Additional Function Evaluations Research Articles

A New Functional Threshold for Minimally Invasive Lobectomy.

To assess the performance of a lower predicted postoperative (ppo) forced expiratory volume in 1 second (FEV1) or diffusion capacity of the lung for carbon monoxide (DLCO) (ppoFEV1/ppoDLCO) threshold to predict cardiopulmonary complications after minimally invasive surgery (MIS) lobectomy. Although MIS is associated with better postoperative outcomes than open surgery, MIS uses risk-assessment algorithms developed for open surgery. Moreover, several different definitions of cardiopulmonary complications are used for assessment. All patients who underwent MIS lobectomy for clinical stage I-II lung cancer from 2018 to 2022 at our institution were considered. The performance of a ppoFEV1/ppoDLCO threshold of <45% was compared against that of the current guideline threshold of <60%. Three different definitions of cardiopulmonary complications were compared: Society of Thoracic Surgeons (STS), European Society of Thoracic Surgeons (ESTS), and Berry et al. In 946 patients, the ppoFEV1/ppoDLCO threshold of <45% was associated with a higher proportion correctly classified (79% [95% CI, 76%-81%] vs. 65% [95% CI, 62%-68%]; P<0.001). The complication with the biggest difference in incidence between ppoFEV1/ppoDLCO of 45%-60% and >60% was prolonged air leak (33 [13%] vs. 34 [6%]; P<0.001). The predicted probability curves for cardiopulmonary complications were higher for the STS definition than for the ESTS or Berry definitions across ppoFEV1 and ppoDLCO values. The ppoFEV1/ppoDLCO threshold of <45% more accurately classified patients for cardiopulmonary complications after MIS lobectomy, emphasizing the need for updated risk-assessment guidelines for MIS lobectomy to optimize additional cardiopulmonary function evaluation.

Increasing in three units the order of convergence of iterative methods for solving nonlinear systems

In this paper, we present a technique aimed at the implementation of high accuracy iterative procedures in nonlinear systems. The main result of this research is a general proof that guarantees an increase in the order of convergence to order p+3 units for any method of order p. This methodology has as its main objective not only this increase, but also the optimization in the use of resources by requiring only one additional functional evaluation per iteration, thus keeping a tight control over the computational cost. Although there are previous proposals in the literature that approach this goal, in this work we consider such approaches as particular cases of our general proposal. This approach encompasses many methods described by other authors and can be applied to any p-order scheme to increase it by three units. Numerical evaluations, supported by a rigorous theoretical framework, validate the robustness of the proposal, highlighting its advantage over conventional techniques and demonstrating its effectiveness in the face of numerical challenges of considerable magnitude. Summing up, with this research, we offer a substantial contribution to the field of numerical methods, proposing a paradigm that consistently enhances the convergence order of iterative methods.

Extended Higher Order Iterative Method for Nonlinear Equations and its Convergence Analysis in Banach Spaces

In this article, a novel higher order iterative method for solving nonlinear equations is developed. The new iterative method obtained from fifth order Newton-Özban method attains eighth order of convergence by adding a single step with only one additional function evaluation. The method is extended to Banach spaces and its local as well as semi-local convergence analysis is done under generalized continuity conditions. The existence and uniqueness results of solution are also provided along with radii of convergence balls. From the numerical experiments, it can be inferred that the proposed method is more accurate and effective in high precision computations than existing eighth order methods. The computation of error analysis and norm of functions demonstrate that proposed method takes a lead over the considered methods.

Generalized high-order iterative methods for solutions of nonlinear systems and their applications

&lt;abstract&gt;&lt;p&gt;In this paper, we have constructed a family of three-step methods with sixth-order convergence and a novel approach to enhance the convergence order $ p $ of iterative methods for systems of nonlinear equations. Additionally, we propose a three-step scheme with convergence order $ p+3 $ (for $ p\geq3 $) and have extended it to a generalized $ (m+2) $-step scheme by merely incorporating one additional function evaluation, thus achieving convergence orders up to $ p+3m $, $ m\in\mathbb{N} $. We also provide a thorough local convergence analysis in Banach spaces, including the convergence radius and uniqueness results, under the assumption of a Lipschitz-continuous Fréchet derivative. Theoretical findings have been validated through numerical experiments. Lastly, the performance of these methods is showcased through the analysis of their basins of attraction and their application to systems of nonlinear equations.&lt;/p&gt;&lt;/abstract&gt;

P-AK-MCS: Parallel AK-MCS method for structural reliability analysis

In recent years, the active learning reliability method that combines the Kriging model and Monte Carlo simulation (AK-MCS) has emerged as a promising approach due to its computational efficiency and accuracy. However, the commonly used learning functions, such as the expected feasibility function (EFF), U function, H function, and expected risk function (ERF), can only select one training point at each iteration which is time-wasteful when parallel computing is available. Therefore, this paper proposes a parallel active learning Kriging strategy, namely P-AK-MCS, for structural reliability analysis. By introducing an influence function that reflects the impact of the added point on the original learning function, four parallel learning functions are constructed: pseudo-U (PU) function, pseudo-EFF (PEFF), pseudo-H (PH) function, and pseudo-ERF (PERF). These functions aim to identify multiple training points at each iteration without requiring additional functional evaluations. The effectiveness of the proposed method is validated using four examples. The results demonstrate that compared to the standard AK-MCS, the proposed P-AK-MCS significantly reduces the number of computation loops and greatly decreases computational costs. Moreover, the total number of functional evaluations required is similar to that of the standard AK-MCS and remains insensitive to the number of multiple training points.

An Efficient Bi-Parametric With-Memory Iterative Method for Solving Nonlinear Equations

New three-step with-memory iterative methods for solving nonlinear equations are presented. We have enhanced the convergence order of an existing eighth-order memory-less iterative method by transforming it into a with-memory method. Enhanced acceleration of the convergence order is achieved by introducing two self-accelerating parameters computed using the Hermite interpolating polynomial. The corresponding R-order of convergence of the proposed uni- and bi-parametric with-memory methods is increased from 8 to 9 and 10, respectively. This increase in convergence order is accomplished without requiring additional function evaluations, making the with-memory method computationally efficient. The efficiency of our with-memory methods NWM9 and NWM10 increases from 1.6818 to 1.7320 and 1.7783, respectively. Numeric testing confirms the theoretical findings and emphasizes the superior efficacy of suggested methods when compared to some well-known methods in the existing literature.

Derivative-Free Families of With- and Without-Memory Iterative Methods for Solving Nonlinear Equations and Their Engineering Applications

In this paper, we propose a new fifth-order family of derivative-free iterative methods for solving nonlinear equations. Numerous iterative schemes found in the existing literature either exhibit divergence or fail to work when the function derivative is zero. However, the proposed family of methods successfully works even in such scenarios. We extended this idea to memory-based iterative methods by utilizing self-accelerating parameters derived from the current and previous approximations. As a result, we increased the convergence order from five to ten without requiring additional function evaluations. Analytical proofs of the proposed family of derivative-free methods, both with and without memory, are provided. Furthermore, numerical experimentation on diverse problems reveals the effectiveness and good performance of the proposed methods when compared with well-known existing methods.

Iterative Backpropagation Method for Efficient Gradient Estimation in Bilevel Network Equilibrium Optimization Problems

Network optimization or network design with an embedded traffic assignment (TA) to model user equilibrium principle, sometimes expressed as bilevel problems or mathematical programs with equilibrium constraints (MPEC), is at the heart of transportation planning and operations. For applications to large-scale multimodal networks with high dimensional decision variables, the problem is nontrivial, to say the least. General-purpose algorithms and problem-specific bilevel formulations have been proposed in the past to solve small problems for demonstration purposes. Research gap, however, exists in developing efficient solution methods for large-scale problems in both static and dynamic contexts. This paper proposes an efficient gradient estimation method called Iterative Backpropagation (IB) for network optimization problems with an embedded static TA model. IB exploits the iterative structure of the TA solution procedure and simultaneously calculates the gradients while the TA process converges. IB does not require any additional function evaluation and consequently scales very well with higher dimensions. We apply the proposed approach to origin-destination (OD) estimation, an MPEC problem, of the Hong Kong multimodal network with 49,806 decision variables, 8,797 nodes, 18,207 links, 2,684 transit routes, and 165,509 OD pairs. The calibrated model performs well in matching the link counts. Specifically, the IB-gradient based optimization technique reduces the link volume squared error by 98%, mean absolute percentage error (MAPE) from 95.29% to 21.23%, and the average GEH statistics from 24.18 to 6.09 compared with the noncalibrated case. The framework, even though applied to OD estimation in this paper, is applicable to a wide variety of optimization problems with an embedded TA model, opening up an efficient way to solve large-scale MPEC or bilevel problems. Funding: The study is supported by IVADO Postdoctoral Fellowship scheme 2021, HSBC 150th Anniversary Charity Programme HKBF17RG01, National Science Foundation of China (71890970, 71890974), General Research Fund (16212819, 16207920) of the HKSAR Government, and the Hong Kong PhD Fellowship.

An automatic quadrature method for semi-infinite integrals of exponentially decaying functions and its Matlab code

For the integral of a function f over a semi-infinite interval [0,∞) with f(x) decaying exponentially as x→∞, an automatic quadrature scheme is constructed and implemented in Matlab. Three limit formulae of Clenshaw–Curtis-type rules in our recent work (Numerical Algorithms, vol.90, 2022, pp.3–30) are exploited that converge fast for analytic functions. Each formula is expressed by a weighted infinite sum of function values at nodes and so its truncated finite sum is used. The best of the three formulae in accuracy is used as an approximation whose error is estimated with other two formulae and no additional function evaluations, since nodes of the other formulae are included in those of the best one. Numerical results indicate that our automatic scheme is better in performance than the QUADPACK routine for several types of test integrals. Particularly, it copes quite well with oscillatory integrals including the sinusoidal function and/or the Bessel function that are known to be difficult to approximate.

Extension of King’s Iterative Scheme by Means of Memory for Nonlinear Equations

We developed a new family of optimal eighth-order derivative-free iterative methods for finding simple roots of nonlinear equations based on King’s scheme and Lagrange interpolation. By incorporating four self-accelerating parameters and a weight function in a single variable, we extend the proposed family to an efficient iterative scheme with memory. Without performing additional functional evaluations, the order of convergence is boosted from 8 to 15.51560, and the efficiency index is raised from 1.6817 to 1.9847. To compare the performance of the proposed and existing schemes, some real-world problems are selected, such as the eigenvalue problem, continuous stirred-tank reactor problem, and energy distribution for Planck’s radiation. The stability and regions of convergence of the proposed iterative schemes are investigated through graphical tools, such as 2D symmetric basins of attractions for the case of memory-based schemes and 3D stereographic projections in the case of schemes without memory. The stability analysis demonstrates that our newly developed schemes have wider symmetric regions of convergence than the existing schemes in their respective domains.

Higher order Traub–Steffensen type methods and their convergence analysis in Banach spaces

Abstract In this paper, we consider two-step fourth-order and three-step sixth-order derivative free iterative methods and study their convergence in Banach spaces to approximate a locally-unique solution of nonlinear equations. Study of convergence analysis provides radius of convergence, error bounds and estimates on the uniqueness of the solution. Such estimates are not provided in the approaches that use Taylor expansions using higher order derivatives. Furthermore, in quest of fast algorithms, a generalized q-step scheme with increasing convergence order 2q + 2 is proposed and analyzed. Novelty of the q-step algorithm is that, in each step, order of convergence is increased by an amount of two at the cost of only one additional function evaluation. To maximize the computational efficiency, the optimal number of steps is calculated. Theoretical results regarding convergence and computational efficiency are verified through numerical experimentation.

Development of Optimal Iterative Methods with Their Applications and Basins of Attraction

In this paper, we construct variants of Bawazir’s iterative methods for solving nonlinear equations having simple roots. The proposed methods are two-step and three-step methods, with and without memory. The Newton method, weight function and divided differences are used to develop the optimal fourth- and eighth-order without-memory methods while the methods with memory are derivative-free and use two accelerating parameters to increase the order of convergence without any additional function evaluations. The methods without memory satisfy the Kung–Traub conjecture. The convergence properties of the proposed methods are thoroughly investigated using the main theorems that demonstrate the convergence order. We demonstrate the convergence speed of the introduced methods as compared with existing methods by applying the methods to various nonlinear functions and engineering problems. Numerical comparisons specify that the proposed methods are efficient and give tough competition to some well known existing methods.

Topological Analysis of Functions on Arbitrary Grids: Applications to Quantum Chemistry.

Algorithms are presented for performing a topological analysis of an arbitrary function, evaluated on an arbitrary grid of points. These algorithms work strictly by post-processing the data and require no additional function evaluations. This is achieved by connecting the grid points with a neighborhood graph, allowing the topological analysis to be recast as a problem in the graph theory. The flexibility of the approach is demonstrated for various applications involving analysis of the charge and magnetically induced current densities in molecules, where features of the neighborhood graph are found to correspond to chemically relevant topographical properties, such as Bader charges. These properties converge using orders of magnitude fewer grid points than uniform-grid approaches while exhibiting an appealing O[N log(N)] scaling of the computational cost. The issue of grid bias is discussed in the context of graph-based algorithms and strategies for avoiding this bias are presented. Python implementations of the algorithms are provided.

Generalized pigeon-inspired optimization algorithm for balancing exploration and exploitation

<p indent="0mm">A generalized pigeon-inspired optimization (GPIO) algorithm for balancing the exploration and exploitation abilities is proposed herein. The traditional pigeon-inspired optimization algorithm includes two operators, namely the map and compass operator and the landmark operator. These two operators are implemented only for one round at a single run. In the GPIO algorithm, the search process is divided into multiple stages, and two operators are implemented in each stage. These two operators are implemented for multiple rounds at one single run. The map and compass operator focuses on the exploration ability, while the landmark operator focuses on the exploitation ability. The GPIO algorithm changes the execution order of the two operators without additional objective function evaluation. Moreover, the structure of the solutions and the parameter settings are extended in the GPIO algorithm, which is beneficial to search quality improvement. The simulation results show that the GPIO algorithm improves the search efficiency and the search results of the algorithm.

Higher-order Taylor series expansion for uncertainty quantification with efficient local sensitivity

An efficient numerical approach for uncertainty quantification using a higher-order Taylor series expansion is presented. Moreover, local sensitivities in the Taylor series are evaluated using a high-accuracy and computationally efficient approach called modified forward finite difference (ModFFD). Once the sensitivities are estimated, with ModFFD, the stochastic response can be obtained for different realizations of the random inputs without incurring additional function evaluations. The main advantage of this approach is that it is applicable for any input distributions and is unrestricted by the nature of random input variables (correlated and uncorrelated). To test the presented approach, several analytical and engineering problems were considered with up to twenty-two random variables. For the engineering problem, a ten-bar truss problem with twenty-two random variables and buckling of a composite laminate with twenty random variables is considered. The comparison of the results with that of a large number of Monte Carlo Simulations demonstrated its high accuracy and computational efficiency for random inputs with non-standard random inputs and varying correlation.

Approximating the diagonal of a Hessian: which sample set of points should be used

An explicit formula based on matrix algebra to approximate the diagonal entries of a Hessian matrix with any number of sample points is introduced. When the derivative-free technique called generalized centered simplex gradient is used to approximate the gradient, then the formula can be computed for only one additional function evaluation. An error bound is introduced and provides information on the form of the sample set of points that should be used to approximate the diagonal of a Hessian matrix. If the sample set of points is built in a specific manner, it is shown that the technique is a \(\mathcal {O}({{\Delta }_{S}^{2}})\) accurate approximation of the diagonal entries of the Hessian matrix, where ΔS is the radius of the sample set.

Prehospital Electrolyte Care: A Review of Symptoms, Evaluation, and Management

Ongoing evolution of prehospital medical care continues to advance beyond tactical field care scenarios in the consideration of prolonged field care. This is even more important to consider in theaters with extended evacuation times and limited local medical assets. The critical regulatory functions of electrolytes such as sodium, potassium, calcium, and glucose require medics operating in these environments to have a strong, fundamental knowledge of the principles, manifestations, and initial stabilization measures to aid their patients prior to, or in lieu of evacuation. Continued development and access to point of care testing in increasingly forward deployed settings further enables medics to perform these tasks. Here, we provide a brief review of these vital electrolytes, as well as additional kidney function evaluation considerations, to assist medics in their treatment efforts. Specific concerns for battlefield and atraumatic presentations are addressed.

A Class of Higher-Order Newton-Like Methods for Systems of Nonlinear Equations

In this paper, a class of efficient iterative methods with increasing order of convergence for solving systems of nonlinear equations is developed and analyzed. The methodology uses well-known third-order Potra–Pták iteration in the first step and Newton-like iterations in the subsequent steps. Novelty of the methods is the increase in convergence order by an amount three per step at the cost of only one additional function evaluation. In addition, the algorithm uses a single inverse operator in each iteration, which makes it computationally more efficient and attractive. Local convergence is studied in the more general setting of a Banach space under suitable assumptions. Theoretical results of convergence and computational efficiency are verified through numerical experimentation. Comparison of numerical results indicates that the developed algorithms outperform the other similar algorithms available in the literature, particularly when applied to solve the large systems of equations. The basins of attraction of some of the existing methods along with the proposed method are given to exhibit their performance.

A class of computationally efficient Newton-like methods with frozen inverse operator for nonlinear systems

Abstract We propose a class of Newton-like methods with increasing convergence order for approximating the solutions of systems of nonlinear equations. Novelty of the methods is that in each step the order of convergence is increased by an amount of three at the cost of only one additional function evaluation. Another important feature is the single use of an inverse operator in each iteration, which makes the schemes attractive and computationally more efficient. Theoretical results regarding convergence and computational efficiency are verified through numerical examples including those arising from boundary value problems and integral equations. By way of comparison, it is shown that the present methods are more efficient than their existing counterparts, particularly when applied to solve the large systems of equations.

The moving particles method for reliability estimation: recent advances

AbstractIn the following, recent advances of the moving particles method are highlighted. A multi level version of the estimator is introduced that balances the statistical error and the numerical approximation error by computing a telescoping sum of estimates for the number of moves. An extension to general model classes in the sense of a multi fidelity method is obtained based on the estimation of bivariate Poisson distributions for censored data. Finally, the local sensitivity of the reliability estimate with respect to the parameters of the probability characteristics of the model input is considered. It is shown how to obtain estimates of the local sensitivity without any additional function evaluations.