What Color Is Your Jacobian? Graph Coloring for Computing Derivatives

Gebremedhin, Assefaw H.; Manne, Fredrik; Pothen, Alex

doi:10.1137/s0036144504444711

Cited by 223 publications

(171 citation statements)

References 87 publications

(144 reference statements)

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“…15 our task, it is sufficient to find a coloring that yields a small number of partitions relative to the highest degree node in the data (well over 1,000). To that end, we implement the greedy sequential coloring algorithm described in Gebremedhin et al (2005). Briefly, the algorithm sorts network nodes from highest to lowest degree (that is, sorting employers in descending order by the number of job-to-job separations).…”

Section: E Online Appendix: Estimation and Data Details E1 Parallelimentioning

confidence: 99%

Modeling Endogenous Mobility in Earnings Determination

Abowd

McKinney

Schmutte

2018

Journal of Business & Economic Statistics

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We evaluate the bias from endogenous job mobility in fixed-effects estimates of worker-and firm-specific earnings heterogeneity using longitudinally linked employer-employee data from the LEHD infrastructure file system of the U.S. Census Bureau. First, we propose two new residual diagnostic tests of the assumption that mobility is exogenous to unmodeled determinants of earnings. Both tests reject exogenous mobility. We relax the exogenous mobility assumptions by modeling the evolution of the matched data as an evolving bipartite graph using a Bayesian latent class framework. Our results suggest that endogenous mobility biases estimated firm effects toward zero. To assess validity, we match our estimates of the wage components to out-of-sample estimates of revenue per worker. The corrected estimates attribute much more of the variation in revenue per worker to variation in match quality and worker quality than the uncorrected estimates. Replication code can be found at DOI:10.5281/zenodo.zenodo.376600 and our Github repository endogenous-mobility-replication .Abowd acknowledges direct support from NSF Grants SES-0339191, CNS-0627680, SES-0922005, TC-1012593, and SES-1131848. This research uses data from the Census Bureau's Longitudinal EmployerHousehold Dynamics Program, which was partially supported by National Science Foundation Grants SES-9978093, SES-0339191 and ITR-0427889; National Institute on Aging Grant AG018854; and grants from the Alfred P. Sloan Foundation. AbstractWe evaluate the bias from endogenous job mobility in fixed-effects estimates of worker-and firm-specific earnings heterogeneity using longitudinally linked employer-employee data from the LEHD infrastructure file system of the U.S. Census Bureau. First, we propose two new residual diagnostic tests of the assumption that mobility is exogenous to unmodeled determinants of earnings. Both tests reject exogenous mobility. We relax exogenous mobility by modeling the matched data as an evolving bipartite graph using a Bayesian latent-type framework. Our results suggest that allowing endogenous mobility increases the variation in earnings explained by individual heterogeneity and reduces the proportion due to employer and match effects. To assess external validity, we match our estimates of the wage components to out-ofsample estimates of revenue per worker. The mobility-bias corrected estimates attribute much more of the variation in revenue per worker to variation in match quality and worker quality than the uncorrected estimates.

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Section: E Online Appendix: Estimation and Data Details E1 Parallelimentioning

confidence: 99%

Modeling Endogenous Mobility in Earnings Determination

Abowd

McKinney

Schmutte

2018

Journal of Business & Economic Statistics

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“…A centered formula is two times more expensive, and it is not clear that the improved accuracy would significantly affect the nonlinear convergence, especially within a quasi-Newton strategy. As in Viallet et al (2011), we use the colored finite differencing algorithm (hereafter CFD, see Curtis et al 1974;Gebremedhin et al 2005). Since the main cost is in evaluating F, CFD minimizes the number of function evaluations by grouping independent columns of the Jacobian in a "compressed" representation of the matrix.…”

Section: Jacobian Matrix Computationmentioning

confidence: 99%

Comparison of different nonlinear solvers for 2D time-implicit stellar hydrodynamics

Viallet

Baraffe

Walder

2013

A&A

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Time-implicit schemes are attractive since they allow numerical time steps that are much larger than those permitted by the CourantFriedrich-Lewy criterion characterizing time-explicit methods. This advantage comes, however, at a cost: the solution of a system of nonlinear equations is required at each time step. In this work, the nonlinear system results from the discretization of the hydrodynamical equations with the Crank-Nicholson scheme. We compare the cost of different methods, based on Newton-Raphson iterations, to solve this nonlinear system, and benchmark their performances against time-explicit schemes. Since our general scientific objective is to model stellar interiors, we use as test cases two realistic models for the convective envelope of a red giant and a young Sun. Focusing on 2D simulations, we show that the best performances are obtained with the quasi-Newton method proposed by Broyden. Another important concern is the accuracy of implicit calculations. Based on the study of an idealized problem, namely the advection of a single vortex by a uniform flow, we show that there are two aspects: i) the nonlinear solver has to be accurate enough to resolve the truncation error of the numerical discretization; and ii) the time step has be small enough to resolve the advection of eddies. We show that with these two conditions fulfilled, our implicit methods exhibit similar accuracy to time-explicit schemes, which have lower values for the time step and higher computational costs. Finally, we discuss in the conclusion the applicability of these methods to fully implicit 3D calculations.

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“…The advantage of this approach for computing higher derivatives over interpolation approaches [10] is that they can be carried over to interval enclosures without additional wrapping. For second and third derivatives graph-coloring like techniques [24] can be used to effectively compute the whole Hessian and tensor of third derivatives in the sparse case.…”

Section: Backward Evaluation Schemementioning

confidence: 99%

Algorithmic differentiation techniques for global optimization in the COCONUT environment

Schichl

Markót

2012

Optimization Methods and Software

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We describe algorithmic differentiation as it can be used in algorithms for global optimization. We focus on the algorithmic differentiation methods implemented in the COCONUT Environment for global nonlinear optimization.The COCONUT Environment represents each factorable optimization problem as a directed acyclic graph (DAG). Various inference modules implemented in this software environment can serve as building blocks for solution algorithms. Many of them use techniques based on various forms of algorithmic differentiation for computing approximations or enclosures of functions or their derivatives.The algorithmic differentiation in the COCONUT Environment does not only provide point evaluations but also range enclosures of derivatives up to order 3, as well as slopes up to second order. Care is taken to ensure that rounding errors are treated correctly. The ranges of the enclosures can be tightened by combining the evaluation routines with constraint propagation. Advantages and pitfalls of this method are also outlined.

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What Color Is Your Jacobian? Graph Coloring for Computing Derivatives

Cited by 223 publications

References 87 publications

Modeling Endogenous Mobility in Earnings Determination

Modeling Endogenous Mobility in Earnings Determination

Comparison of different nonlinear solvers for 2D time-implicit stellar hydrodynamics

Algorithmic differentiation techniques for global optimization in the COCONUT environment

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