Concerning Multivariate Bernstein Polynomials and Stochastic Logic

Uniform approximation and Bernstein polynomials with coefficients in the unit interval

In this paper, the multivariate Bernstein polynomials defined on a simplex are viewed as sampling operators, and a generalization by allowing the sampling operators to take place at scattered sites is studied. Both stochastic and deterministic aspects are applied in the study. On the stochastic aspect, a Chebyshev type estimate for the sampling operators is established. On the deterministic aspect, combining the theory of uniform distribution and the discrepancy method, the rate of approximating continuous function and L p convergence for these operators are studied, respectively.

We answer and generalize an open problem on the two-dimensional Bernstein polynomials on the unit triangle.

Division algorithms for Bernstein polynomials

The <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> -out-of- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> : G(F) majority voter consists of n components (or modules) and a number of the components are required to be operating correctly for the overall system to be correct. As per the state discretization of the components, such a system is usually classified as either a binary system or a multi-state system. In practice, the operating conditions of different components may contribute differently to the operation of the entire system. In this manuscript, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> -out-of- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> : G(F) majority voter is generalized as a consecutive-weighted- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> -out-of- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> : G(F) voter with either binary states or multiple states. To overcome the drawbacks of existing approaches, a stochastic analysis is proposed for assessing the system reliability. In the stochastic analysis, the input signal probabilities are encoded into non-Bernoulli sequences with fixed numbers of 0s and 1s for the Boolean case, or randomly permuted sequences for the multi-state scenario. By using stochastic logic, the reliability of a general system consisting of consecutive-weighted- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> -out-of- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> majority voters is efficiently and accurately predicted. The results are validated by an analysis of several case studies. Although the accuracy of the stochastic analysis is closely related with the employed sequence length, it is shown that a stochastic approach is more efficient than a universal generating function (UGF) method, while still retaining an acceptable accuracy.

In stochastic computing (SC) systems numbers are represented with mean values of random binary sequences. This paper introduces a novel fuzzy inference architecture, in which the computational mechanism is based on stochastic logic (SL). First, the basic concept of SL is described, then the architecture of the SL-based fuzzy logic controller (SFLC) is built up systematically using the derived stochastic elements. The second part of the paper demonstrates the application of the proposed techniques, where the SFLC-based control performance is evaluated on a real mechatronic system. The results show that the SL-based approach provides effective and robust control performance, simple architecture and high noise tolerance. The proposed method is also benchmarked against conventional FLCs indicating that the robustness of the stochastic architecture allowed to outperform the benchmark controllers in noisy environments.

An identity for multivariate Bernstein polynomials

This study develops the method of constructing a multi-parametric distribution on the unit interval using Bernstein polynomials and investigates its key statistical properties. The performance of the proposed distribution’s parameter estimators was evaluated by Monte Carlo simulation under both maximum-likelihood (MLE) and minimum-distance frameworks, with the MLE approach generally yielding lower estimation error. A real data example illustrates the advantages of the construction method. The results reveal that the Bernstein distribution provides superior goodness-of-fit to the dataset compared with competing models.

Robust arithmetic for multivariate Bernstein-form polynomials

Polynomial operators and local smoothness classes on the unit interval

A general method for rendering isosurfaces of multivariate rational and polynomial tensor products is described. The method is robust up to degree 15, handling singularities without introducing spurious rendering artifacts. The approach does not solve the problem of singularities in general, but it removes the problem from the rendering domain to the interpolation/approximation domain. It is based on finding real roots of a polynomial in Bernstein form. This makes it particularly suitable for parallel and pipelined processing. It is envisioned that the tensor products will be used as approximants or interpolants for empirical data or scalar fields. An interpolation scheme is given as an example. >

In this paper, we present a new approximation method for non-linear activation functions including tanh and sigmoid functions using stochastic computing (SC) logic based on the piecewise-linear approximation (PWL) for the full range of [-1, 1]. SC implementations with PWL approximation expansions for non-linear functions are based on a 90nm CMOS process. The implementation results shown that the proposed SC circuits can provide better performance compared with the previous methods such as the well-known Maclaurin expansions based, Bernstein polynomial based and finite-state-machine (FSM) based implementations. The implementation results are also presented and discussed.

We present an open source toolbox in Scilab for multivariate polynomial optimization based on the Bernstein form. The developed toolbox finds the global minimum of unconstrained polynomial optimization problems. We first describe the method used by the toolbox, and then demonstrate its performance on several standard optimization examples. We also compare the quality of the results obtained using the developed toolbox with those of popular global optimization methods, such as genetic algorithms, simulated annealing, pattern search, and global search interior point methods. The tests show the superior performance of the presented toolbox.

V.9 - SUBDIVIDING SIMPLICES

Stochastic logic allows complex arithmetic to be performed with very simple logic, but it suffers from high latency and poor precision. Furthermore, the results are always somewhat inaccurate due to random fluctuations. The random or pseudorandom sources required to generate the representation are costly, consuming a majority of the circuit area (and diminishing the overall gains in area). This chapter reexamines the foundations of stochastic computing and comes to some surprising conclusions. It demonstrates that one can compute deterministically using the same structures that are used to compute stochastically. In doing so, the latency is reduced by an exponential factor; also, the area is reduced significantly (and this correlates with a reduction in power); and finally, one obtains completely accurate results, with no errors or uncertainty. This chapter also explores an alternate view of this deterministic approach. Instead of viewing signals as digital bit streams, we can view them as periodic signals, with the value encoded as the fraction of the time that the signal is in the high (on) state compared to the low (off) state in each cycle. Thus we have a time-based encoding. All of the constructs developed for stochastic computing can be used to compute on these periodic signals, so the designs are very efficient in terms of area and power. Given how precisely values can be encoded in the time, the method could produce designs that have much lower latency that conventional ones.