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What color is your Jacobian? Graph coloring for computing derivatives
 SIAM REV
, 2005
"... 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 specific ..."
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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 vertexcoloring 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 matrixestimation problems. The framework is based upon the viewpoint that a partition of a matrixinto structurally orthogonal groups of columns corresponds to distance2 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 matrixas 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.
Matrixfree preconditioning using partial matrix estimation
, 2004
"... We consider matrixfree 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 us ..."
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We consider matrixfree 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 matrixfree 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.
and
, 2003
"... It is well known that a sparse Jacobian matrix can be determined with fewer function evaluations or automatic differentiation passes than the number of independent variables of the underlying function. In this paper we show that by grouping together rows into blocks one can reduce this number furthe ..."
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It is well known that a sparse Jacobian matrix can be determined with fewer function evaluations or automatic differentiation passes than the number of independent variables of the underlying function. In this paper we show that by grouping together rows into blocks one can reduce this number further. We propose a graph coloring technique for row partitioned Jacobian matrices to efficiently determine the nonzero entries using a direct method. We characterize optimal direct determination and derive results on the optimality of any direct determination technique based on column computation. The computational results from coloring experiments on HarwellBoeing test matrix collection demonstrate that our row partitioned direct determination approach can yields considerable savings in function evaluations or AD passes over methods based on the Curtis, Powell, and Reid technique.
Partial Jacobian Computation in the Domainspecific Program Transformation System ADiCape
"... Abstract—Sensitivities of functions given in the form of computer models are crucial in various areas of computational science and engineering. We consider computer models written in CapeML, a domainspecific XMLbased language used in process engineering. Rather than computing all nonzero entries o ..."
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Abstract—Sensitivities of functions given in the form of computer models are crucial in various areas of computational science and engineering. We consider computer models written in CapeML, a domainspecific XMLbased language used in process engineering. Rather than computing all nonzero entries of a sparse Jacobian matrix, we are interested in obtaining only a subset of these entries. For the solution of this problem called partial Jacobian computation, we transform a CapeML model of an industrial distillation column using the automatic differentiation system ADiCape. I.
The CPR Method and Beyond: Prologue
"... a seminal paper on the estimation of Jacobian matrices which was later coined as the CPR method. Central to the CPR method is the effective utilization of a priori known sparsity information. It is only recently that the optimal CPR method in its general form is characterized and the theoretical und ..."
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a seminal paper on the estimation of Jacobian matrices which was later coined as the CPR method. Central to the CPR method is the effective utilization of a priori known sparsity information. It is only recently that the optimal CPR method in its general form is characterized and the theoretical underpinning for the optimality is shown. In this short note we provide an overview of the development of computational techniques and software tools for the estimation of Jacobian matrices.