Parallel monotone spline interpolation and approximation on GPUs

Monotonicity preserving interpolation and approximation have received substantial attention in the last thirty years because of their numerous applications in computer aided-design, statistics, and machine learning [9, 10, 19]. Constrained splines are particularly popular because of their flexibility in modeling different geometrical shapes, sound theoretical properties, and availability of numerically stable algorithms [9,10,26]. In this work we examine parallelization and adaptation for GPUs of a few algorithms of monotone spline interpolation and data smoothing, which arose in the context of estimating probability distributions.

Approximation of centroid end-points and switch points for replacing type reduction algorithms

Despite several years of research, type reduction (TR) operation in interval type-2 fuzzy logic system (IT2FLS) cannot perform as fast as a type-1 defuzzifier. In particular, widely used Karnik-Mendel (KM) TR algorithm is computationally much more demanding than alternative TR approaches. In this work, a data driven framework is proposed to quickly, yet accurately, estimate the output of the KM TR algorithm using simple regression models. Comprehensive simulation performed in this study shows that the centroid end-points of KM algorithm can be approximated with a mean absolute percentage error as low as 0.4%. Also, switch point prediction accuracy can be as high as 100%. In conjunction with the fact that simple regression model can be trained with data generated using exhaustive defuzzification method, this work shows the potential of proposed method to provide highly accurate, yet extremely fast, TR approximation method. Speed of the proposed method should theoretically outperform all available TR methods while keeping the uncertainty information intact in the process.

Linear approximation of Karnik-Mendel type reduction algorithm

Karnik-Mendel (KM) algorithm is the most used and researched type reduction (TR) algorithm in literature. This algorithm is iterative in nature and despite consistent long term effort, no general closed form formula has been found to replace this computationally expensive algorithm. In this research work, we demonstrate that the outcome of KM algorithm can be approximated by simple linear regression techniques. Since most of the applications will have a fixed range of inputs with small scale variations, it is possible to handle those complexities in design phase and build a fuzzy logic system (FLS) with low run time computational burden. This objective can be well served by the application of regression techniques. This work presents an overview of feasibility of regression techniques for design of data-driven type reducers while keeping the uncertainty bound in FLS intact. Simulation results demonstrates the approximation error is less than 2%. Thus our work preserve the essence of Karnik-Mendel algorithm and serves the requirement of low
computational complexities.

Approximation of riemanns zeta function by finite dirichlet series: A multiprecision numerical approach

The finite Dirichlet series of the title are defined by the condition that they vanish at as many initial zeros of the zeta function as possible. It turns out that such series can produce extremely good approximations to the values of Riemanns zeta function inside the critical strip. In addition, the coefficients of these series have remarkable number-theoretical properties discovered in large-scale high-precision numerical experiments. So far, we have found no theoretical explanation for the observed phenomena.

Tail conditional expectation for multivariate distributions : a game theory approach

This paper proposes using the Shapley values in allocating the total tail conditional expectation (TCE) to each business line (X j, j = 1, ... , n) when there are n correlated business lines. The joint distributions of X j and S (S = X1 + X2 + ⋯ + X n) are needed in the existing methods, but they are not required in the proposed method.