New Method For Fast Evaluation Research Articles

Compressed sparse tensor based quadrature for vibrational quantum mechanics integrals

A new method for fast evaluation of high dimensional integrals arising in quantum mechanics is proposed. The method is based on sparse approximation of a high dimensional function followed by a low-rank compression. In the first step, we interpret the high dimensional integrand as a tensor in a suitable tensor product space and determine its entries by a compressed sensing based algorithm using only a few function evaluations. Secondly, we implement a rank reduction strategy to compress this tensor in a suitable low-rank tensor format using standard tensor compression tools. This allows representing a high dimensional integrand function as a small sum of products of low dimensional functions. Finally, a low dimensional Gauss–Hermite quadrature rule is used to integrate this low-rank representation, thus alleviating the curse of dimensionality. Numerical tests on synthetic functions, as well as on energy correction integrals for water and formaldehyde molecules demonstrate the efficiency of this method using very few function evaluations as compared to other integration strategies.

A New Method for Fast Evaluation of the Cooray–Rubinstein Formula in Time Domain

The Cooray–Rubinstein formula is most widely used to evaluate the horizontal electric field generated by a lightning return stroke over a lossy ground. In this paper, a new method for fast evaluation of the Cooray–Rubinstein formula in time domain is proposed, by means of approximating the integral form of the modified Bessel function of the first kind using Gauss numerical integral. In the proposed method, the integral form of the modified Bessel function of the first kind is expressed by a summation of exponential functions, and a recursive convolution is achieved, which makes the calculation of the Cooray–Rubinstein formula in time domain much more efficient. Besides, the residues and poles due to the expression of the kernel function K ( s ) can be obtained directly for any ground conductivity and permittivity which makes the method proposed very simple and easy to program. The numerical examples demonstrate the validity and efficiency of the proposed method.

Faster fast evaluation of thin plate splines in two dimensions

A new method for fast evaluation of thin plate splines in two dimensions is presented. The paper first develops exponential approximations to thin plate splines. These are the analytical basis for an improved fast multipole evaluator. Analytic error bounds are supplemented by offline parallel numerical computation of the underlying error constants. These error constants enable adaptive selection of series lengths as a function of the weights associated with a source panel, and the desired accuracy. Numerical comparisons with a competing algorithm show that the new method is significantly faster when moderate to high precision is required.

Fast evaluation of cutting forces in milling, applying no approximate models

A new method for fast evaluation of cutting forces in milling is introduced and tested experimentally. Unlike all existing procedures, which include the use of cutting models and approximate assumptions, in this method, the elementary functions of the cutting force are obtained from measured values only. The basic force functions for the whole feed range are acquired from one experiment using a single-tooth full-diameter (slot) milling, applying a specially developed procedure. The milling experiment is conducted under low-impact conditions, enabling accurate measurement and convenient signal processing. The basic force functions are then integrated and superimposed, using known procedures, to combine the total force in any multitooth milling combination. In this work the method is explained and tested experimentally. The suggested method enables a reliable evaluation of the cutting forces, while demanding minimal experimental work, the method applies to cutters having complicated edge geometry, and to high speed milling.