This paper proposes a new algorithm for Bayesian model determination in Gaussian graphical models under G-Wishart prior distributions. We first review recent development in sampling from G-Wishart distributions for given graphs, with a particular interest in the efficiency of the block Gibbs samplers and other competing methods. We generalize the maximum clique block Gibbs samplers to a class of flexible block Gibbs samplers and prove its convergence. This class of block Gibbs samplers substantially outperforms its competitors along a variety of dimensions. We next develop the theory and computational details of a novel Markov chain Monte Carlo sampling scheme for Gaussian graphical model determination. Our method relies on the partial analytic structure of G-Wishart distributions integrated with the exchange algorithm. Unlike existing methods, the new method requires neither proposal tuning nor evaluation of normalizing constants of G-Wishart distributions.
We present Bayesian analyses of matrix-variate normal data with conditional independencies induced by graphical model structuring of the characterizing covariance matrix parameters. This framework of matrix normal graphical models includes prior specifications, posterior computation using Markov chain Monte Carlo methods, evaluation of graphical model uncertainty and model structure search. Extensions to matrix-variate time series embed matrix normal graphs in dynamic models. Examples highlight questions of graphical model uncertainty, search and comparison in matrix data contexts. These models may be applied in a number of areas of multivariate analysis, time series and also spatial modelling.
In this paper, we present new a posteriori and a priori error bounds for the Krylov subspace methods for computing e −τ A v for a given τ > 0 and v ∈ C n , where A is a large sparse non-Hermitian matrix. The a priori error bounds relate the convergence to λ min(the smallest and the largest eigenvalue of the Hermitian part of A) and |λ max A−A * 2 | (the largest eigenvalue in absolute value of the skew-Hermitian part of A), which define a rectangular region enclosing the field of values of A. In particular, our bounds explain an observed superlinear convergence behavior where the error may first stagnate for certain iterations before it starts to converge. The special case that A is skew-Hermitian is also considered. Numerical examples are given to demonstrate the theoretical bounds.
We consider a 1D periodic atomistic model, for which we formulate and analyze an adaptive variant of a quasicontinuum method. We establish a posteriori error estimates for the energy norm and for the energy, based on a posteriori residual and stability estimates. We formulate adaptive mesh refinement algorithms based on these error estimators. Our numerical experiments indicate optimal convergence rates of these algorithms.
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Atomistic/continuum coupling methods aim to achieve optimal balance between accuracy and efficiency. Adaptivity is the key for the efficient implementation of such methods. In this paper, we carry out a rigorous a posteriori analysis of the residual, the stability constant, and the error bound, for a consistent atomistic/continuum coupling method in 2D. We design and implement the corresponding adaptive mesh refinement algorithm, and the convergence rate with respect to degrees of freedom is optimal compare with a priori error estimates.
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