Victor J. Law

Computation of the gradient and sensitivity coefficients in sum of squares minimization problems with differential equation models

Abstract In parameter estimation problems where the system model consists of differential equations, methods for minimizing a sum of squares of residuals objective function require derivatives of the residuals with respect to the parameters being estimated (sensitivity coefficients) or the gradient of the objective function (depending on the numerical optimization method). This paper considers two methods for generating such derivatives: (1) the adjoint equation — gradient formula; and (2) complimentary sensitivity coefficient differential equations. Particular attention is given to the consistency between the method used to solve the model equations and the proper formulation of the additional equations required by the two methods. Two example problems illustrate computational experience using a modified quasi-Newton method with the adjoint method used to generate gradients and applying a modified Gauss-Newton approach with the sensitivity coefficient equations to calculate both the Gauss-Newton matrix and the objective function gradient. Results indicate the superiority of the sensitivity coefficient approach. When comparing the computational effort required by the two methods and the results from the simple examples, it appears that the use of complimentary sensitivity coefficient equations is much more efficient than using only the gradient of the sum of squares function.

Tutorial on structure charts as an algorithm design tool

Structure charts have been presented via seminars to groups of computing professionals. Some of these software designers have adopted structure charts as their standard design tool. These practitioners are providing impressive anecdotal evidence that structure charts have significant practical value and are not merely of academic interest.n The precise origin of structure charts is not accurately recorded. Their first appearance in a textbook was in Bowles(1) who cites Doran and Tate(2) as their originators. Bowles used the name structure diagram. Jensen and Tonies(3) presented some very similar design notation which they called schematic logic. They claimed that their work was a modification of some previous design methodology due to Jackson.(4) Jensen later presented a modification of schematic logic and called the new notation processing logic trees.(5) The particular geometric shapes used in the structure charts of this paper are the same as those suggested in a forthcoming textbook by one of the authors(6). This tutorial presents structure charts as a design tool which has many desirable characteristics for students of computer science as well as for practicing software designers.

PARAMETER ESTIMATION OF MONOD KINETICS WITHOUT BIOMASS DATA AND INITIAL SUBSTRATE CONCENTRATION

Abstract The uniqueness of the parameters in a Monod kinetics model estimated from substrate depletion and product growth data were analyzed using nonlinear regression. The initial substrate and biomass concentrations were considered as unknown (unmeasured) parameters. Simulated data containing known measurement errors were generated first using specified parameters and then these data were used for the analysis. Sensitivity coefficients were determined by differentiating the original differential equations of the Monod kinetics. A procedure for determining the scale factors for the parameters (used in the nonlinear regression method) has also been developed. The number of parameters that can be determined uniquely depends on the region of substrate depletion (first order, mixed order or zero order). The availability of product data leads to additional estimation of parameters in the mixed order region. The number of parameters that could be estimated are identified qualitatively by the sensitivity coeffi...

SLAW (abstract only): a language free environment—future directions and research

SLAW is a language free environment based on structure chart techniques presented in the Tulane University Department of Computer Science introductory classes [LAW83] [LAW86]. SLAW is currently in its third generation. The first two generations were implemented on a DECSYSTEM-2060 and a VAX 11/780, respectively, by Cook [C0083]. The current implementation, which is the most advanced, is written for the IBM-PC in Turbo Pascal [DOR86a] [DOR86b].nA structure chart can be translated into different programming languages due in part to the dictionary of variables [DOR85]. Because SLAW is a language free environment, variable types are not easily discernible. With the dictionary of variables, it becomes easier to declare variables correctly in the target translation language and to allow for code generation. Currently, the SLAW environment allows for translation into PASCAL, FORTRAN, and BASIC.nA structure chart is represented as a forest of trees. The main program is one such tree. Subroutines and functions are the siblings of the main program, themselves forming trees. Since they have a dictionary of variables, including variables passed as parameters, subroutines and functions are as easily codable as the main program. This representation also allows for recursion. In the main program, subroutines and functions are called by a certain type of node: the CALL box.

Structured programming environments (abstract only)

Structure charts are a graphic algorithm design tool. An on-line editor has been implemented that can be used to create and maintain structure charts. In addition to the graphic nature of structure charts, a data dictionary is necessary to more completely define the design. A current project in this area involves incorporating the data dictionary into the on-line environment. This additional feature enables automatic coding in a high-level programming language.