Iain Duff

• Doctor of Philosophy OXON
• Researcher at Science and Technology Facilities Council

The 1996 conference on the State of the Art in Numerical Analysis was organized to provide the numerical analysis community, and users of numerical methods, with a forum where an account of the important recent developments in the subject could be presented in a coherent and concentrated way in a manner accessible to the non-specialist in the sub-a...

The hybrid scheme block row-projection method implemented in the ABCD Solver is designed for solving large sparse unsymmetric systems of equations on distributed memory parallel computers. The method implements a block Cimmino iterative scheme, accelerated with a stabilized block conjugate gradient algorithm. An augmented pseudo-direct variant has...

We present a multistage procedure to cluster directed and undirected weighted graphs by finding the block structure of their adjacency matrices. A central part of the process is to scale the adjacency matrix into a doubly-stochastic form, which permits detection of the whole matrix block structure with minimal spectral information (theoretically a...

The factorization of sparse symmetric indefinite systems is particularly challenging since pivoting is required to maintain stability of the factorization. Pivoting techniques generally offer limited parallelism and are associated with significant data movement hindering the scalability of these methods. Variants of the Threshold Partial Pivoting (...

We describe the parallelization of the solve phase in the sparse Cholesky solver SpLLT when using a sequential task flow model. In the context of direct methods, the solution of a sparse linear system is achieved through three main phases: the analyse, the factorization and the solve phases. In the last two phases, which involve numerical computati...

public NLAFET deliverable

The factorization of sparse symmetric indefinite systems is particularly challenging since pivoting is required to maintain stability of the factorization. Pivoting techniques generally offer limited parallelism and are associated with significant data movement hindering the scalability of these methods. Variants of the Threshold Partial Pivoting (...

We describe the parallelization of the solve phase in the sparse Cholesky solver SpLLT [Duff, Hogg, and Lopez. Numerical Algebra, Control and Optimization. Volume 8, 235- 237, 2018] when using a sequential task flow (STF) model. In the context of direct methods, the solution of a sparse linear system is achieved through three main phases: the analy...

We describe the development of a prototype code for the solution of large sparse symmetric positive definite systems that is efficient on parallel architectures. We implement a DAG-based Cholesky factorization that offers good performance and scalability on multicore architectures. Our approach uses a runtime system to execute the DAG. The runtime...

We describe our recent work on designing algorithms and software for solving sparse systems using direct methods on parallel computers. This work has been conducted within an EU Horizon 2020 Project called NLAFET. We first discuss the solution of large sparse symmetric positive definite systems. We use a runtime system to express and execute a DAG-...

We describe the design of a sparse direct solver for symmetric positive-definite systems using the PaRSEC runtime system. In this approach the application is represented as a DAG of tasks and the runtime system runs the DAG on the target architecture. Portability of the code across different architectures is enabled by delegating to the runtime sys...

The sparse triangular solve kernels, SpTRSV and SpTRSM, are important building blocks for a number of numerical linear algebra routines. Parallelizing SpTRSV and SpTRSM on today's manycore platforms, such as GPUs, is not an easy task since computing a component of the solution may depend on previously computed components, enforcing a degree of sequ...

We describe the design of a sparse direct solver for symmetric positive-definite systems using the PaRSEC runtime system. In this approach the application is represented as a DAG of tasks and the runtime system is in charge of running the DAG on the target architecture. Portability of the code across different architectures is enabled by delegating...

Direct Methods for Sparse Matrices, second edition, is a complete rewrite of the first edition published 30 years ago. Much has changed since that time. Problems have grown greatly in size and complexity; nearly all our examples were of order less than 5,000 in the first edition, and are often more than a million in the second edition. Computer arc...

Direct Methods for Sparse Matrices, second edition, is a complete rewrite of the first edition published 30 years ago. Much has changed since that time. Problems have grown greatly in size and complexity; nearly all our examples were of order less than 5,000 in the first edition, and are often more than a million in the second edition. Computer arc...

Direct Methods for Sparse Matrices, second edition, is a complete rewrite of the first edition published 30 years ago. Much has changed since that time. Problems have grown greatly in size and complexity; nearly all our examples were of order less than 5,000 in the first edition, and are often more than a million in the second edition. Computer arc...

Direct Methods for Sparse Matrices, second edition, is a complete rewrite of the first edition published 30 years ago. Much has changed since that time. Problems have grown greatly in size and complexity; nearly all our examples were of order less than 5,000 in the first edition, and are often more than a million in the second edition. Computer arc...

Direct Methods for Sparse Matrices, second edition, is a complete rewrite of the first edition published 30 years ago. Much has changed since that time. Problems have grown greatly in size and complexity; nearly all our examples were of order less than 5,000 in the first edition, and are often more than a million in the second edition. Computer arc...

Direct Methods for Sparse Matrices, second edition, is a complete rewrite of the first edition published 30 years ago. Much has changed since that time. Problems have grown greatly in size and complexity; nearly all our examples were of order less than 5,000 in the first edition, and are often more than a million in the second edition. Computer arc...

Direct Methods for Sparse Matrices, second edition, is a complete rewrite of the first edition published 30 years ago. Much has changed since that time. Problems have grown greatly in size and complexity; nearly all our examples were of order less than 5,000 in the first edition, and are often more than a million in the second edition. Computer arc...

Direct Methods for Sparse Matrices, second edition, is a complete rewrite of the first edition published 30 years ago. Much has changed since that time. Problems have grown greatly in size and complexity; nearly all our examples were of order less than 5,000 in the first edition, and are often more than a million in the second edition. Computer arc...

Direct Methods for Sparse Matrices, second edition, is a complete rewrite of the first edition published 30 years ago. Much has changed since that time. Problems have grown greatly in size and complexity; nearly all our examples were of order less than 5,000 in the first edition, and are often more than a million in the second edition. Computer arc...

Direct Methods for Sparse Matrices, second edition, is a complete rewrite of the first edition published 30 years ago. Much has changed since that time. Problems have grown greatly in size and complexity; nearly all our examples were of order less than 5,000 in the first edition, and are often more than a million in the second edition. Computer arc...

Direct Methods for Sparse Matrices, second edition, is a complete rewrite of the first edition published 30 years ago. Much has changed since that time. Problems have grown greatly in size and complexity; nearly all our examples were of order less than 5,000 in the first edition, and are often more than a million in the second edition. Computer arc...

Direct Methods for Sparse Matrices, second edition, is a complete rewrite of the first edition published 30 years ago. Much has changed since that time. Problems have grown greatly in size and complexity; nearly all our examples were of order less than 5,000 in the first edition, and are often more than a million in the second edition. Computer arc...

Direct Methods for Sparse Matrices, second edition, is a complete rewrite of the first edition published 30 years ago. Much has changed since that time. Problems have grown greatly in size and complexity; nearly all our examples were of order less than 5,000 in the first edition, and are often more than a million in the second edition. Computer arc...

Direct Methods for Sparse Matrices, second edition, is a complete rewrite of the first edition published 30 years ago. Much has changed since that time. Problems have grown greatly in size and complexity; nearly all our examples were of order less than 5,000 in the first edition, and are often more than a million in the second edition. Computer arc...

Direct Methods for Sparse Matrices, second edition, is a complete rewrite of the first edition published 30 years ago. Much has changed since that time. Problems have grown greatly in size and complexity; nearly all our examples were of order less than 5,000 in the first edition, and are often more than a million in the second edition. Computer arc...

Direct Methods for Sparse Matrices, second edition, is a complete rewrite of the first edition published 30 years ago. Much has changed since that time. Problems have grown greatly in size and complexity; nearly all our examples were of order less than 5,000 in the first edition, and are often more than a million in the second edition. Computer arc...

Direct Methods for Sparse Matrices, second edition, is a complete rewrite of the first edition published 30 years ago. Much has changed since that time. Problems have grown greatly in size and complexity; nearly all our examples were of order less than 5,000 in the first edition, and are often more than a million in the second edition. Computer arc...

Direct Methods for Sparse Matrices, second edition, is a complete rewrite of the first edition published 30 years ago. Much has changed since that time. Problems have grown greatly in size and complexity; nearly all our examples were of order less than 5,000 in the first edition, and are often more than a million in the second edition. Computer arc...

We describe the development of a prototype code for the solution of large sparse symmetric positive definite systems that is efficient on parallel architectures. We implement a DAG-based Cholesky factorization that offers good performance and scalability on multicore architectures. Our approach uses a runtime system to execute the DAG. The runtime...

The sparse triangular solve kernel, SpTRSV, is an important building block for a number of numerical linear algebra routines. Parallelizing SpTRSV on today’s manycore platforms, such as GPUs, is not an easy task since computing a component of the solution may depend on previously computed components, enforcing a degree of sequential processing. As...

We study the solution of the linear least-squares problem minx ∥b−Ax∥2 where the matrix A ∈ IRm×n (m ≥ n) has rank n and is large and sparse. We assume that A is available as a matrix, not an operator. The preconditioning of this problem is difficult because the matrix A does not have the properties of differential problems that make standard preco...

We study the solution of the linear least-squares problem minx ∥b−Ax∥_2 where the matrix A ∈ IR^{m×n} (m ≥ n) has rank n and is large and sparse. We assume that A is available as a matrix, not an operator. The preconditioning of this problem is difficult because the matrix A does not have the properties of differential problems that make standard p...

We introduce and study a novel way of accelerating the convergence of the block Cimmino method by augmenting the matrix so that the subspaces corresponding to the partitions are orthogonal. This results in a requirement to solve a relatively smaller symmetric positive definite system. We discuss several issues involved in a parallel implementation...

In the context of the block Cimmino algorithm, we study preprocessing strategies to obtain block partitionings that can be applied to general linear systems of equations \(\mathbf{A}\mathbf{x}= \mathbf{b}\). We study strategies that transform the matrix \(\mathbf{A}\mathbf{A}^\mathrm{{T}}\) into a matrix with a block tridiagonal structure. This pro...

The Fourth International Conference on Numerical Algebra and Scientific Computing (NASC 2012) was held in Dalian from 20 to 24 October, 2012. More than \(160\) participants attended the conference, coming from many countries including Canada, China, Czech Republic, France, Italy, Japan, Russia, Serbia, Sweden, the UK, and the USA. This issue of the...

We discuss the use of hypergraph partitioning based methods in fill-reducing orderings of sparse matrices for Cholesky, LU and QR factorizations. For the Cholesky factorization, we investigate a recent result on pattern-wise decomposition of sparse matrices, generalize the result, and develop algorithmic tools to obtain more effective ordering meth...

This paper proposes an approach for obtaining block diagonal and block triangular preconditioners that can be used for solving a linear system Ax = b, where A is a large, nonsingular, real, n × n sparse matrix. The proposed approach uses Tarjan's algorithm for hierarchically decomposing a digraph into its strong subgraphs. To the best of our knowle...

In this paper, we are concerned about computing in parallel several entries of the inverse of a large sparse matrix. We assume that the matrix has already been factorized by a direct method and that the factors are distributed. Entries are efficiently computed by exploiting sparsity of the right-hand sides and the solution vectors in the triangular...

The solution of large-scale problems in Computational Science and Engineering relies on the availability of accurate, robust and efficient numerical algorithms and software that are able to exploit the power offered by modern computer architectures. Such algorithms and software provide building blocks for prototyping and developing novel applicatio...

The inverse of an irreducible sparse matrix is structurally full, so that it is impractical to think of computing or storing it. However, there are several applications where a subset of the entries of the inverse is required. Given a factorization of the sparse matrix held in outof- core storage, we show how to compute such a subset efficiently, b...

We report on careful implementations of seven algorithms for solving the problem of finding a maximum transversal of a sparse matrix. We analyse the algorithms and discuss the design choices. To the best of our knowledge, this is the most comprehensive comparison of maximum transversal algorithms based on augmenting paths. Previous papers with the...

Computers with sustained Petascale performance are now available and it is expected that hardware will be developed with a peak capability in the Exascale range by around 2018. However, the complexity, hierarchical nature, and probable heterogeneity of these machines pose great challenges for the development of software to exploit these architectur...

Newton-HSS methods, which are variants of inexact Newton methods different from the Newton–Krylov methods, have been shown to be competitive methods for solving large sparse systems of nonlinear equations with positive-definite Jacobian matrices (J. Comp. Math. 2010; 28:235–260). In that paper, only local convergence was proved. In this paper, we p...

Numerical linear algebra and combinatorial optimization are vast subjects; as is their interaction. In virtually all cases there should be a notion of sparsity for a combinatorial problem to arise. Sparse matrices therefore form the basis of the interaction of these two seemingly disparate subjects. As the core of many of today's numerical linear a...

The biennial Copper Mountain Conference on Iterative Methods was held April 4-9, 2010. This meeting included more than 140 presentations covering many scientific computing areas, such as uncertainty quantification, optimization, Markov chains, saddle-point systems, inverse problems, direct factorizations, Krylov methods, algebraic multigrid, softwa...