Local Polynomial Approximation Research Articles

Maximally smooth cubic spline quasi-interpolants on arbitrary triangulations

We investigate the construction of C2 cubic spline quasi-interpolants on a given arbitrary triangulation T to approximate a sufficiently smooth function f. The proposed quasi-interpolants are locally represented in terms of a simplex spline basis defined on the cubic Wang–Shi refinement of the triangulation. This basis behaves like a B-spline basis within each triangle of T and like a Bernstein basis for imposing smoothness across the edges of T. Any element of the cubic Wang–Shi spline space can be uniquely identified by considering a local Hermite interpolation problem on every triangle of T. Different C2 cubic spline quasi-interpolants are then obtained by feeding different sets of Hermite data to this Hermite interpolation problem, possibly reconstructed via local polynomial approximation. All the proposed quasi-interpolants reproduce cubic polynomials and their performance is illustrated with various numerical examples.

Optimization assisted autoregressive technique with deep convolution neural network-based entropy filter for image demosaicing

ABSTRACT This paper presents an image demosaicing based on an optimization-driven deep learning model, namely the Autoregressive Water Wave Optimization algorithm (Autoregressive-WWO). The proposed method is devised by assimilating the Wave Optimization algorithm (WWO), and the Conditional autoregressive value at risk (CAViaR) model. Here, the input images are subjected to Autoregressive WWO-based local polynomial approximation and intersection of confidence intervals (LPA-ICI) filter, and Deep Convolution neural network (Deep CNN) in a concurrent manner. The filter coefficients are obtained from the proposed Autoregressive WWO-based LPA-ICI filter and the residual image is obtained from Deep CNN. In order to create the demosaiced image, these two outputs are combined using an entropy measure. The proposed method offered superior performance with the highest Peak signal to noise ratio (PSNR) of 40.049dB, the highest Second derivative measure of enhancement (SDME) of 50.168dB, and highest Structural Index Similarity (SSIM) of 0.9056.

Local theory for 2‐microlocal Besov and Triebel–Lizorkin spaces of Jaffard type

AbstractIn this paper, we introduce new 2‐microlocal Besov and Triebel–Lizorkin spaces of Jaffard type, which contain many classical functions. We establish local theory of these function spaces by the φ‐transform and the wavelet decomposition coefficients. Moreover, we give a characterization of these local spaces by a local polynomial approximation.

Wild Bootstrap-Based Bias Correction for Spatial Quantile Panel Data Models with Varying Coefficients

This paper studies quantile regression for spatial panel data models with varying coefficients, taking the time and location effects of the impacts of the covariates into account, i.e., the implications of covariates may change over time and location. Smoothing methods are employed for approximating varying coefficients, including B-spline and local polynomial approximation. A fixed-effects quantile regression (FEQR) estimator is typically biased in the presence of the spatial lag variable. The wild bootstrap method is employed to attenuate the estimation bias. Simulations are conducted to study the performance of the proposed method and show that the proposed methods are stable and efficient. Further, the estimators based on the B-spline method perform much better than those of the local polynomial approximation method, especially for location-varying coefficients. Real data about economic development in China are also analyzed to illustrate application of the proposed procedure.

An hp‐adaptive multi‐element stochastic collocation method for surrogate modeling with information re‐use

AbstractThis article introduces an ‐adaptive multi‐element stochastic collocation method, which additionally allows to re‐use existing model evaluations during either ‐ or ‐refinement. The collocation method is based on weighted Leja nodes. After ‐refinement, local interpolations are stabilized by adding and sorting Leja nodes on each newly created sub‐element in a hierarchical manner. For ‐refinement, the local polynomial approximations are based on total‐degree or dimension‐adaptive bases. The method is applied in the context of forward and inverse uncertainty quantification to handle non‐smooth or strongly localized response surfaces. The performance of the proposed method is assessed in several test cases, also in comparison to competing methods.

Enhancement of video demosaicing techniques using optimization algorithm

SummaryDemosaicing and denoising are the most significant steps in the imaging process for both images and videos. The correlation noise corrupts the image quality, such that an accurate interpretation of the image is avoided. Thus, demosaicing is utilized to restructure the full‐color image from imperfect color samples. In this article, cat shuffled shepherd optimization‐based local polynomial approximation and intersection of confidence intervals (Cat SSO‐based LPA‐ICI) approach is developed for video demosaicing. The LPA‐ICI model is employed for frame demosaicing in which the second‐order polynomial interpolation filter coefficients are produced using the devised optimization algorithm, named cat SSO technique. Similarly, generative adversarial network classifier is also applied for frame demosaicing. In addition, initialization and flow estimation is performed for the demosaicing process. Besides, spatio temporal demosaicing model is utilized in order to get a noise‐free sequence. However, the developed cat SSO technique is newly devised by the incorporation of the shuffled shepherd optimization algorithm and cat swarm optimization technique. The created video demosaicing model outperformed with peak signal‐to‐noise ratio, root mean square error, and second derivative‐like measure of enhancement values of 56.85 dB, 0.46, and 60.62, respectively.

A power series method for static and dynamic analysis of offshore mooring lines

The mooring line response is described by the nonlinear partial differential equations (PDEs). The nonlinearity arises from the cable geometry through the drag force and the geometric compatibility condition. This paper develops a power series method (PSM) and applies this semi-analytical approach to solve the PDEs of moorling line motion, which often have been dealt with fully numerically. This technique permits solutions for some differential equations through approximation with polynomial series. Besides being less computationally intensive, PSM-based analyses are straightforward to implement. For the present model, vector components are approximated with infinite polynomial series being functions of spatial and temporal variables. The paper addresses a two-dimensional mooring line model with a fixed bottom end and subject to hydrodynamic and hydrostatic forces. A harmonic excitation at the top end is approximated by a local polynomial approximation enabling the inference of wave parameters. The analysis highlights the effects of pretension, mass per unit length, and offset on the mooring line response. The dynamic analysis enables the evaluation of dynamic tensions for various polynomial orders of temporal and spatial coordinates. It is noticed that the numerical convergence occurs when increasing the degree of polynomials to be up to at least the seventh degree.

Response Surface Method for Optimization of Synchronous Reluctance Motor Rotor

In order to define the best design structure of the synchronous reluctance motor (SynRM) rotor, optimization must be carried out, implying the selection of the best alternative for each specific criterion. The optimization of an electrical machine is a complicated work involving meeting different criteria requirements while dealing with a range of constraints. In order to implement the optimization, it is necessary to process a huge number of options, changing the combinations of the factors affecting criteria and restrictions, which is a time-consuming process. This research presents the optimization technique that gives a mathematically proven solution of the optimal rotor design of a synchronous reluctance machine obtained by using metamodels in the form of local polynomial approximations. Analysis of the results of numerical modeling and experimental investigation has been performed in order to validate the developed technique and recommendations. SynRM rotor was manufactured, with the stator to be taken from the 1.1 kW W21 WEG induction motor, which makes possible the relevant experimental study. The performance analysis of the developed SynRM is shown in the paper.

ESTIMATES OF DERIVATIVES OF (LOG) DENSITIES AND RELATED OBJECTS

We estimate the density and its derivatives using a local polynomial approximation to the logarithm of an unknown density function f. The estimator is guaranteed to be non-negative and achieves the same optimal rate of convergence in the interior as on the boundary of the support of f. The estimator is therefore well-suited to applications in which non-negative density estimates are required, such as in semiparametric maximum likelihood estimation. In addition, we show that our estimator compares favorably with other kernel-based methods, both in terms of asymptotic performance and computational ease. Simulation results confirm that our method can perform similarly or better in finite samples compared to these alternative methods when they are used with optimal inputs, that is, an Epanechnikov kernel and optimally chosen bandwidth sequence. We provide code in several languages.

Efficient Surrogate Modeling and Design Optimization of Compact Integrated On-Chip Inductors Based on Multi-Fidelity EM Simulation Models

High-performance and small-size on-chip inductors play a critical role in contemporary radio-frequency integrated circuits. This work presents a reliable surrogate modeling technique combining low-fidelity EM simulation models, response surface approximations based on kriging interpolation, and space mapping technology. The reported method is useful for the development of broadband and highly accurate data-driven models of integrated inductors within a practical timeframe, especially in terms of the computational expense of training data acquisition. Application of the constructed surrogate model for rapid design optimization of a compact on-chip inductor is demonstrated. The optimized EM-validated design solution can be reached at a low computational cost, which is a considerable improvement over existing approaches. In addition, this work provides a description and illustrates the usefulness of a multi-fidelity design optimization method incorporating EM computational models of graduated complexity and local polynomial approximations managed by an output space mapping optimization framework. As shown by the application example, the final design solution is obtained at the cost of a few high-fidelity EM simulations of a small-size integrated coil. A supplementary description of variable-fidelity EM computational models and a trade-off between model accuracy and its processing time complements the work.

Sparse Brudnyi and John–Nirenberg Spaces

A generalization of the theory of Y. Brudnyi [7], and A. and Y. Brudnyi [5, 6], is presented. Our construction connects Brudnyi’s theory, which relies on local polynomial approximation, with new results on sparse domination. In particular, we find an analogue of the maximal theorem for the fractional maximal function, solving a problem proposed by Kruglyak–Kuznetsov. Our spaces shed light on the structure of the John–Nirenberg spaces. We show that SJN p (sparse John–Nirenberg space) coincides with L p ,1<p<∞. This characterization yields the John–Nirenberg inequality by extrapolation and is useful in the theory of commutators.

Wheelchair Movement Based on Convolution Neural Network

This paper intends to develop a methodology for helping amputees and crippled people old, by ongoing voice direction and association between patient and personal computer (PC) where these blends offer a promising response for helping the debilitated people. The major objective of this work is accurately detected audio orders via a microphone of an English language (go, stop, right and left) in a noisy environment by the proposed system. Thus, a patient that utilizes the proposed system can be controlling a wheelchair movement. The venture depends on preparing an off-line dataset of audio files are included 10000 orders and background noise. The proposed system has two important steps of preprocessing to get accurate of specific audio orders, accordingly, the accurate direction of wheelchair movement. Firstly, a dataset was preprocessed to reduce ambient noise by using Butterworth (cutoff 500-5000 Hz) and Wiener filter. Secondly, in the input (a microphone) of the proposed discriminative model put a procedure of infinite impulse response filter (Butterworth), passband filter for cutoff input microphone from 150-7000 Hz for back-off the loud and environment noise and local polynomial approximation (Savitzky-Golay) smoothing filter that plays out a polynomial regression on the signal values. Thus, a better for filtering from ambient noise and keeping on a waveform from distortion that makes the discriminative model accurate when voice orders were recognized. The proposed system can work with various situations and speeds for steering; forward, stop, left and right. All datasets are trained by using deep learning with specific parameters of...

Anisotropic Denoising of 3D Point Clouds by Aggregation of Multiple Surface-Adaptive Estimates.

3D point clouds commonly contain positional errors which can be regarded as noise. We propose a point cloud denoising algorithm based on aggregation of multiple anisotropic estimates computed on local coordinate systems. These local estimates are adaptive to the shape of the surface underlying the point cloud, leveraging an extension of the Local Polynomial Approximation (LPA) - Intersection of Confidence Intervals (ICI) technique to 3D point clouds. The adaptivity due to LPA-ICI is further strengthened by the dense aggregation with data-driven weights. Experimental results demonstrate state-of-the-art restoration quality of both sharp features and smooth areas.

A 2D SWE meshless model with fictitious water level at dry nodes

In this study, a two-dimensional shallow-water-equation numerical model is developed for simulating free surface water flows with internal moving wet/dry front boundaries. It is based on the Eulerian aspect and meshless approach. The predictor-corrector process is used for the time marching. The spatial derivatives of physical quantities are computed using the local polynomial approximation with the weighted-least-squares approach. A formula for specifying the fictitious water level, which does not exist physically, at dry nodes is proposed. This is the main novelty of the present study and it significantly reduces computation time compared with the authors’ previous work. Four dam break test cases are employed to validate the capability of the present model.

Improved filtering approach for identification of protein-coding regions in eukaryotes by background noise reduction using S–G filter

Unlike prokaryotic genomes, the arrangement of protein-coding regions in eukaryotic genomes is not continuous. It is interrupted by the non-coding DNA sequences called introns. Therefore, the identification of protein-coding regions in a eukaryotic DNA sequence is one of the challenging research issues in bioinformatics. Signal processing-based computational tools such as short-time discreet Fourier transforms (STDFT) and narrow bandpass digital filters has been successfully used to resolve this problem. Filtering techniques are popularly used because of its faster response than the transform techniques. However, the prediction accuracy of the filtering approach is still limited due to background noise present in its spectrum. Background noise masks the discriminative three base periodicity (TBP) features and increases the chances of false prediction. Several de-noising techniques have been proposed so far. Recently, second-order moving average filter has been used to diminish the effect of background noise. However, the problem with moving average filter is that it operates similarly on coding and non-coding regions, and therefore, along with noise reduction, it also affects the spectral features of protein-coding regions which lead to inaccurate prediction results. In this work, we used the Savitzky–Golay (SG) filter for background noise reduction and compared the performance with other existing de-noising techniques. S–G filter works on the local least-squares polynomial approximation principal and act as a weighted moving average filter. This investigation shows that along with noise reduction, S–G filter can preserve the spectral values of coding regions with a greater extent, and therefore, provide more accurate prediction results than other de-noising techniques.

On the Solution of Fredholm Integral Equations of the First Kind

As it is well known the problem of solving the Fredholm integral equation of the first kind belongs to the class of ill-posed problems. The Tikhonov regularization method is well known. This method is usually applied to an integral equation and a system of linear algebraic equations. The authors firstly propose to reduce the integral equation of the first kind to a system of linear algebraic equations. This system is usually extremely ill-posed. Therefore, it is necessary to carry out the Tikhonov regularization for the system of equations. In this paper, to form a system of linear algebraic equations, local polynomial and non-polynomial spline approximations of the second order of approximation are used. The results of numerical experiments are presented.

Phase-unwrapping method based on local polynomial models and a maximum a posteriori model correction.

Recently, a theory on local polynomial approximations for phase-unwrapping algorithms, considering a state space analysis, has been proposed in Appl. Opt.56, 29 (2017)APOPAI0003-693510.1364/AO.56.000029. Although this work is a suitable methodology to deal with relatively low signal to noise ratios observed in the wrapped phase, the methodology has been developed only for local-polynomial phase models of order 1. The resultant proposal is an interesting Kalman filter approach for estimating the coefficient or state vectors of these local plane models. Thus, motivated by this approach and simple Bayesian theory, and considering our previous research on local polynomial models up to the third order [Appl. Opt.58, 436 (2019)APOPAI0003-693510.1364/AO.58.000436], we propose an equivalent methodology based on a simple maximum a posteriori estimation, but considering a different state space: difference vectors of coefficients for the current high-order polynomial models. Specific estimations of the covariance matrices for difference vectors, as well as noise covariance matrices involved with the correct estimation of coefficient vectors, are proposed and reconstructions with synthetic and real data are provided.

Phase unwrapping based on accumulation of residual maps with local denoising

Phase unwrapping aims to reconstruct the absolute phase from its values mod 2π, which is a key problem in many non-contact optical metrologies and is still a challenging issue in the presence of noise. We present a robust phase unwrapping method with two steps based on local denoising and the accumulation of residual map (ARM). In the first step, based on the local polynomial approximations and intersection of confidence intervals, we adopt a sine–cosine filtering and iteration algorithm to denoise the wrapped phase, which uses the periodicity of the trigonometric function to avoid the influence of phase jumps effectively. In the second step, the denoised wrapped phase is unwrapped by the improved ARMs method, in which we modify the edge regular term to avoid over-fitting and maintain edge information. The algorithm is tested with a set of numerical simulations under various noise levels, and the experimental results confirm its effectiveness and robustness.

Gravitational-Wave Burst Signals Denoising Based on the Adaptive Modification of the Intersection of Confidence Intervals Rule.

Gravitational-wave data (discovered first in 2015 by the Advanced LIGO interferometers and awarded by the Nobel Prize in 2017) are characterized by non-Gaussian and non-stationary noise. The ever-increasing amount of acquired data requires the development of efficient denoising algorithms that will enable the detection of gravitational-wave events embedded in low signal-to-noise-ratio (SNR) environments. In this paper, an algorithm based on the local polynomial approximation (LPA) combined with the relative intersection of confidence intervals (RICI) rule for the filter support selection is proposed to denoise the gravitational-wave burst signals from core collapse supernovae. The LPA-RICI denoising method’s performance is tested on three different burst signals, numerically generated and injected into the real-life noise data collected by the Advanced LIGO detector. The analysis of the experimental results obtained by several case studies (conducted at different signal source distances corresponding to the different SNR values) indicates that the LPA-RICI method efficiently removes the noise and simultaneously preserves the morphology of the gravitational-wave burst signals. The technique offers reliable denoising performance even at the very low SNR values. Moreover, the analysis shows that the LPA-RICI method outperforms the approach combining LPA and the original intersection of confidence intervals (ICI) rule, total-variation (TV) based method, the method based on the neighboring thresholding in the short-time Fourier transform (STFT) domain, and three wavelet-based denoising techniques by increasing the improvement in the SNR by up to 118.94% and the peak SNR by up to 138.52%, as well as by reducing the root mean squared error by up to 64.59%, the mean absolute error by up to 55.60%, and the maximum absolute error by up to 84.79%.

Mollified finite element approximants of arbitrary order and smoothness

The approximation properties of the finite element method can often be substantially improved by choosing smooth high-order basis functions. It is extremely difficult to devise such basis functions for partitions consisting of arbitrarily shaped polytopes. We propose the mollified basis functions of arbitrary order and smoothness for partitions consisting of convex polytopes. On each polytope an independent local polynomial approximant of arbitrary order is assumed. The basis functions are defined as the convolutions of the local approximants with a mollifier. The mollifier is chosen to be smooth, to have a compact support and a unit volume. The approximation properties of the obtained basis functions are governed by the local polynomial approximation order and mollifier smoothness. The convolution integrals are evaluated numerically first by computing the boolean intersection between the mollifier and the polytope and then applying the divergence theorem to reduce the dimension of the integrals. The support of a basis function is given as the Minkowski sum of the respective polytope and the mollifier. The breakpoints of the basis functions, i.e. locations with non-infinite smoothness, are not necessarily aligned with polytope boundaries. Furthermore, the basis functions are not boundary interpolating so that we apply boundary conditions with the non-symmetric Nitsche method as in immersed/embedded finite elements. The presented numerical examples confirm the optimal convergence of the proposed approximation scheme for Poisson and elasticity problems.