Nelder-Mead simplex optimization algorithm

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. 

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. 

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. 

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).

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 packages, e.g., BMDP ASCL-Optimize and Matlab. 

Commercially available software developed to process individual impedance spectra use few general algorithms such as Levenberg-Marquardt algorithm, the Nelder-Mead downhill simplex method or genetic algorithms [3-7]. The software is optimized to process only... 

The rigidity that prevented an accurate optimal point from being obtained was solved by Nelder and Mead [17] in 1965. They proposed a modification of the algorithm that allowed the size of the simplex to be varied to adapt it to the experimental response. It expanded when the experimental result was far of the optimum - to reach it with more rapidly - and it contracted when it approached a maximum value, so as to detect its position more accurately. This algorithm was termed the modifiedsimplex method. Deming and it co-workers published the method in the journal Analytical Chemistry and in 1991 they published a book on this method and its applications. 

The resulting set of differential equations was solved numerically by means of an improved Euler method with variable step size control [9]. Parameter optimization was accomplished by a Mead Nelder simplex algorithm [10]. 

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