Mathematical models involve matrix analysis, and matrix transformations, for example 3D rotations, but the solution is normally found using optimisation.

Why use optimisation?

You may recall a from school-days that it could be tricky to solve 2 simultaneous equations. In engineering we may be dealing with 10 or even 100 such equations which may be non-linear. To solve this type of problem requires an optimisation strategy.

Optimisation involves locating the minimum or maximum point of a multi-variable function. This can be easily done in 2D or 3D with simple differentiation. However, when the function is dependent on many variables the problem becomes more complicated.

A multi-variable function often has a huge number of local minima, that can mislead the optimisation program. A starting point which is fairly close to the absolute minima needs to be carefully selected. Multi-variable systems often require statistical analysis, and this can be built into an optimisation routine. This offers improved algorithm stability and fewer local minima. The choice of such a technique is dependent on the application.

Reality

Even the Chancellor needs to check his economic forecasts against reality, the same is true for any mathematical model. In engineering it is important to have a eye on the practicalities of the solution suggested by any model.

Some algorithms that DIVERSE have utilised for optimisation are DH Simplex method, Powell's method as well as classics techniques like Newtons method and Conjugate-Gradient. Each of them has its own characteristics and is suitable for a certain type of functions. For example, DH Simplex is unique in a way that it does not need to calculate the differential of the function, and unlike the others, this method does not need any linear (2D) optimisation algorithms to support it.

For a particular physical scenario, it is important that the most appropriate method is chosen. Determination of the choice of algorithm can only be made by having a realistic model in place which allows each to be tested in a number of physically realistic situations.

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