Anal. Chem. 7992. 64. 1199-1200
which is valid for all points that lie on the interior boundary r = a. We substitute eq 52 into eq 15, and the result is substituted into eq 56 evaluated at r = d to obtain
which is valid for all points that lie on the exterior boundary r = d. We then substituted eq 54 into eq 14 to obtain
which is valid for all points that lie at the interface between the two regions, i.e. at r = b. Boundary condition eq 7 is simply
c1 in
K C 2 in
(60)
and is also valid for all points that lie at r = b. The initial conditions are thus
c1:
=
ct
(61)
and
c,: = czo Now we determine the concentrations at intervals of Ar along the radius. We thus need to determine the concentrations C1l,C12,CIS, ...,CIM-I, CIM,C ~ MC2~+1, , ...,C ~ N Thus . there are N + 1 concentrations to determine using N + 1 equations (1 from eq 57,M - 2 from eq 56,l from eq 59,lfrom eq 60, N - M - 1 from eq 56,and 1 from eq 58). For each value of n (i.e. at each time), we have to solve a system of N + 1 simultaneous linear equations. A t each time we know that Cn-% and we want to solve for the Cn's. Rearranging eq 57 we obtain eq 16,rea~angingeq 56 we obtain eq 17,rearranging eq 59 we obtain eq 18,rearranging eq 60 we obtain eq 19, rearranging eq 56 we obtain eq 20,and rearranging eq 58 we obtain eq 21.
Registry
NO.
1199
C&, 71-43-2;C~H~CHB, 108-88-3;HsCCeHd-
p-CH,, 106-42-3; HzO, 7732-18-5; fused silica, 60676-86-0.
REFERENCES (1) Suffet. I. H.. Maiaiyandi, M., Eds. Advances In Chem&by, Serk*r, Vol. 2 1 4 : Organic Pohtants In Water: San@hg, Analysb. and Toxicby Testing; Amerlcan Chemical Society: Washington, D.C., 1987. (2) Ciesceri, L. S., Greenberg. A. E., TmsseW, R. R., Eds.; SmnderdMethod$ for the Examlnadbn of Water and Wastewater, 17th ed.;Am& can Public Health Association: Washington, D.C., 1989. (3) Arthur, C. L.; Pawllszyn, J. Anel. Chem. 1990, 6 2 , 2145-2148. (4) Beiardl, R. G.; Pawllszyn, J. Water Pollut. Res. J . Can. 1989, 2 4 , 179. ( 5 ) Crank, J. The Methemtics of Diffusion, 2nd ed.; Ciarendon Press: Oxford, U.K., 1989. (6) Newns, A. C.; Park, 0. S. J . pdvm. Scl.: part C 1989, 22. 927-937. (7) Wendt, J. 0. L.; Frarler, 0. C., Jr. Ind. Eng. Chem., Funohm. 1973, 12 (2), 239-243. (8) Van Krevelen, D. W. properties of Pdymers: 73ek Estknetkn and m t b n wlth Chemlcal Structure,2nd ed.; Elsevler Sclentlflc Publlshlng Company: Amsterdam, The Netherlands, 1976. (9) Brandrup, J.; Immrgut, E. H. polvmer Hendbodc, 3rd ed.;John Wiley and Sons Inc.: New York, 1989. (10) Ohlberg, S. M.; Alexander, L. E.; Warrlck, E. L. J . pdvm. Scl. 1958, 2 7 , 1. (11) M I k . M. Michele et ai. Envkon. S d . Technol. 1985, 19 (6), 522-529. (12) Keith, L. H. Advances in the IdenMfkaNOn and Analysis of Organlc Pollut8nts In Water, Vol. 2 ; Butterworths: Sevenoaks, U.K., 1981. (13) Van Hall, C. E.; Editor ASTh4 Speck?/ TeChnlcelFubkabbn 686: Measurement of Organic Pollutants In Water and Wastewater. Proceedings of a Symposium Held in Denver. 19-20 June I978\; ASTM: Phliadelphia, PA, 1979. (14) Keith, Lawrence H. Advances in the IdenMflcatlOn andAna&sb of Organic Pollutants In Water, Vol. 1 ; Butterworths: Sevenoaks, U.K., 1QS1. ..
(15) Llska, I.; Krupcik, J.; Leclercq, P. A. J . High Resolut. Chrometogr. 1989. 12 (9). 577-90. (18) Leithe, w.' Ana/ysb of Organic Po//utants in water and waste water Ann Arbor Sclence Publ.: Ann Arbor, MI, 1973. (17) . . McGregor, R. Diffusion and Sorption In Flbers and Fllms; Academic Press: -London, U.K., 1974. (18) Boyce, W. E.; DIPrima, R. C. Elementary Dltferential E9uaHOns and Boundery Value problems, 4th ed.;John Wlley and Sons, Inc.: New York. 1986. (19) Sp&el, M. R. Applled DlfXwentiel Equations, 3rd ed.; Prentlce-Hall, Inc.: Englewood Cliffs, NJ, 1981. (20) Rlggs, J. B. An Introducdbn to Numerial Methods for Chemical Englnews; Texas Tech University Press: Lubbock, TX, 1988.
RECEIVED for review October 1,1991. Accepted January 21, 1992.
CORRESPONDENCE Exchange of Comments on Convergence of Generalized Simulated Annealing with Variable Step Size with Application toward Parameter Estimations of Linear and Nonlinear Models Sir: In a recent article in this journal, Sutter and Kalivas have suggested that the technique of generalized simulated annealing (GSA) can be modified for good use in fitting data to functions.' The GSA technique was developed to find global minima or maxima associated with response surfaces. These authors show that the addition of a variable step size (designated as VSGSA) to GSA results in improved fitting of two examples previously treated with a simplex optimization by Zupan and R i m 2 The present communication is not concemed with response surface methodology. We specifically address the issue of whether VSGSA converges to optimum parameter values in the two examples of data fitting given by Sutter and Kalivas. It is somewhat surprising that neither pair of authors 0003-2700/92/0384-1199$03.00/0
compared the results of their fitting techniques with the well-established procedure of using nonlinear least squares to fit data to a given function. Nonlinear least squares has a substantial history of being extremely useful for this purpose and is almost certainly the most generally applicable technique. A number of examples and pertinent references are given in the review article by Rusling.s We have refit the data used by Sutter and Kalivas by two related processes. The first process uses a general purpose nonlinear least squares approach (using a MarquardtLevenberg algorithm), and the second u888 an approach which considers the problem as a set of simultaneous equations to be s ~ l v e d . ~The ? ~ best-fit criterion for our treatment is the same as that used in refs 1 and 2 (designated by OC there); 0 1992 American Chemical Society
Anal. Chem. 1992, 6 4 , 1200
1200 Table I. Results of Fitting the Function Y = a
exp(-cX)
least squares-3" SS = 0.000604 a = 0.0056 (0.oos9) b = 0.97 (0.01) c = 4.77 (0.12)
least squares-2b
eq solving'
+b VSGSAd
SS = 0.000671 SS = 0.000604 SS = 0.000767 a = 0.0 a = 0.00558 a = 0.0
b = 0.98 (0.01) b = 0.9737 c = 4.70 (0.08) c = 4.767
b = 0.97 = 4.62
c
"Three parameter least squares with parameter errors in parentheses. b T w o parameter least squares with parameter errors in parentheses. Simultaneous equation solution. Note that the agreement in parameter values between least squares and equation solving, although not shown, is exact to more than six digits. The least squares answers (and errors) are rounded while the equation solving answers are given to more digits. dResult of three parameter fit reported by Sutter and Kalivas.'
+ bZ2
Table 11. Results of Fitting the Function Y = ax2
+ cXZ + dX + e 2 + f least squares"
eq solving
VSGSAb
SS = 0.2305
SS = 0.2305
SS = 9.44
a = 0.018 (0.002)
a = 0.01795
a = -0.0022
b = -1.42 (0.79) c = 0.014 (0.028) d = -0,995 (0.172) e = 16.5 (8.7) f = 33.4 (23.9)
b = -1.420 c = 0.0145 d = 0.9952 e = 16.51 f = 33.455
b = -0.0041 c = -0.0209 d = 0.002 e = 2.3226 f = -5.1236
Least squares and equation solving values are expressed in the same manner as in Table I. Parameter agreement is extended to at least six digits between these fits. *Result of six parameter fit reported by Sutter and Kalivas.' a minimum in the sum of squares of residuals between observed and calculated Y values, for all data points, is expressed by eq 1. Our results and the VSGSA results obtained by Sutter and Kalivas are given in Tables I and 11.
Sir: Tucker has directly shown that variable step size generalized simulated annealing (VSGSA) in its present form is not necessarily the best way to estimate model parameters for two data sets.l Because Sutter and Kalivas2 showed the VSGSA optimization method to perform better than the popular simplex method? Tucker has also shown that simplex is not reasonable. The VSGSA procedure is based on generalized simulated annealing (GSA).4 In a recent publication, it was shown that GSA does not always converge to the global optimum? Thus, one would expect s i m i problems for VSGSA. As with many new methodologies, modifications become necessary as they are tested by other scientists. To this end, it appears that VSGSA does indeed require adjustments for parameter estimations and we are thankful to Tucker for pointing this out to us and the scientific community. A primary question raised by Tucker is the influence the optimization criterion behavior has on convergence of VSGSA. Additionally, are particular problems more agreeable to VSGSA, and if so, how are they recognized? Some work has been accomplished toward this goal for simulated annealing: but to date, we are not aware of any studies of this kind for GSA or VSGSA. In the paper commented on by Tucker, VSGSA was also shown to operate quite successfully toward optimizing the conditions for a given model. In that situation, VSGSA was capable of converging to the exact global optimum. In another paper, VSGSA has been verified as able to locate the global optimum on a rather complex response surface.' The authors 0003-2700/92/0364-1200$03.00/0
The results of fitting both sample sets of data show that the least squares procedure obtains SS values which are substantially lower than the SS values for the VSGSA fits. The least squares parameters are therefore more closely related to globally optimum parameter values than are the parameters from VSGSA. Although nonlinear least squares has been used to fit both models here, the second model could be fit by using linear least squares. A linear least squares fit of the second model should produce essentially the same parameter and SS values shown in Table I1 for the nonlinear fit. To be useful,a new technique for fitting data to well-defined functions or models should have one or more redeeming features. Characteristics such as finding optimum parameter values more quickly or more accurately or working extremely well for difficult problems would be among the primary reasons for employing a new fitting technique. In the limited cases examined here, the VSGSA technique appears to offer none of these advantages when compared with traditional least squares procedures.
REFERENCES (1) (2) (3) (4) (5)
Sutter, J. M.; Kalivas, J. H. Anal. Chem. 1991, 63, 2383. Zupan, J.; Rlus. F. X. Anal. Chlm. Acta 1990, 239. 311. Rusllng, J. F. Crlt. Rev. Anal. Chem. 1989, 27 (l), 49. Christian, S. D.; Tucker, E. E. Am. Lab. 1982, 74 (9), 31. Christian, S. D.; Tucker, E. E.; Enwall, E. Am. l a b . 1986, 78(6), 41.
Edwin E. Tucker CET Research Group, Ltd. P.O.Box 2029 Norman, Oklahoma 73070 RECEIVED for review October 28,1991. Accepted February 21, 1992.
have developed a tutorial version of GSA and VSGSA capable of running on IBM/AT compatible computers. A copy of the package is available by writing the authors. In summary, we are still confident in VSGSA; to restate the last sentence of the preliminary VSGSA article,2 "These applications of VSGSA express the ability of the algorithm to converge onto global or near-global conditions".
REFERENCES (1) Tucker, E. E. Anal. Chem., precedlng paper in this issue. (2) Sutter,J. M.; Kalivas, J. H. Anal. Chem. 1901, 63, 2383. (3) tupan, J.; Ruis, F. X. Anal. Chim. Acta IWO, 239, 311. (4) Bohechevsky, I. 0.;Johnson, M. E.; Stein, M. L. Technometrics 1986, 28, 209. (5) Kaihras, J. H.; Roberts, N.; Sutter, J. M. Anal. Chem. 1989, 61, 2024. (6) Sorkln, 0. B. A@otWtm/ca 1991. 6 , 367. (7) Hkchcock, K.; Kalivas. J. H.; Sutter, J. M. J . Chemom., in press.
John H.Kalivas* Jon M. Sutter
Department of Chemistry Idaho State University Pocatello, Idaho 83209
RECEIVED for review February 10,1992. Accepted February 21, 1992. 0 1992 Amerlcan Chemical Society
