Abstract
We present a projection based multiscale optimization method for eigenvalue problems. In multiscale optimization, optimization steps using approximations at a coarse scale alternate with corrections by occasional calculations at a finer scale. We study an example in the context of electronic structure optimization. Theoretical analysis and numerical experiments provide estimates of the expected efficiency and guidelines for parameter selection.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Canning A., Wang L.-W., Williamson A. and Zunger A. (2000). Parallel empirical pseudopotential electronic structure calculations for million atom systems. J. Comput. Phys. 160: 29–41
Cwik, T., Klimeck, G.: Genetically engineered microelectronic infrared filters. In: Proceedings of the First NASA/DoD Workshop on Evolvable Hardware, IEEE, pp. 242–246 (1999)
Demmel J. (1997). Applied Numerical Linear Algebra. SIAM, Philadelphia
Eiben A.E. and Smith J.E. (2003). Introduct Evol Comput. Springer, Berlin
Eriksson K., Estep D., Hansbo P. and Johnson C. (1996). Computational Differential Equations. Cambridge University Press, Cambridge
Franceschetti A. and Zunger A. (1999). The inverse band-structure problem of finding an atomic configuration with given electronic properties. Nature 402: 60–63
Graf, P.: Optimization of model reduction for linear ordinary differential equations. PhD Thesis, University of California at Berkeley, Department of Mathematics (2003)
Hager and W. (2001). Minimizing a quadratic over a sphere. SIAM J. Optimiz. 12: 188–208
Janssen M. (2001). Sequential optimization of integrated climate change models. Math. Comput. Simulat. 54: 477–489
Kim J.: Materials Computation Center. University of Illinois (2003)
Kim K., Graf P. and Jones W. (2005). A genetic algorithm based inverse band structure method for semiconductor alloys. J. Comput. Phys. 208: 735
Kittel C. (1996). Introduction to Solid State Physics. Wiley, New York
Landshoff P., Metherell A. and Rees G. (1997). Essential Quantum Mechanics. Cambridge University Press, Cambridge
Lewis R. and Nash S.G. (2002). Practical aspects of multiscale optimization methods for VLSICAD. In: Cong, J. and Shinnerl, J.R. (eds) Multiscale Optimization and VLSI/CAD, pp 265–291. Kluwer, Boston
Ravindran S.S. (2002). Adaptive reduced-order controllers for a thermal flow system using proper orthogonal decomposition. SIAM J. Sci. Comput. 23(6): 1924–1942
Saad Y. (1992). Numerical Methods for Large Eigenvalue Problems. Manchester University Press, Manchester
Siah, E.S., Sasena, M., Volakis, J.L., Papalambros, P.Y., Wiese R.W.: Fast parameter optimization of large-scale electromagnetic objects using DIRECT with kriging metamodeling. IEEE Trans. Microwave Theory Tech. 52(1) (2004)
Torczon V., Trosset, M.W.: Using approximations to accelerate engineering design optimizations. In: Proceedings of the 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, St. Louis, Missouri (1998)
Vazirani V.V. (2003). Approximation Algorithms. Springer, Berlin
Wang L.-W. and Zunger A. (1994). Solving Schrödinger’s equation around a desired energy: application to silicon quantum dots. J. Chem. Phys. 100: 2394
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Graf, P.A., Jones, W.B. A projection based multiscale optimization method for eigenvalue problems. J Glob Optim 39, 235–245 (2007). https://doi.org/10.1007/s10898-007-9135-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-007-9135-3