7 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
Discretization error estimates in maximum norm for convergent splittings of matrices with a monotone preconditioning part
For finite difference matrices that are monotone, a discretization error estimate in maximum
norm follows from the truncation errors of the discretization. It enables also discretization error
estimates for derivatives of the solution. These results are extended to convergent operator
splittings of the difference matrix where the major, preconditioning part is monotone but the
whole operator is not necessarily monotone
A scalable and robust vertex-star relaxation for high-order FEM
Pavarino proved that the additive Schwarz method with vertex patches and a
low-order coarse space gives a -robust solver for symmetric and coercive
problems. However, for very high polynomial degree it is not feasible to
assemble or factorize the matrices for each patch. In this work we introduce a
direct solver for separable patch problems that scales to very high polynomial
degree on tensor product cells. The solver constructs a tensor product basis
that diagonalizes the blocks in the stiffness matrix for the internal degrees
of freedom of each individual cell. As a result, the non-zero structure of the
cell matrices is that of the graph connecting internal degrees of freedom to
their projection onto the facets. In the new basis, the patch problem is as
sparse as a low-order finite difference discretization, while having a sparser
Cholesky factorization. We can thus afford to assemble and factorize the
matrices for the vertex-patch problems, even for very high polynomial degree.
In the non-separable case, the method can be applied as a preconditioner by
approximating the problem with a separable surrogate. We demonstrate the
approach by solving the Poisson equation and a -conforming
interior penalty discretization of linear elasticity in three dimensions at