Nelder-Mead simplex algorithm

Figure 53. Example of GenOpt calibration run, using a Nelder Mead Simplex algorithm...

We do not design our own algorithm here but use the fin Insearch. m function supplied by Matlab. It is based on the original Nelder, Mead simplex algorithm. As an example, we re-analyse our exponential decay data Data Decay. m (see p. 106], this time fitting both parameters, the rate constant and the amplitude. Compare the results with those from the linearisation of the exponential curve, followed by a linear least-squares fit, as performed in Linearisation of Non-Linear Problems, (p.127). 

The Nelder-Mead SIMPLEX algorithm has been frequently used in Analytical Chemistry as well as in other areas of science and engineering. Assessment and further development of the method remains an active field of research (4.). 

The Nelder-Mead simplex algorithm was published already on 1965, and it has become a classic (Nelder Mead, 1965). Several variants and applications of it have been published since then. It is often also called the flexible polyhedron method. It should be noted that it has nothing to do with the so-called Dantzig s simplex method used in linear programming. It can be used both in mathematical and empirical optimization. 

The Nelder-Mead simplex algorithm [13] was used to fit the experimental data to the LF equation. The goodness-of-fit was determined using the following standard deviation equation... 

If the numerical computation of the gradient of an objective function shall be avoided, and if accuracy requirements are not too high, a direct method such as the Nelder-Mead simplex algorithm [12] implemented in the Scilab function fminsearch () may be used that allows for noise in the cost function. 

To perform the maximization over (X,t), we need an algorithm such as the Nelder-Mead simplex search (14). An alternative that is adequate in many cases is a simple search over a (X,t) grid. The critical value XX has an interpretation of its own. It is the upper bound on a simultaneous prediction interval for ng as yet unobserved observations from the background population. 

Basically two search procedures for non-linear parameter estimation applications apply. (Nash and Walker-Smith, 1987). The first of these is derived from Newton s gradient method and numerous improvements on this method have been developed. The second method uses direct search techniques, one of which, the Nelder-Mead search algorithm, is derived from a simplex-like approach. Many of these methods are part of important mathematical computer-based program packages (e.g., IMSL, BMDP, MATLAB) or are available through other important mathematical program packages (e.g., IMSL). 

SimSim performs a pressure match of measured and calculated reservoir or compartment pressures with an automatic, non-linear optimization technique, called the Nelder-Mead simplex algorithm3. During pressure matching SimSim s parameters (e.g. hydrocarbons in place, aquifer size and eigentime, etc.) are varied in a systematic manner according to the simplex algorithm to achieve pressure match. In mathematical terms the residuals sum of squares (least squares) between measured and calculated pressures is minimized. The parameters to be optimized can be freely selected by the user. 

There are two basic types of unconstrained optimization algorithms (I) those reqmring function derivatives and (2) those that do not. The nonderivative methods are of interest in optimization applications because these methods can be readily adapted to the case in which experiments are carried out directly on the process. In such cases, an ac tual process measurement (such as yield) can be the objec tive function, and no mathematical model for the process is required. Methods that do not reqmre derivatives are called direc t methods and include sequential simplex (Nelder-Meade) and Powell s method. The sequential simplex method is quite satisfac tory for optimization with two or three independent variables, is simple to understand, and is fairly easy to execute. Powell s method is more efficient than the simplex method and is based on the concept of conjugate search directions. 

To offer more flexibility we adopt an approach, based on the transient simulation model TRNSYS (Klein et al., 1976), making use of the Lund DST borehole model (Hellstrom, 1989). The parameter estimation procedure is carried out using the GenOPT (Wetter, 2004) package with the Nelder and Mead Simplex minimization algorithm (Nelder and Mead, 1965) or Hooke and Jeeves minimization algorithm (Hooke and Jeeves, 1961). 

Figure 17 Schematic illustrating the concept of adaptive simplex optimization using the Nelder-Mead algorithm described in Olsson and Nelson (1975). The simplex initially expands in size and so makes rapid progress toward the minimum. It then contracts repeatedly, allowing it to converge on the minimum at (3,2).

A more rewarding solution to this problem is the use of modified Simplex procedures, such as first described by Nelder and Mead [507], Such modified algorithms allow other operations besides reflecting the triangle, such as contraction or expansion. The manner in which such a modified Simplex algorithm proceeds is illustrated in figure 5.8 for a... 

The different mixing rules and nomenclature used are described in table 1. The simplex algorithm modified by Nelder-Mead (10) is used to fit the model parameter to experimental solubility. 

A polynomial was fit to the calibration curve for the thermocouple by means of a minimization of the maximum deviation technique using the Nelder Mead sequential simplex minimization algorithm method.( 5,6, 7) The coefficients of this polynomial are stored in the analysis program and are used to convert thermocouple voltages to temperature values. Y values are converted to dH(t,T)/dt, the heat flow into and out of the sample in mcal/sec. The operator selects a baseline for the analysis by entering the temperatures of the beginning and end points of the baseline. A plot is produced of the raw data with the operator selected baseline shown as illustrated in Figure A. 

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