Fabio Boschetti,
Research Scientist,
CSIRO Ocean & Atmosphere, Australia
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Optimisation

Mathematical knowledge mostly expresses itself via 'forward modeling', that is the ability to describe the result of a process acting under certain initial conditions (cause -> effect framework). However, scientific quest usually strives for the opposite, that is, to define the causes which lead to specific effects (cause <- effects). Roughly, this is what inverse theory is about. For only very few problems a direct solution to the effect->cause problem can be found, so inversion is mostly about more or less cleverly designed heuristics.

I have applied inversion to very different mathematical problems with the purpose of building a tool for expert users as well as for users with little mathematical background. The inversion strategy is based mainly on the four components described below; depending on the expertise, the various components can be used differently:

 

Optimization routines

A set of optimization routines, based mostly on global search strategies, but complemented by local search engines, when needed. They include Genetic algorithms, Swarm optimization, Direct Algorithm, Lipswitch search, plus local searches with and without use of derivatives. The choice of the routine to use depends on the kind and dimensionality of the inverse problem.

Interactive Inversion

This is a surprising powerful technique in which the traditional numerical cost function is replaced by a subjective user evaluation. This is designed to deal with problems which are non numerical or which can not return a numerical solution. Examples are geological problems for which a measure of geological similarity to structures seen in Nature is not available or natural resource management problems in which we goal may not be easy to define or agree upon. This is dealt with by asking expert users to rank results according to subjective criteria. This method is also used to include expert knowledge in numerical inversion.

Parameter space visualisation

We consider this an integral part of an optimisation process. See here.

Applications