Abstract
Four examples are presented on the applications of least squares optimization to real problems. The first three deal with physical chemistry and are: 1. the determination of the molecular force constants of a molecular species from the observed vibrational frequencies of that species; 2. the determination of the parameters describing the interaction of a neptunium 237 nucleus with its environment from the recoilless absorption spectrum of gamma rays, or Mossbauer spectrum, by a neptunium containing substance. 3. the determination of the electronic hamiltonian parameters describing the unfilled f-shell of a paramagnetic ion from the electronic absorption spectrum and/or the paramagnetic susceptibility.
In each of these examples, the model may have to be simplified or enriched in order to properly account for the observed spectrum or curve.
Conversely, the orbit determination of double stars, which pertains to celestial mechanics, relies on an invariable model of ephemeris calculation. The analytical partial derivatives are therefore used in the optimization, contrary to the three other examples.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Morbey, C. L.: A synthesis of the solutions of spectroscopic and visual binary orbits. Publications of the Astronomical Society of the Pacific 87 (1975), 689–693.
Wilson, E. Bright, Jr.; Decius, J. C. and Cross, Paul C.: Molecular vibrations; the theory of infrared and Raman spectra. McGraw-Hill Book Cy, New York (1955).
Rousson, Roger, Tantot, Georges and Tournarie, Max: Numerical determination of vibrational force constants by means of a simplex minimizing method. Journal of Molecular Spectroscopy 59 (1976), 1–7.
Chandler, J. P.: STEPIT - Minimum of a function of several (N) variables. QCPE No. 66.1, Quantum Chemistry Program Exchange, Bloomington, Indiana, USA.
Tantot, Georges: Etude structurale de quelques dérivés oxyûuorés des halogènes. Thèse de doctorat d’état, Université de Paris-6 (1976).
Soulie, Edgar: Etude de quelques composes à électrons f non appariés. Rapport CEA-R-4849 (1977).
Joubert, Philippe: Synthèses, études dynamiques et structurales de quelques dérivés d’addition de l’oxytétraûuorure d’uranium et de fluorures monovalents. Thèse de doctorat d’état, Université de Limoges (1979).
Powell, M. J. D.: VA05A, a subroutine to minimize a sum of squares of m functions in n variables (m x n), without analytical derivatives. In: “Harwell subroutine library, a catalogue of subroutines” complied by M. J. Hopper, AERE-R-9185, fourth edition, November 1981, p. 51, last subroutine revision on 10/09/1974.
Marquardt, Donald W.: An algorithm for least squares estimation of nonlinear parameters. J. Soc. Indust. Appl. Math. 11 (2) (1963), 431 - 441; Jorge J. Moré: “The Levenberg-Marquardt algorithm: implementation and theory” in: Numerical Analysis; G. A. Watson (ed.), Lecture Notes in Mathematics 630 (1977), 105–116, Springer Verlag, Berlin.
Pillinger, W. L. and Stone, J. A.: Methodology of the 237 Np Mössbauer effect. In: Mössbauer effect methodology 4 (1968), 217–236, Irwin J. Gruverman (ed.), Plenum Press, New York.
Ball, W. E.: BSOLVE: Solution to a set of nonlinear equations by Marquardt’s method. In: Material and energy balance computations by Ernest J. Henley and Edward M. Rosen, John Wiley & Sons, Inc., New York (1969), Appendix E 560–566.
Ruby, Stanley L.: Why MISFIT when you already have x 2? In: Mössbauer effect methodology 8 (1973), 263–276, Irwin J. Gruverman and Carl W. Seidel (eds.), Plenum Press, New York.
Caro, Paul: Structure électronique des éléments de transition; l’atome dans le cristal. Presses Universitaires de France, Paris (1976).
Tournarie, Max: Evaluations optimales des inconnues d’un système statistique non linéaire I. Principe et théorie. Journal de Physique 30 (10) (1969), 737–751.
Soulie, Edgar and Goodman, Gordon: Niveaux d’énergie électronique et susceptibilité magnetique des ions de conûguration f 2 en champ cristallin cubique. Theoret. Chim. Acta (Berl.)41 (1976), 17–36; Erratum, Theoret. Chim. Acta (Berl.) 51 (1979), 259–260.
Soulie, Edgar: Champ de coordinats, anisotropic de susceptibilité magnétique et déplacement chimique dans le tetrakis-(acetylacetonato) uranium (IV). Inst. Phys. Conf. Ser. No. 37 (1978), 166–172.
Biaise, Alain; Genet, Michel; Song, Chong-Li and Soulie, Edgar: Structure électronique du tetrakis-(1,1,1-trifluoro -4-phénylbutane-2,4-dionato) uranium (IV): étude magnétochimique. Journal of the Less Common Metals (sous presse).
Couteau, Paul: L’observation des étoiles doubles visuelles. Flammarion, Paris (1978).
Heintz, Wulff Dieter: Doppelsterne. Wilhelm Goldmann Verlag, München (1971).
Yvon Villarceau, Antoine: Méthode pour calculer les orbites des étoiles doubles, déduite de considerations géométriques. In: Additions à la Connaissance des Temps, Gauthier Villars (ed.), Paris (1877).
Soulie, Edgar: L’amélioration de l’orbite d’une étoile double visuelle. Astron. Astro phys. 164 (1986), 408–414.
Lemarechal, Claude: Nonsmooth optimization: use of the code DYNEPS. Communication No. WP-80-36 to the International Institute for Applied Systems Analysis, Laxenburg, Austria (1980).
Baize, Paul: Les masses des étoiles variables à longue Période. L’Astronomie 94 (1980), 71–84. note en bas de page 72.
Dennis, Jr. John E.; Gay, David, M. and Welsch, Roy E.: An adaptive nonlinear least squares algorithm. ACM Transactions on Mathematical Software 7 (3)(1981), 348–368; Algorithm 573 NL2SOL - An adaptive nonlinear least squares algorithm, ACM Transactions on Mathematical Software 7 (3) (1981), 369–383.
Traitement de l’information: Langage de programmation FORTRAN. Norme AFNOR NF-Z 65-110, Paris La Defense (juin 1983 ).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1988 Birkhaürser Verlag Basel
About this chapter
Cite this chapter
Soulie, E. (1988). A Few Examples of Least Squares Optimization in Physical Chemistry and Astronomy. In: Hoffmann, KH., Zowe, J., Hiriart-Urruty, JB., Lemarechal, C. (eds) Trends in Mathematical Optimization. International Series of Numerical Mathematics/Internationale Schriftenreihe zur Numerischen Mathematik/Série internationale d’Analyse numérique, vol 84. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9297-1_21
Download citation
DOI: https://doi.org/10.1007/978-3-0348-9297-1_21
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-0348-9984-0
Online ISBN: 978-3-0348-9297-1
eBook Packages: Springer Book Archive