Convergence Properties of Piecewise Power Approximations

We address the problem of convergence of approximations obtained from two versions of the piecewise power-law representations arisen in Systems Biology. The most important cases of mean-square and uniform convergence are studied in detail. Advantages and drawbacks of the representations as well as properties of both kinds of convergence are discussed. Numerical approximation algorithms related to piecewise power-law representations are described in Appendix.

Born-series approximation to volume-scattering wave for piecewise heterogeneous media

An efficient approximate scheme is presented for wave-propagation simulation in piecewise heterogeneous media by applying the Born-series approximation to volume-scattering waves. The numerical scheme is tested for dimensionless frequency responses to a heterogeneous alluvial valley where the velocity is perturbed randomly in the range of 5 %–25 %,compared with the full-waveform numerical solution. Then,the scheme is extended to a heterogeneous multilayered model by calculating synthetic seismograms to evaluate approximation accuracies Numerical experiments indicate that the convergence rate of this method decreases gradually with increasing velocity perturbations. The method has a fast convergence for velocity perturbations less than 15 %. However,the convergence becomes slow drastically when the velocity perturbation increases to 20 %. The method can hardly converge for the velocity perturbation up to 25 %.

PIECEWISE RATIONAL APPROXIMATIONS OF REAL ALGEBRAIC CURVES

We use a combination of both algebraic and numerical techniques to construct a C-1-continuous, piecewise (m, n) rational epsilon-approximation of a real algebraic plane curve of degree d. At singular points we use the classical Weierstrass Preparation Theorem and Newton power series factorizations, based on the technique of Hensel lifting. These, together with modified rational Pade approximations, are used to efficiently construct locally approximate, rational parametric representations for all real branches of an algebraic plane curve. Besides singular points we obtain an adaptive selection of simple points about which the curve approximations yield a small number of pieces yet achieve C-1 continuity between pieces. The simpler cases of C-1 and C-0 continuity are also handled in a similar manner. The computation of singularity, the approximation error bounds and details of the implementation of these algorithms are also provided.

Smooth interpolation on homogeneous matrix groups for computer animation

Homogeneous matrices are widely used to represent geometric transformations in computer graphics, with interpo- lation between those matrices being of high interest for computer animation. Many approaches have been proposed to address this problem, including computing matrix curves from curves in Euclidean space by registration, representing one-parameter curves on manifold by rational representations, changing subdivisional methods generating curves in Euclidean space to corresponding methods working for matrix curve generation, and variational methods. In this paper, we propose a scheme to generate rational one-parameter matrix curves based on exponential map for interpolation, and demonstrate how to obtain higher smoothness from existing curves. We also give an iterative technique for rapid computing of these curves. We take the computation as solving an ordinary differential equation on manifold numerically by a generalized Euler method. Furthermore, we give this algorithm’s bound of the error and prove that the bound is proportional to the shift length when the shift length is sufficiently small. Compared to direct computation of the matrix functions, our Euler solution is faster.

Modified min-sum SCAN decoding algorithm for polar codes

The state-of-the-art soft-output decoder of polar codes is the soft cancellation(SCAN) decoding algorithm, which performs well at the cost of plentiful computations. Based on the SCAN decoding algorithm, a modified method with revised iterative formula is proposed, marked modified min-sum SCAN(MMS-SCAN). The proposed algorithm simplifies the update formula of nodes and reduces the complexity of iterative decoding process by the piecewise approximation function. Meanwhile, the bit error rate(BER) of the proposed method can approach the performance of original SCAN decoding method without performance loss. The simulation reveals that the MMS-SCAN decoding algorithm can achieve the effect that the BER curve almost coincides with the original SCAN decoding curve.