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Using Discrete and Continuous Models to Solve Nanoporous Flow Optimization Problems

Published online by Cambridge University Press:  28 May 2015

Paul T. Boggs ,
David M. Gay  and
Stephen G. Nash
Paul T. Boggs
Affiliation:
Sandia National Laboratories (retired), M.S. 9159, P.O. Box 969, Livermore, CA 94551, U.S.A.
David M. Gay
Affiliation:
AMPL Optimization, Inc., 900 Sierra Place SE, Albuquerque, NM 87108-3379 U.S.A.
Stephen G. Nash*
Affiliation:
George Mason University, M.S. 5C8, Fairfax, VA 22030 U.S.A.
*
*Email addresses: paulboggs@comcast.net (Paul T. Boggs), dmg@ampl.com (David M. Gay), snash@gmu.edu (Stephen G. Nash)
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Abstract

We consider using a discrete network model in combination with continuous nonlinear optimization models to solve the problem of optimizing channels in nanoporous materials. The problem and the hierarchical optimization algorithm are described in [2]. A key feature of the model is the fact that we use the edges of the finite element grid as the locations of the channels. The focus here is on the use of the discrete model within that algorithm. We develop several approximations to the relevant flow and a greedy algorithm for quickly generating a “good” tree connecting all of the nodes in the finite-element mesh to a designated root node. We also consider Metropolis-Hastings (MH) improvements to the greedy result. We consider both a regular triangulation and a Delaunay triangulation of the region, and present some numerical results.

Keywords

49M3765K0490C08Multigrid OptimizationMultilevel OptimizationNonlinear OptimizationComplex Hierarchical Optimization
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 8 , Issue 2 , May 2015 , pp. 149 - 167
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Banavar, J.R., Colairoi, F., Flammini, A., Maritan, A., and Rinaldo, A., Topology of the Fittest Transportation Network, Physics Review Letters, 84 (2000), pp. 4745–4748.CrossRefGoogle ScholarPubMed
[2]Boggs, P.T., Gay, D.M., Griffiths, S., Lewis, R.M., Long, K.R., Nash, S., and Nilson, R.H., Optimization Algorithms for Hierarchical Problems, with Application to Nanoporous Materials, SIAM Journal on Optimization, 22 (2012), pp. 1285–1308.CrossRefGoogle Scholar
[3]Boggs, P.T., Gay, D.M., and Nilson, R.H., Network heuristics for initial guesses to nanoporous flow optimization problems, Tech. Report 2012-3681, Sandia National Laboratories, Albuquerque NM and Livermore CA, 2012.Google Scholar
[4]Borzì, A. and Hohenester, U., Multigrid optimization schemes for solving Bose-Einstein condensate control problems, SIAM Journal on Scientific Computing, 30 (2008), pp. 441–462.CrossRefGoogle Scholar
[5]Burer, S. and Letchford, A.N., Non-convex mixed-integer nonlinear programming: A survey, Surveys in Operations Research and Management Science, 17 (2012), pp. 97–106.CrossRefGoogle Scholar
[6]Di, Z., Emelianenko, M., and Nash, S.G., Truncated newton-based multigrid algorithm for centroidal Voronoi diagram calculation, Numerical Mathematics: Theory, Methods & Applications, 5 (2012), pp. 242–259.Google Scholar
[7]Duran, M. and Grossmann, I., An outer-approximation algorithm for a class of mixed-integer nonlinear programs, Mathematical Programming, 36 (1986), pp. 307–339.CrossRefGoogle Scholar
[8]Fredman, M.L. and Tarjan, R.E., Fibonacci heaps and their uses in improved network optimization algorithms, J. ACM, 34 (1987), pp. 596–615.CrossRefGoogle Scholar
[9]Gupta, O. and Ravindran, V., Branch and bound experiments in convex nonlinear integer programming, Management Science, 31 (1985), pp. 1533–1546.CrossRefGoogle Scholar
[10]Lewis, R.M. and Nash, S.G., Factors affecting the performance of an optimization-based multigrid method, in Multiscale Optimization Methods and Applications, Hager, W.W., Huang, S.-J., Pardalos, P.M., and Prokopyev, O.A., eds., Springer, Berlin, 2005, pp. 151–172.Google Scholar
[11]Liu, D.C. and Nocedal, J., On the limited memory method for large scale optimization, Mathematical Programming B, 45 (1989), pp. 503–528.Google Scholar
[12]Nash, S.G., Amultigrid approach to discretized optimization problems, Optimization Methods and Software, 14 (2000), pp. 99–116.CrossRefGoogle Scholar
[13]Nash, S.G., Properties of a class of multilevel optimization algorithms for equality-constrained problems, Optimization Methods and Software, 29 (2014), pp. 137–159.CrossRefGoogle Scholar
[14]Nash, S.G. and Lewis, R.M., Assessing the performance of an optimization-based multigrid method, Optimization Methods and Software, 26 (2011), pp. 693–717.CrossRefGoogle Scholar
[15]Nilson, R.H. and Griffiths, S.K., Optimizing transport in materials having two scales of porosity, Phys. Rev. E, 79 (2009), p. 036304.CrossRefGoogle ScholarPubMed
[16]Rumpfkeil, M.P. and Mavriplis, D.J., Optimization-based multigrid applied to aerodynamic shape design, tech. report, Department of Mechanical Engineering, University of Wyoming, Laramie, 2009.Google Scholar