Distributed Approximating Functionals: a Robust, New Approach to Computational Chemistry and Physics

Abstract

Distributed approximating functionals (DAFs) result from a general approach to describing a class of functions (the “DAF-class”) by expanding them at a point x’ in the neighborhood of x in terms of a complete basis set constructed using the point x as the origin of coordinates:

$$f\left( {x'} \right) = {\sum\nolimits_n {{a_n}\left( x \right)\phi } _n}\left( {x' - x} \right).$$

One may use any convenient complete set {ø_n} for expanding f (x’)with x’ near x and furthermore different bases can be used for different neighborhoods. This results in an extremely general basis set approximation. In fact, we use the truncated (on the summation index n) approximation above only at the single pointx’ ≡ x. The specific form of the expansion coefficients, a_n (x), is determined by a variational optimization. We shall give a detailed derivation in the lecture, and show how a particularly useful approximation to the a _n (x) can be obtained. In the course of the discussion, we shall obtain the resulting DAFs. We also show the relationship of DAFs to “two-parameter Delta sequences”. The theory will be illustrated in terms of the DAF that results from choosing Hermite polynomials as the expansion basis. The properties of these DAFs will be explicated by considering their structure both in coordinate space and in Fourier space. Of greatest importance is the so-called “well-tempered” property of the DAFs. DAFs have been widely applied to solving partial differential equations (with particular emphasis on quantum scattering), as well as the more general problem of constructing approximations to functions based on a finite, discrete sampling. The original derivation of a DAF was for the quantum mechanical free propagator. General, non-product sampling in multidimensional systems can be employed, including Monte Carlo and number theoretic methods. Finally, DAFs have an intimate connection to wavelets. In addition to quantum mechanics, example potential applications (some of which have been realized) include solving nonlinear partial differential equations in situations where instabilities are encountered by other methods, signal denoising, signal enhancement, signal inversion, data compression (both lossless and lossy), imaging, pattern recognition and characterization, medical imaging, teleradiology, target acquisition, periodic and non-periodic extensions of functions in 1D, 2D, 3D, 4D,..., filling in gaps in data (including noisy experimental data), imposition of general boundary conditions onto experimental or computational data, etc.