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

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

Preconditioned fast solvers for large linear systems with specific sparse and/or Toeplitz-like structures and applications

In this thesis, the design of the preconditioners we propose starts from applications instead of treating the problem in a completely general way. The reason is that not all types of linear systems can be addressed with the same tools. In this sense, the techniques for designing efficient iterative solvers depends mostly on properties inherited from the continuous problem, that has originated the discretized sequence of matrices. Classical examples are locality, isotropy in the PDE context, whose discrete counterparts are sparsity and matrices constant along the diagonals, respectively. Therefore, it is often important to take into account the properties of the originating continuous model for obtaining better performances and for providing an accurate convergence analysis. We consider linear systems that arise in the solution of both linear and nonlinear partial differential equation of both integer and fractional type. For the latter case, an introduction to both the theory and the numerical treatment is given. All the algorithms and the strategies presented in this thesis are developed having in mind their parallel implementation. In particular, we consider the processor-co-processor framework, in which the main part of the computation is performed on a Graphics Processing Unit (GPU) accelerator. In Part I we introduce our proposal for sparse approximate inverse preconditioners for either the solution of time-dependent Partial Differential Equations (PDEs), Chapter 3, and Fractional Differential Equations (FDEs), containing both classical and fractional terms, Chapter 5. More precisely, we propose a new technique for updating preconditioners for dealing with sequences of linear systems for PDEs and FDEs, that can be used also to compute matrix functions of large matrices via quadrature formula in Chapter 4 and for optimal control of FDEs in Chapter 6. At last, in Part II, we consider structured preconditioners for quasi-Toeplitz systems. The focus is towards the numerical treatment of discretized convection-diffusion equations in Chapter 7 and on the solution of FDEs with linear multistep formula in boundary value form in Chapter 8

Efficient approximation of functions of some large matrices by partial fraction expansions

Some important applicative problems require the evaluation of functions $\Psi$ of large and sparse and/or \emph{localized} matrices $A$. Popular and interesting techniques for computing $\Psi(A)$ and $\Psi(A)\mathbf{v}$, where $\mathbf{v}$ is a vector, are based on partial fraction expansions. However, some of these techniques require solving several linear systems whose matrices differ from $A$ by a complex multiple of the identity matrix $I$ for computing $\Psi(A)\mathbf{v}$ or require inverting sequences of matrices with the same characteristics for computing $\Psi(A)$. Here we study the use and the convergence of a recent technique for generating sequences of incomplete factorizations of matrices in order to face with both these issues. The solution of the sequences of linear systems and approximate matrix inversions above can be computed efficiently provided that $A^{-1}$ shows certain decay properties. These strategies have good parallel potentialities. Our claims are confirmed by numerical tests

Preconditioned fast solvers for large linear systems with specific sparse and/or Toeplitz-like structures and applications

In this thesis, the design of the preconditioners we propose starts from applications instead of treating the problem in a completely general way. The reason is that not all types of linear systems can be addressed with the same tools. In this sense, the techniques for designing efficient iterative solvers depends mostly on properties inherited from the continuous problem, that has originated the discretized sequence of matrices. Classical examples are locality, isotropy in the PDE context, whose discrete counterparts are sparsity and matrices constant along the diagonals, respectively. Therefore, it is often important to take into account the properties of the originating continuous model for obtaining better performances and for providing an accurate convergence analysis. We consider linear systems that arise in the solution of both linear and nonlinear partial differential equation of both integer and fractional type. For the latter case, an introduction to both the theory and the numerical treatment is given. All the algorithms and the strategies presented in this thesis are developed having in mind their parallel implementation. In particular, we consider the processor-co-processor framework, in which the main part of the computation is performed on a Graphics Processing Unit (GPU) accelerator. In Part I we introduce our proposal for sparse approximate inverse preconditioners for either the solution of time-dependent Partial Differential Equations (PDEs), Chapter 3, and Fractional Differential Equations (FDEs), containing both classical and fractional terms, Chapter 5. More precisely, we propose a new technique for updating preconditioners for dealing with sequences of linear systems for PDEs and FDEs, that can be used also to compute matrix functions of large matrices via quadrature formula in Chapter 4 and for optimal control of FDEs in Chapter 6. At last, in Part II, we consider structured preconditioners for quasi-Toeplitz systems. The focus is towards the numerical treatment of discretized convection-diffusion equations in Chapter 7 and on the solution of FDEs with linear multistep formula in boundary value form in Chapter 8