Point Approximation Method Research Articles

Grid Point Approximation for Distributed Nonparametric Smoothing and Prediction

Kernel smoothing is a widely used nonparametric method in modern statistical analysis. The problem of efficiently conducting kernel smoothing for a massive dataset on a distributed system is a problem of great importance. In this work, we find that the popularly used one-shot type estimator is highly inefficient for prediction purposes. To this end, we propose a novel grid point approximation (GPA) method, which has the following advantages. First, the resulting GPA estimator is as statistically efficient as the global estimator under mild conditions. Second, it requires no communication and is extremely efficient in terms of computation for prediction. Third, it is applicable to the case where the data are not randomly distributed across different machines. To select a suitable bandwidth, two novel bandwidth selectors are further developed and theoretically supported. Extensive numerical studies are conducted to corroborate our theoretical findings. Two real data examples are also provided to demonstrate the usefulness of our GPA method.

Co-design of an unmanned cable shovel for structural and control integrated optimization: A highly heterogeneous constrained multi-objective optimization algorithm

Unmanned cable shovel is an intelligence-based large machine used for excavating in open-pit coal mines, with significant implications for the security and development of energy resources. The co-design of unmanned cable shovels, considering elements from both structural and control systems, has demonstrated superior performance in obtaining optimal solutions compared to the traditional approaches that focus on individual systems. However, the inherent complexities of diverse systems often lead to heterogeneous function evaluations, posing challenges for the existing optimization algorithms. To address highly heterogeneous multi-objective optimization problems characterized by surrogate-approximated expensive functions and explicit inexpensive functions, this study proposes a highly heterogeneous constrained multi-objective optimization (HHCMO) algorithm. Leveraging NSGA-III as an evolutionary optimizer and the Kriging model for approximating expensive functions, HHCMO strategically addresses the challenges posed by highly heterogeneous objective evaluations. A Monte Carlo sampling-based expected hypervolume improvement (EHVI) criterion, employing a new nearest point approximation method in a unified sampling space, effectively handles the impact of heterogeneous objective evaluations on the sequential infill of the Kriging model. Thorough evaluations on both unconstrained and constrained multi-objective benchmark problems validate the correctness of HHCMO and its superiority relative to state-of-the-art algorithms. Finally, the effectiveness of the proposed method is successfully demonstrated in an engineering scenario of an unmanned cable shovel, where it adeptly handles expensive functions from the structural system and inexpensive functions from the control system.

Convergence of the Two Point Flux Approximation method and the fitted Two Point Flux Approximation method for options pricing with local volatility function

In this paper, we deal with numerical approximations for solving the Black–Scholes Partial Differential Equation (PDE) for European and American options pricing with local volatility. This PDE is well-known to be degenerated. Local volatility model is a model where the volatility depends locally of both stock price and time. In contrast to constant volatility or time-dependent volatility models for which analytical representations of the exact solution is known for European Call options, there is no analytical solution for local volatility. The space discretization is performed using the classical finite volume method with Two-Point Flux Approximation (TPFA) and a novel scheme called Fitted Two-Point Flux Approximation (FTPFA). The Fitted Two-Point Flux Approximation (FTPFA) combines the fitted finite volume method and the standard TPFA method. More precisely the fitted finite volume method is used when the stock price approaches zero with the goal to handle the degeneracy of the PDE while the TPFA method is used on the rest of space domain. This combination yields our fitted TPFA scheme. The Euler method is used for the time discretization. We provide the rigorous convergence proofs of the two fully discretized schemes. Numerical experiments to support theoretical results are provided.

A new fixed point approximation method for solving third-order BVPs based on Green's function

A new fixed point approximation method for solving third-order BVPs based on Green's function

Simplified Uncertainty Bounding: An Approach for Estimating Flood Hazard Uncertainty

Deterministic flood hazard estimates neglect the inherent uncertainty associated with model estimates and can substantially underestimate flood risk. Monte Carlo simulation (MCS) has been a valuable tool for conducting uncertainty analysis. However, its application has primarily been limited to a single research setting. Recent development of a point approximation method, simplified uncertainty bounding (SUB), simulated the uncertainty from MCS with high accuracy (e.g., a critical success index of 0.75). However, an evaluation of additional flood hazard metrics and hydro-climate settings that impact the distribution of uncertainty is required. We evaluated SUB at two contrasting study sites by comparing their results with MCS and identified scenarios where performance increased and decreased. The SUB method accurately matched aerial inundation metrics, but performance was reduced for relative errors in flood depth and top width. Hydraulic structures had a heterogeneous impact on accuracy, and the confinement ratio had a positive relationship with the top width error. While SUB generally performed well with relative errors of approximately ±10% for a 90% confidence interval, some outliers did exist. The acceptability of the approach will depend on the specific application. Though SUB overestimated uncertainty, it provides a conservative estimate and is a cost-effective alternative to MCS.

Riemannian stochastic fixed point optimization algorithm

This paper considers a stochastic optimization problem over the fixed point sets of quasinonexpansive mappings on Riemannian manifolds. The problem enables us to consider Riemannian hierarchical optimization problems over complicated sets, such as the intersection of many closed convex sets, the set of all minimizers of a nonsmooth convex function, and the intersection of sublevel sets of nonsmooth convex functions. We focus on adaptive learning rate optimization algorithms, which adapt step-sizes (referred to as learning rates in the machine learning field) to find optimal solutions quickly. We then propose a Riemannian stochastic fixed point optimization algorithm, which combines fixed point approximation methods on Riemannian manifolds with the adaptive learning rate optimization algorithms. We also give convergence analyses of the proposed algorithm for nonsmooth convex and smooth nonconvex optimization. The analysis results indicate that, with small constant step-sizes, the proposed algorithm approximates a solution to the problem. Consideration of the case in which step-size sequences are diminishing demonstrates that the proposed algorithm solves the problem with a guaranteed convergence rate. This paper also provides numerical comparisons that demonstrate the effectiveness of the proposed algorithms with formulas based on the adaptive learning rate optimization algorithms, such as Adam and AMSGrad.

Saddle Point Approximation of Mutual Information for Finite-Alphabet Inputs over Doubly Correlated MIMO Rayleigh Fading Channels

Given the mutual information of finite-alphabet inputs cannot be calculated concisely and accurately over fading channels, this paper proposes a new method to calculate the mutual information. First, the applicability of the saddle point method is studied, and then the mutual information is estimated by the saddle point approximation method with known channel state information. Furthermore, we induce the expectation of mutual information over doubly correlated multiple-input multiple-output (MIMO) Rayleigh fading channels. The validity of the saddle point approximation method is verified by comparing the numerical results of the Monte Carlo method and the saddle point approximation method under different doubly correlated MIMO fading channel scenarios.

A system reliability estimation method by the fourth moment saddle point approximation and copula functions

AbstractDespite many advances in the field of computational system reliability analysis, estimating the joint probability distribution of correlated non‐normal state variables on the basis of incomplete statistical data brings great challenges for engineers. To avoid multidimensional integration, system reliability estimation usually requires the calculation of marginal failure probability and joint failure probability. The current article proposed an integrated approach for estimating system reliability on the basis of the high moment method, saddle point approximation, and copulas. First, the statistic moment estimation based on the stochastic perturbation theory is presented. Thereafter, by constructing CGF (concise cumulant generating function) for the state variable with its first four statistical moments, a fourth moment saddle point approximation method is established for the component reliability estimation. Second, the copula theory is briefly introduced and extensively utilized two‐dimensional copulas are presented. The best fit copula for estimating the probability of system failure is selected according to the AIC (Akaike Information Criterion). Finally, the derived method is applied to three numerical examples for the sake of a comprehensive validation.

Uncertainty-Based Design and Optimization Using First Order Saddle Point Approximation Method for Multidisciplinary Engineering Systems

AbstractThe widely used First Order Reliability Method (FORM) is efficient for uncertainty quantification and safety assessment in Uncertainty-based Multidisciplinary Design Optimization (UBMDO). H...

Novel probabilistic model for searching most probable point in structural reliability analysis

The Most Probable Point (MPP) is typically searched by transforming the basic random variables from original to standard normal space in the traditional first order reliability method (FORM). However, the FORM formulation may provide inaccurate MPP for those reliability problems with several local MPPs. The FORM based probabilistic model is a computational method and it may provide inaccuracy and unstable results for complex problems. In this paper, a probabilistic model is proposed to evaluate the MPP using cumulative distribution function (CDF) of basic random variables. The MPP is determined by solving an optimization problem in the original space which maximizes a multiplier function in terms of the CDF of the random variables. The performance of the proposed probabilistic model, usual FORM and first order saddle point approximation method (FOSAM) using SQP optimization solver is demonstrated by several numerical and engineering problems with normal and non-normal variables. The results of the numerical study illustrate that the proposed model provides an efficient approach to obtain the MPP which is simpler and more accurate than the usual FORM and FOSAM; particularly for reliability problems with non-normal random variables.

Probabilistic Load Flow Based on Holomorphic Embedding, Kernel Density Estimator and Saddle Point Approximation Including Correlated Uncertainty Variables

Probabilistic Load Flow (PLF) methods have been extensively regarded due to the growth of uncertainty and renewable energies penetration in power systems. Given that, the current study aims to propose two novel probabilistic load flow methods based on holomorphic embedding method. Holomorphic method, unlike other iterative methods, introduces a nonlinear equation solver in an independent class, recursively. To that end, Kernel Density Estimator (KDE) and Saddle Point Approximation (SPA) methods are used to efficiently estimate probabilistic characteristics of load flow outputs. The correlations between random input variables with non-normal/normal probability distributions have been considered. The proposed algorithms have been examined on the modified IEEE 14- and 118-bus systems. Finally, to consider the uncorrelated and correlated conditions, the researcher compared the simulation results of the two proposed methods with some well-known and published probabilistic iterative load flow methods such as Monte Carlo Simulation (MCS), Parzen Window (PW), Diffusion method, and 2n+1 Point Estimate Method (PEM). MCS is always a dependable solution but time-consuming that make it useless for large power systems. In the present study, the accurate results obtained from MCS are regarded as a reference. The comparison of results shows high efficiency and low computational burden of proposed methods, and also the accuracy of these methods in density function estimation of output random variables.

Protocol-Based Unscented Kalman Filtering in the Presence of Stochastic Uncertainties

In this paper, the unscented Kalman filtering (UKF) problem is investigated for a class of general nonlinear systems with stochastic uncertainties under communication protocols. A modified unscented transformation is put forward to account for stochastic uncertainties caused by modeling errors. For preventing data collisions and mitigating communication burden, the round-robin protocol and the weighted try-once-discard protocol are, respectively, introduced to regulate the data transmission order from sensors to the filter. Then, by employing two kinds of data-holding strategies (i.e., zero-order holder and zero input) for those nodes without transmission privilege, two novel protocol-based measurement models are formulated. Subsequently, by resorting to the sigma point approximation method, two resource-saving UKF algorithms are developed, where the impact from the underlying protocols on the filter design is explicitly quantified. Finally, compared with the protocol-based extended Kalman filtering algorithms, a simulation example is presented to demonstrate the effectiveness of the proposed protocol-based UKF algorithms.

On the formulation of Adams-Bashforth scheme with Atangana-Baleanu-Caputo fractional derivative to model chaotic problems.

Mathematical analysis with the numerical simulation of the newly formulated fractional version of the Adams-Bashforth method using the Atangana-Baleanu operator which has both nonlocal and nonsingular properties is considered in this paper. We adopt the fixed point theory and approximation method to prove the existence and uniqueness of the solution via general two-component time fractional differential equations. The method is tested with three nonlinear chaotic dynamical systems in which the integer-order derivative is modeled with the proposed fractional-order case. The simulation result for different α values in (0,1] is presented.

A Structure Topology Optimization With the First-Order Saddlepoint Approximation

In practical engineering, uncertain loads usually cause the variations of structure stiffness to affect the security of the structure. To enhance the reliability and find the optimum structure in engineering, a topology optimization method is developed for the optimization of structural stiffness in this paper. The first-order saddle point approximation method is introduced into the topology optimization process. A multi-constrained stiffness optimization model considering uncertain loads is proposed. The corresponding sensitivity mathematical formula for stiffness is also given. The complete optimization iteration process is given. The structural stiffness optimization problems with the joint constraints of volume and displacement are solved here to illustrate the application of the proposed method. In the examples, a U-shape groove in different working situations is considered which includes the single node displacement constraint and the multi-node displacement constraint. The corresponding optimization results of different working situations are compared, and the results accord with the law of engineering.

Reliability Analysis of Alumina-Based Ceramic Cutting Tool's Wear Life by Saddle Point Approximation

Under continuous cutting conditions, the reliability of alumina-based ceramic cutting tools wear life with abrasion wear was discussed extensively. An ultimate state equation of alumina-based ceramic cutting tools wear life was proposed according to stress-strength interference theory. Based on the premise that the probability distribution of random parameters has been known, the fracture toughness KIC and the hardness H which obey Weibull distribution were transformed into normal distribution by means of the equivalent normal distribution method, and then the reliability of alumina-based ceramic cutting tools wear life was analyzed by saddle point approximation method. The probability density function and cumulative distribution function of the ultimate state equation were accurately and quickly obtained by way of saddle point approximation, as a result, the saddle point approximation method was proved accurate with high computing speed in comparison with the Monte-Carlo method. Therefore, the application of the saddle point approximation method developed the reliability analysis theory of alumina-based ceramic cutting tools wear life.

Approximate analysis of non-stationary loss queues and networks of loss queues with general service time distributions

A Fixed Point Approximation (FPA) method has recently been suggested for non-stationary analysis of loss queues and networks of loss queues with Exponential service times. Deriving exact equations relating time-dependent mean numbers of busy servers to blocking probabilities, we generalize the FPA method to loss systems with general service time distributions. These equations are combined with associated formulae for stationary analysis of loss systems in steady state through a carried load to offered load transformation. The accuracy and speed of the generalized methods are illustrated through a wide set of examples.

Asymptotes in SU(2) Recoupling Theory: Wigner Matrices, 3j Symbols, and Character Localization

In this paper we employ a novel technique combining the Euler Maclaurin formula with the saddle point approximation method to obtain the asymptotic behavior (in the limit of large representation index $J$) of generic Wigner matrix elements $D^{J}_{MM'}(g)$. We use this result to derive asymptotic formulae for the character $\chi^J(g)$ of an SU(2) group element and for Wigner's $3j$ symbol. Surprisingly, given that we perform five successive layers of approximations, the asymptotic formula we obtain for $\chi^J(g)$ is in fact exact. This result provides a non trivial example of a Duistermaat-Heckman like localization property for discrete sums.

Two positive solutions to three-point singular boundary value problems

In this article, we consider the existence of two positive solutions to nonlinear second order three-point singular boundary value problem: − u″(t)=λ f( t,u( t)) for all t ∈ (0,1) subjecting to u(0)=0 and α u(η)= u(1), where η ∈ (0,1),α ∈[0,1), and λ is a positive parameter. The nonlinear term f(t,u) is nonnegative, and may be singular at t=0, t=1, and u=0. By the fixed point index theory and approximation method, we establish that there exists λ *∈(0,+∞], such that the above problem has at least two positive solutions for any λ ∈(0,λ *) under certain conditions on the nonlinear term f.

Solution Methods for Robust Pricing of Transportation Networks under Uncertain Demand

Toll prices on traffic networks have been traditionally determined using a single expected demand value or deterministic demand supply relationships. Previous work by Gardner, Unnikrishnan, and Waller (2008) show that marginal social cost prices obtained using the expected value of demand can significantly deteriorate system performance especially when the actual system state deviates from the planned forecasted conditions. Determining the globally optimal tolls which are resilient to demand uncertainty entails a significantly high number of system performance evaluations which is a computationally intensive process. This work presents two practical methods to arrive at near optimal tolls – single point approximation methods and multiple point inflation/deflation approximation methods – and compares their performance in terms of computational efficiency and proximity to the optimal solution with two other commonly used meta-heuristics – Genetic Algorithm and Adaptive Simulated Annealing. Computational tests reveal that inflation/deflation methods can provide “near to optimal solutions” using a lower number of system performances in comparison to the meta-heuristics and single point approximation methods.

Mathematical Modeling and Statistical Analysis of Second Harmonic Generation Effects With Thin and Thick Crystals in Ultrahigh-Speed Optically Amplified Digital Lightwave Communication Systems

In this paper, we present an in depth analysis and discussions on the behavior and performance of using either thin or thick second harmonic generation (SHG) crystals prior to photodetector in ultrahigh-speed optically amplified digital lightwave communication systems. Our study begins by considering a conventional low-speed optically amplified lightwave receiver and study its performance in ultrahigh-speed regime and show its substantially degraded behavior, indicating the need to use advance optical nonlinear elements such as SHG prior to photodetection. In studying SHG effects, we begin by discussing the mathematical models of thin and thick SHG crystals in the context of statistical digital lightwave communication systems. In the case of thin SHG crystals and in the regime of high-speed digital systems, where Gaussian assumption is not necessarily valid, we use advanced numerical simulation based on Monte Carlo simulation to obtain its corresponding performance. However, for thick SHG crystals by invoking an advanced theorem due to Papoulis on Gaussianity approximation and a lemma due to Turin, we obtain the moment generating function of the sampled output with which, we obtain its corresponding performance using saddle point approximation method. Finally in our performance analysis, we consider electronically induced thermal noise and sketch the performance for various system parameters. The results in this paper indicate that the use of SHG crystals can enhance the system performance, when compared with the conventional receiver, by a substantial amount, i.e., five orders of magnitude. Furthermore, we highlight the existence of an optimum length for thick SHG crystals in the presence of thermal and optical amplifier noise.