Best accessible estimation: Convergence properties and limiting forms of the direct and reduced versions

Abstract

Best accessible estimation is an iterative numerical procedure based on the stochastic extension concept that an integral equation of the first kind is improperly posed because it ignores background noise on the unknown function and measurement or computational error in the known function. Certain restrictions on the “reduced” version of the method, introduced originally to combat extreme noise levels, without resort to manipulation of very large matrices, have been removed. Alternate computational methods are presented, together with a basis for choosing between them according to the relative seriousness of different sources of error in a particular problem. It is shown that, in the absence of computational error, the iterative procedure would converge to the minimum-residue approximate solution of minimum norm, when appropriate ellipsoidal norms are employed which are defined by the noise and error processes present. There is no restriction on the form or correlation of these processes. Practical tests to detect cumulative numerical error and terminate iteration have been evaluated. An example, using actual experimental data, is given of the ability of the method to enhance the effective resolution of measurements. It is noteworthy that, when the equation to be solved represents inference of the input to a physical instrument from its measured output, then the iteration reveals most rapidly those sources to which the instrument responds most strongly and for the measurement of which it would most logically be employed. The concept of strength of response is sharpened by identification with the singular value belonging to a pair of basis functions for input and output, measured in terms of signal to noise ratio.