Estimation of Sparse Jacobian Matrices

Newsam, G.N.; Ramsdell, John D.

doi:10.1137/0604041

Cited by 43 publications

(30 citation statements)

References 3 publications

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“…However, approaches that rely on more "fine-grained" partitioning have also been suggested. The element isolation method of Newsam and Ramsdell [103] and the segmented column method of Hossain and Steihaug [61,65] are examples of such approaches.…”

Section: Other Variationsmentioning

confidence: 99%

“…8. We formulate the element isolation technique of Newsam and Ramsdell [103] for computing a Jacobian as an edge coloring problem in the associated bipartite graph (section 10).…”

Section: New Contributionsmentioning

confidence: 99%

“…A second common feature is that the system of equations is produced from a partition of a set of columns or rows (or both) of a matrix; i.e., a set is divided into groups in which an element of the set belongs to one and only one group. Newsam and Ramsdell [103] suggested a method for computing a Jacobian that relies on solving an overdetermined rectangular system of equations, a system produced from a column grouping that is not necessarily a partition. The approach of Newsam and Ramsdell has been applied within the context of AD by Geitner, Utke, and Griewank [49].…”

Section: Other Variationsmentioning

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: Other Variationsmentioning

confidence: 99%

“…8. We formulate the element isolation technique of Newsam and Ramsdell [103] for computing a Jacobian as an edge coloring problem in the associated bipartite graph (section 10).…”

Section: New Contributionsmentioning

confidence: 99%

Section: Other Variationsmentioning

confidence: 99%

See 1 more Smart Citation

What Color Is Your Jacobian? Graph Coloring for Computing Derivatives

Gebremedhin¹,

Manne²,

Pothen³

2005

SIAM Rev.

233

174

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show abstract

“…It is also quite easy to verify that the maximum number of nonzero elements on a single row is a lower bound on the number of groups, and hence of differences, that will be needed to estimate a particular sparse Jacobian matrix. Moreover Newsam and Ramsdell [9] have shown that it is always possible (mathematically) to estimate a Jacobian matrix in this number of differences. In [5], Curtis, Powell and Reid proposed forming groups by considering successively the columns of J in their natural order.…”

mentioning

confidence: 99%

“…Let S(i, k) be defined as the set of elements with row index i in a column 7 in group k, where 7 < /'; i.e. (9) S(i,k)= {(ij) | j < i,HtJ *0andk(j) = k], where k(j) is the group number of column7 in our a priori grouping. If for some /', the set S(i, k) contains more than one element, then group k is not CPR valid and we need more information to estimate the elements in S(i, k).…”

mentioning

confidence: 99%

Optimal Estimation of Jacobian and Hessian Matrices That Arise in Finite Difference Calculations

Goldfarb¹,

Toint²

1984

Mathematics of Computation

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Abstract. In this paper, the problem of estimating Jacobian and Hessian matrices arising in the finite difference approximation of partial differential equations is considered. Using the notion of computational molecule or stencil, schemes are developed that require the minimum number of differences to estimate these matrices. A procedure applicable to more complicated structures is also given.1. Introduction. In the past few years, there has been a growing interest in studying efficient ways to obtain good estimates of sparse Jacobian or Hessian matrices. Two basic approaches have been followed. In the first approach, it is assumed that one has an estimate of the Jacobian (Hessian) which one wishes to revise given "arbitrary" changes in the variables and the corresponding changes in the functions (gradients). This has led to the development of several sparse quasi-Newton methods (see [11], [14], [12], for example). In the second approach, it is assumed that one can specify the changes in the variables. One then forms an estimate of the Jacobian (Hessian) matrix from the observed changes in the functions (gradients). When the dimension of the matrix is large and the evaluation of each function (gradient) vector is expensive, it is necessary to take the sparsity structure of the matrix into account, in order to make this approach efficient.

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Vorticity-Velocity Formulation for Three-Dimensional Steady Compressible Flows

Ern

Smooke

1993

Journal of Computational Physics

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Estimation of Sparse Jacobian Matrices

Cited by 43 publications

References 3 publications

What Color Is Your Jacobian? Graph Coloring for Computing Derivatives

What Color Is Your Jacobian? Graph Coloring for Computing Derivatives

Optimal Estimation of Jacobian and Hessian Matrices That Arise in Finite Difference Calculations

Vorticity-Velocity Formulation for Three-Dimensional Steady Compressible Flows

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