The Efficient Computation of Sparse Jacobian Matrices Using Automatic Differentiation

Coleman, Thomas F.; Verma, Arun

doi:10.1137/s1064827595295349

Cited by 49 publications

(68 citation statements)

References 11 publications

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“…A discussion of the technical details of AD is beyond the scope of this paper. The book by Nocedal and Wright [104] contains an introductory discussion of AD and how graph coloring can be used to reduce the number of passes Bipartition [32] Row-column partition [62] needed to compute the Jacobian and the Hessian. The recent book by Griewank [53] has an excellent discussion of the principles and techniques of AD.…”

Section: Scope and Further Readingmentioning

confidence: 99%

“…Their formulation relies on the notion of a consistent row-column partition in which the entire set of rows and columns is partitioned into two respective sets of groups. Coleman and Verma [32] also studied the same problem and identified a similar bidirectional partitioning problem. Their notion of partition differs from that of Hossain and Steihaug in that it partitions only a subset of the columns and the rows of the matrix that suffices for the direct determination of the entries.…”

Section: The Matrix Partitioning Problemmentioning

confidence: 99%

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What Color Is Your Jacobian? Graph Coloring for Computing Derivatives

Gebremedhin¹,

Manne²,

Pothen³

2005

SIAM Rev.

233

174

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Abstract. Graph coloring has been employed since the 1980s to efficiently compute sparse Jacobian and Hessian matrices using either finite differences or automatic differentiation. Several coloring problems occur in this context, depending on whether the matrix is a Jacobian or a Hessian, and on the specifics of the computational techniques employed. We consider eight variant vertex coloring problems here. This article begins with a gentle introduction to the problem of computing a sparse Jacobian, followed by an overview of the historical development of the research area. Then we present a unifying framework for the graph models of the variant matrix estimation problems. The framework is based upon the viewpoint that a partition of a matrix into structurally orthogonal groups of columns corresponds to distance-2 coloring an appropriate graph representation. The unified framework helps integrate earlier work and leads to fresh insights; enables the design of more efficient algorithms for many problems; leads to new algorithms for others; and eases the task of building graph models for new problems. We report computational results on two of the coloring problems to support our claims. Most of the methods for these problems treat a column or a row of a matrix as an atomic entity, and partition the columns or rows (or both). A brief review of methods that do not fit these criteria is provided. We also discuss results in discrete mathematics and theoretical computer science that intersect with the topics considered here.

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Section: Scope and Further Readingmentioning

confidence: 99%

Section: The Matrix Partitioning Problemmentioning

confidence: 99%

What Color Is Your Jacobian? Graph Coloring for Computing Derivatives

Gebremedhin¹,

Manne²,

Pothen³

2005

SIAM Rev.

233

174

View full text Add to dashboard Cite

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“…Matrix entries may be obtained by solving a triangular system (substitution methods) or even by the solution of a general linear algebraic system (elimination methods). For additional details see, for example, [38], [14], [25], and [26]. Matrix estimation algorithms can be designed to exploit any symmetry of the matrix efficiently, [12], [11].…”

Section: Matrix Estimation Using Graph Coloringmentioning

confidence: 99%

“…If matrix vector computations with both the matrix A and its transpose A T can be performed, their overall number required for the estimation of the matrix A can be decreased even more [13], [14]. Operations with A T can be achieved, for example, using the reverse mode of automatic differentiation [21], [20].…”

Section: Matrix Estimation Using Graph Coloringmentioning

confidence: 99%

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Matrix-free preconditioning using partial matrix estimation

Cullum

Tůma

2006

Bit Numer Math

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Abstract.We consider matrix-free solver environments where information about the underlying matrix is available only through matrix vector computations which do not have access to a fully assembled matrix. We introduce the notion of partial matrix estimation for constructing good algebraic preconditioners used in Krylov iterative methods in such matrix-free environments, and formulate three new graph coloring problems for partial matrix estimation. Numerical experiments utilizing one of these formulations demonstrate the viability of this approach. (2000): 65F10, 65F50, 49M37, 90C06. AMS subject classification

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The independent set perturbation adjoint method: A new method of differentiating mesh‐based fluids models

Fang

Pain

Navon

et al. 2011

Numerical Methods in Fluids

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SUMMARYA new scheme for differentiating complex mesh-based numerical models (e.g. finite element models), the Independent Set Perturbation Adjoint method (ISP-Adjoint), is presented. Differentiation of the matrices and source terms making up the discrete forward model is realized by a graph coloring approach (forming independent sets of variables) combined with a perturbation method to obtain gradients in numerical discretizations. This information is then convolved with the 'mathematical adjoint', which uses the transpose matrix of the discrete forward model. The adjoint code is simple to implement even with complex governing equations, discretization methods and non-linear parameterizations. Importantly, the adjoint code is independent of the implementation of the forward code. This greatly reduces the effort required to implement the adjoint model and maintain it as the forward model continues to be developed; as compared with more traditional approaches such as applying automatic differentiation tools. The approach can be readily extended to reduced-order models. The method is applied to a one-dimensional Burgers' equation problem, with a highly non-linear high-resolution discretization method, and to a two-dimensional, non-linear, reduced-order model of an idealized ocean gyre.

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The Efficient Computation of Sparse Jacobian Matrices Using Automatic Differentiation

Cited by 49 publications

References 11 publications

What Color Is Your Jacobian? Graph Coloring for Computing Derivatives

What Color Is Your Jacobian? Graph Coloring for Computing Derivatives

Matrix-free preconditioning using partial matrix estimation

The independent set perturbation adjoint method: A new method of differentiating mesh‐based fluids models

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