Samples, Symmetries and Extensions: Exploring Parameter Space in Nonlinear Problems

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Andrew Curtis


Many scientific and technological advances require the values of a set of parameters to be constrained or estimated using recorded data. It is often possible to model data that would be recorded for particular parameter values using a computable and in general nonlinear function, where its inverse is unknown and does not exist as a unique-valued mapping. The space of parameter values must then be explored to find values that are consistent with measured data. This may be achieved by sampling values at a sufficiently dense set of points in parameter space, evaluating the forward function for each sample, and testing their modelled data against the measured data. For many-parameter (high-dimensional) systems, achieving sufficient sample density may be infeasible due to the 'curse of dimensionality' - the exponential increase in sampling required to achieve similar sample density in spaces of increasing dimensionality. However, this article shows that for each sample, significantly more information may be available than the forward function evaluation alone, due to symmetries and other properties of the physical system. Call this additional information the extension of a sample. In travel time tomographic imaging, the extension of every sample provides forward function values throughout continuous parameter semi-subspaces. The dimensionality of these subspaces increases exponentially with the dimensionality of the problem, almost matching the rate of information increase required by the so-called curse of dimensionality. The use of extensions within Physics-based sampling schemes may therefore contribute to increase the dimensionality of problems in which one can explore parameter space, solve inverse or inference problems, or provide distributed information about forward function values directly.



Physical Sciences and Mathematics


Inversion, Geophysics, uncertainty, Symmetry, Tomography, Exploration, Monte Carlo, Nonlinear, Sampling, Nullspace, Ray, Linearised, Parameter space, Travel time, Extensions


Published: 2020-12-31 09:31

Last Updated: 2020-12-31 17:31


CC BY Attribution 4.0 International

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