostermann [2009], Caliari et al. [2014], is a well established method to efficiently approximate the action of the matrix exponential. We denote this action by exp(A)b, A ∈ Cn×n, b ∈ Cn. In general, methods that compute exp(A) and in a second step compute the action exp(A)b are not feasible for high dimensional matrices, see Higham [2008]. The Leja method is an efficient method to compute directly exp(A)b for large matrices. A dimension of n ≥ 104 is common. The Leja method was undergoing some changes that focused on a backward error analysis inspired by AlMohy and