Optimal Approximation Research Articles

Joint Optimization Method of Spectrum Resource for UAV Swarm Information Transmission

For the problems brought by malicious interference in the unmanned aerial vehicle (UAV) swarm network, we establish a cluster-based UAV swarm information transmission model. We mainly consider four aspects: cluster head selection, channel allocation, power allocation and UAV position. In order to improve the backhaul information rate of UAV swarm, we propose a joint optimization method of spectrum resource with the goal of maximizing the sum throughput of the cluster head UAV. We decompose the original mixed integer nonlinear programming (MINLP) problem into multiple sub-problems based on the block coordinate descent (BCD) technique, and then solve them based on convex optimization and successive convex approximation (SCA) technique. Simulation results show that the proposed method can obtain the spectrum resource optimization strategy of UAV swarm information transmission, reduce the impact of malicious interference, and effectively improve the backhaul information rate of UAV swarm.

Majority Approximators for Low-Latency Data Bus Inversion

Data bus inversion (DBI) is an encoding technique that saves power in data movement in which the majority function plays an essential role. For a latency optimization, the majority function can be replaced by a majority approximator that allows for a small error in majority voting to obtain a faster encoder that still saves power. In this work, we propose two systematic approaches for finding high-performance majority approximators. First, we perform an exhaustive search of all possible Boolean functions to find an optimal approximator based on a certain circuit structure comprised of fifteen logic gates. The approximator found by the systematic search can be implemented using compound gates, resulting in a latency-efficient design with only two gate levels. Compared with prior works using a heuristic idea, the proposed circuit runs at the same speed but achieves greater switching activity savings. Second, we propose another majority approximator using the average of three randomly permuted copies of the approximator found in the first approach. We show that the second proposed approximator achieves even higher savings in switching activity as its function is closer to a true majority voter. We report various performance metrics of the newly found majority approximators based on syntheses using a 65 nm process.

Angles between subspaces and nearly optimal approximation in GFEM

The Generalized Finite Element Methods (GFEM) are known for accurately resolving local features in heterogeneous media. Recent numerical experiments have shown that the use of local bases made from A-harmonic extensions of well chosen boundary data have nearly optimal approximation properties. In this paper we show that the explanation is geometric and given by the angle between any finite dimensional approximation space and the optimal finite dimensional approximation space introduced in (Babǔska and Lipton, 2011). The effect of the angle is quantified by an upper bound on the convergence rate. This rate is seen in the numerical simulations where local bases with nearly optimal approximation properties are identified.

Classification of Terrestrial Laser Scanner Point Clouds: A Comparison of Methods for Landslide Monitoring from Mathematical Surface Approximation

Terrestrial laser scanners (TLS) are contact-free measuring sensors that record dense point clouds of objects or scenes by acquiring coordinates and an intensity value for each point. The point clouds are scattered and noisy. Performing a mathematical surface approximation instead of working directly on the point cloud is an efficient way to reduce the data storage and structure the point clouds by transforming “data” to “information”. Applications include rigorous statistical testing for deformation analysis within the context of landslide monitoring. In order to reach an optimal approximation, classification and segmentation algorithms can identify and remove inhomogeneous structures, such as trees or bushes, to obtain a smooth and accurate mathematical surface of the ground. In this contribution, we compare methods to perform the classification of TLS point clouds with the aim of guiding the reader through the existing algorithms. Besides the traditional point cloud filtering methods, we will analyze machine learning classification algorithms based on the manual extraction of point cloud features, and a deep learning approach with automatic extraction of features called PointNet++. We have intentionally chosen strategies easy to implement and understand so that our results are reproducible for similar point clouds. We show that each method has advantages and drawbacks, depending on user criteria, such as the computational time, the classification accuracy needed, whether manual extraction is performed or not, and if prior information is required. We highlight that filtering methods are advantageous for the application at hand and perform a mathematical surface approximation as an illustration. Accordingly, we have chosen locally refined B-splines, which were shown to provide an optimal and computationally manageable approximation of TLS point clouds.

Quality-driven video streaming for ultra-dense OFDMA heterogeneous networks

The explosive growth of mobile video traffic has brought enormous traffic pressure to the existing bandwidth-limited cellular network. Ultra-dense heterogeneous networks (HetNets) are deemed as one of the most promising 5G technologies for increasing the cellular network capacity. In this paper, we study quality-driven multiuser video streaming algorithm for two-tier orthogonal frequency division multiple access (OFDMA) HetNets. Specifically, we formulate a stochastic optimization problem for the quality-driven multiuser video streaming in two-tier OFDMA HetNets to maximize the average perceived video quality of small cell users while meeting the average data rate demand of macrocell users. We then decompose the original stochastic problem into a sequence of per-slot deterministic subproblems of adaptive bitrate selection and radio resource management which have no requirement of prior channel knowledge by employing Lyapunov optimization technique. We accordingly propose an online quality-driven video streaming algorithm. In this algorithm, we derive the optimal bitrate decision for the adaptive bitrate selection subproblem, and design an iterative scheduling algorithm of joint subchannel assignment and power control by utilizing alternating optimization and successive convex approximation techniques to solve the radio management subproblem. Extensive simulation results validate the high efficiency of the proposed quality-driven video streaming algorithm in OFDMA HetNets.

A Model of Pixel and Superpixel Clustering for Object Detection.

The paper presents a model of structured objects in a grayscale or color image, described by means of optimal piecewise constant image approximations, which are characterized by the minimum possible approximation errors for a given number of pixel clusters, where the approximation error means the total squared error. An ambiguous image is described as a non-hierarchical structure but is represented as an ordered superposition of object hierarchies, each containing at least one optimal approximation in g0 = 1, 2,..., etc., colors. For the selected hierarchy of pixel clusters, the objects-of-interest are detected as the pixel clusters of optimal approximations, or as their parts, or unions. The paper develops the known idea in cluster analysis of the joint application of Ward’s and K-means methods. At the same time, it is proposed to modernize each of these methods and supplement them with a third method of splitting/merging pixel clusters. This is useful for cluster analysis of big data described by a convex dependence of the optimal approximation error on the cluster number and also for adjustable object detection in digital image processing, using the optimal hierarchical pixel clustering, which is treated as an alternative to the modern informally defined “semantic” segmentation.

Hybrid Model Linearization Predictive Control for DC Bus Stabilization in Spacecraft Combined Power System

In spacecraft, facing different loading conditions of subsystems, especially of permanent magnet synchronous motor (PMSM) system, the bus voltage has large fluctuation, which will seriously affect the power quality of the spacecraft combined power system. Therefore, it is necessary to introduce a bidirectional DC–DC converter to suppress the fluctuation of the bus voltage. Aiming at the need of controlling the bidirectional DC–DC converter, a hybrid model linearization predictive control (HMLPC) algorithm is proposed. According to the strong nonlinear hybrid characteristics of the bidirectional DC–DC converter, an optimal linear approximation model at the sampling point is established using the Jacobian matrix, based on which a piecewise affine (PWA) model is established for Buck and Boost mode, and then, a hybrid linearization predictive model is constructed. Ultimately, by introducing a two-period PWA and multistep prediction strategy, proposing a prediction error, and solving the constrained finite time optimization control (CFTOC) problem online, an optimal control law is derived. In addition, a stability analysis is provided for the proposed control algorithm. In this article, a simulative validation and a comparative analysis are presented; moreover, a combined power test platform is built to validate the proposed control algorithm. Both simulation and experiment results show that the HMLPC algorithm can effectively suppress the fluctuation of bus voltage and enhance the anti-interference ability of the system.

End-to-End Latency Analysis and Optimal Block Size of Proof-of-Work Blockchain Applications

Due to the increasing interest in blockchain technology for fostering secure, auditable, decentralized applications, a set of challenges associated with this technology need to be addressed. In this letter, we focus on the delay associated with Proof-of-Work (PoW)-based blockchains, whereby participants validate the new information to be appended to a distributed ledger via consensus to confirm transactions. We propose a novel end-to-end latency model based on batch-service queuing theory that characterizes timers and forks for the first time. Furthermore, we derive an estimation of the optimal block size analytically. Endorsed by analytical and simulation results, we show that the optimal block size approximation is a consistent method that leads to close-to-optimal performance by significantly reducing the overheads associated with blockchain applications.

Caching and UAV Friendly Jamming for Secure Communications With Active Eavesdropping Attacks

In this paper, we discuss the security and reliability performance of a communication system, where the base station (BS) transmits signals to multiple users and an unmanned aerial vehicle (UAV) jammer sends friendly jamming signals to protect against a full-duplex active eavesdropper with the aid of caching. We derive the closed-form expressions of outage probability (OP) and intercept probability (IP) for the legitimate users and active eavesdropper, respectively. Aimed at minimizing the sum of the IP and the maximum OP among all users, we jointly optimize the BS transmit power, UAV jammer location and jamming power. An efficient algorithm based on alternating optimization (AO) and successive convex approximation (SCA) methods is designed to solve the optimization problem. Numerical results show that compared to other benchmark strategies, the proposed solution improves the reliability and security performance with the aid of caching and UAV friendly jamming.

Inversion of chlorophyll content under the stress of leaf mite for jujube based on model PSO-ELM method.

During the growth season, jujube trees are susceptible to infestation by the leaf mite, which reduces the fruit quality and productivity. Traditional monitoring techniques for mites are time-consuming, difficult, subjective, and result in a time lag. In this study, the method based on a particle swarm optimization (PSO) algorithm extreme learning machine for estimation of leaf chlorophyll content (SPAD) under leaf mite infestation in jujube was proposed. Initially, image data and SPAD values for jujube orchards under four severities of leaf mite infestation were collected for analysis. Six vegetation indices and SPAD value were chosen for correlation analysis to establish the estimation model for SPAD and the vegetation indices. To address the influence of colinearity between spectral bands, the feature band with the highest correlation coefficient was retrieved first using the successive projection algorithm. In the modeling process, the PSO correlation coefficient was initialized with the convergent optimal approximation of the fitness function value; the root mean square error (RMSE) of the predicted and measured values was derived as an indicator of PSO goodness-of-fit to solve the problems of ELM model weights, threshold randomness, and uncertainty of network parameters; and finally, an iterative update method was used to determine the particle fitness value to optimize the minimum error or iteration number. The results reflected that significant differences were observed in the spectral reflectance of the jujube canopy corresponding with the severity of leaf mite infestation, and the infestation severity was negatively correlated with the SPAD value of jujube leaves. The selected vegetation indices NDVI, RVI, PhRI, and MCARI were positively correlated with SPAD, whereas TCARI and GI were negatively correlated with SPAD. The accuracy of the optimized PSO-ELM model (R2 = 0.856, RMSE = 0.796) was superior to that of the ELM model alone (R2 = 0.748, RMSE = 1.689). The PSO-ELM model for remote sensing estimation of relative leaf chlorophyll content of jujube shows high fault tolerance and improved data-processing efficiency. The results provide a reference for the utility of UAV remote sensing for monitoring leaf mite infestation of jujube.

Optimal analytical and numerical approximations to the (un)forced (un)damped parametric pendulum oscillator

The (un)forced (un)damped parametric pendulum oscillator (PPO) is analyzed analytically and numerically using some simple, effective, and more accurate techniques. In the first technique, the ansatz method is employed for analyzing the unforced damped PPO and for deriving some optimal and accurate analytical approximations in the form of angular Mathieu functions. In the second approach, some approximations to (un)forced damped PPO are obtained in the form of trigonometric functions using the ansatz method. In the third approach, He’s frequency-amplitude principle is applied for deriving some approximations to the (un)damped PPO. In the forth approach, He’s homotopy technique is employed for analyzing the forced (un)damped PPO numerically. In the fifth approach, the p-solution Method, which is constructed based on Krylov–Bogoliúbov Mitropolsky method, is introduced for deriving an approximation to the forced damped PPO. In the final approach, the hybrid Padé-finite difference method is carried out for analyzing the damped PPO numerically. All proposed techniques are compared to the fourth-order Runge–Kutta (RK4) numerical solution. Moreover, the global maximum residual distance error is estimated for checking the accuracy of the obtained approximations. The proposed methodologies and approximations can help many researchers in studying and investigating several nonlinear phenomena related to the oscillations that can arise in various branches of science, e.g. waves and oscillations in plasma physics.

MARS: A method for the adaptive removal of stiffness in PDEs

The E(xplicit)I(implicit)N(null) method was developed recently to remove numerical instability from PDEs, adding and subtracting an operator D of arbitrary structure, treating the operator implicitly in one case, and explicitly in the other. Here we extend this idea by devising an adaptive procedure to find an optimal approximation for D. We propose a measure of the numerical error which detects numerical instabilities across all wavelengths, and adjust each Fourier component of D to the smallest value such that numerical instability is suppressed. We show that for a number of nonlinear and non-local PDEs, in one and two dimensions, the spectrum of D adapts automatically and dynamically to the theoretical result for marginal stability. The adaptive implicit part is diagonal in Fourier space, so that our method has the same stability properties as a fully implicit method, with minimal computational overhead coming only from performing the fast Fourier transform.

Inexact Newton method with iterative combined approximations in the topology optimization of geometrically nonlinear elastic structures and compliant mechanisms

Inexact Newton method with iterative combined approximations in the topology optimization of geometrically nonlinear elastic structures and compliant mechanisms

Saphenous Vein Interposition Grafts in Lower Extremity Reconstruction: Appraisal of Technique and Case Series.

Summary:Vascular microanastomosis is technically challenging in patients with calcified recipient and donor vessels. Inside-to-outside suturing can prevent plaque rupture and ensure full-thickness intimal approximation. Although this is the preferred technique for anastomosis of atherosclerotic vessels, direct connection of calcified arteries necessitates outside-to-inside suturing on one side of the anastomosis. Furthermore, it is difficult to achieve optimal vessel wall approximation in the setting of luminal size mismatch and rigid vasculature. We previously reported on the use of a saphenous vein interposition graft as a novel technique to achieve a flow-sparing anastomosis in patients with diffuse atherosclerosis who are undergoing free tissue transfer. This study further assesses outcomes of this technique in a series of patients and demonstrates a flap success rate of over 93% in patients with calcified recipient and donor microvasculature.

Entropy for evaluation of Dempster-Shafer belief function models

Applications of Dempster-Shafer (D-S) belief functions to practical problems involve difficulties arising from their high computational complexity. One can use space-saving factored approximations such as graphical belief function models to solve them. Using an analogy with probability distributions, we represent these approximations in the form of compositional models. Since no theoretical apparatus similar to probabilistic information theory exists for D-S belief functions (e.g., dissimilarity measure analogous to the Kullback-Liebler divergence measure), the problems arise not only in connection with the design of algorithms seeking optimal approximations but also in connection with a criterion comparing two different approximations. In this respect, the application of the analogy with probability theory fails. Therefore, in this paper, we conduct some synthetic experiments and describe the results designed to reveal whether some belief function entropy definitions described in the literature can detect optimal approximations, i.e., that achieve their minimum for an optimal approximation.

CutFEM based on extended finite element spaces

We develop a general framework for construction and analysis of discrete extension operators with application to unfitted finite element approximation of partial differential equations. In unfitted methods so called cut elements intersected by the boundary occur and these elements must in general by stabilized in some way. Discrete extension operators provides such a stabilization by modification of the finite element space close to the boundary. More, precisely the finite element space is extended from the stable interior elements over the boundary in a stable way which also guarantees optimal approximation properties. Our framework is applicable to all standard nodal based finite elements of various order and regularity. We develop an abstract theory for elliptic problems and associated parabolic time dependent partial differential equations and derive a priori error estimates. We finally apply this to some examples of partial differential equations of different order including the interface problems, the biharmonic operator and the sixth order triharmonic operator.

A Homotopy Bayesian Approach for Inverse Problems

In solving Bayesian inverse problems, it is often desirable to use a common density parameterization to characterize the prior and posterior. Typically, we seek an optimal approximation of the true posterior from the same distribution family as the prior. As one of the most important classes of distributions in statistics, the exponential family is considered as the parameterization. The optimal parameter values for representing the approximated posterior are achieved by minimizing the deviation between the parameterized density and a homotopy that deforms the prior density into the posterior density. Instead of the usual moment parameters, we introduce a new parameter system in the process, by which we reduce the dimension of the parameter and therefore the computational cost. The parameters are ruled by a system of explicit first-order differential equations. Solving this system over finite “time” interval yields the desired optimal density parameters. This method is proven to be effective using some numerical examples.

Quasi-interpolation for analysis-suitable T-splines

We propose a novel local approximation method for analysis-suitable T-spline (AS T-spline) spaces via quasi-interpolation. The quasi-interpolants are defined as linear combination of the approximated function's values at appropriately chosen points. Benefited from the inherent nice properties of AS T-splines, the proposed quasi-interpolants can reproduce polynomials up to the same degree of AS T-spline spaces and can provide optimal approximation order. Some numerical examples of specific quasi-interpolants for bi-cubic AS T-splines are investigated to show the stability and efficiency.

Arbitrarily Sparse Polynomial Chaos Expansion for High-Dimensional Parametric Problems: Parametric and Probabilistic Power Flow as an Example

Parameters, no matter whether they are uncontrollable uncertainty factors or controllable variables, have a great impact on the states and performances of continuous engineering systems. Acquiring an explicit expression of the complicated implicit function between these parameters and system states, called parametric problem in this article, will facilitate immediate analysis of parameters’ impact on the system, optimal parameter design, and uncertainty quantification. Polynomial chaos expansion (PCE) is a globally optimal polynomial approximation method for parametric problems, but it suffers from the curse of dimensionality, i.e., the number of basis functions and consequent computational burden increases rapidly with the number of parameters (i.e., dimensions). This article proposes a PCE method characterized by the arbitrarily sparse basis and novel generalized Smolyak sparse grid quadrature and, hence, effectively relieves the curse of dimensionality for high-dimensional problems. This basis is not restricted by any existing fixed-form truncation and can merely incorporate a few important basis functions. The novel generalized sparse grid can remarkably reduce the number of required collocation points compared with the prevailing classic sparse grid while achieving high accuracy. The proposed method is universal for parametric problems in engineering systems and is exemplified by parametric and probabilistic power flow problems of electrical power systems. Its effectiveness is validated by computational results on the IEEE 30-bus, 118-bus, and 500-parameter 9241pegase systems.

A fourth-order optimal numerical approximation and its convergence for singularly perturbed time delayed parabolic problems

A fourth-order optimal numerical approximation and its convergence for singularly perturbed time delayed parabolic problems