Additional Function Evaluations Research Articles

On generalized biparametric multipoint root finding methods with memory

A general family of biparametric n-point methods with memory for solving nonlinear equations is proposed using an original accelerating procedure with two parameters. This family is based on derivative free classes of n-point methods without memory of interpolatory type and Steffensen-like method with two free parameters. The convergence rate of the presented family is considerably increased by self-accelerating parameters which are calculated in each iteration using information from the current and previous iteration and Newton’s interpolating polynomials with divided differences. The improvement of convergence order is achieved without any additional function evaluations so that the proposed family has a high computational efficiency. Numerical examples are included to confirm theoretical results and demonstrate convergence behavior of the proposed methods.

A gradient-based transformation method in multidisciplinary design optimization

Multidisciplinary design optimization (MDO) has become essential for solving the complex engineering design problems. The most common approach is to "divide and conquer" the MDO problem, that is, to decompose the complex problem into several sub-problems and to collect the local solutions to give a new design point for the original problem. In 1990s, researchers have developed some decomposition strategies to find or synthesize the optimal model of the optimization structure in order to evenly distribute the computational workloads to multiple processors. Several MDO methods, such as Collaborative Optimization (CO) and Analytical Target Cascading (ATC), were then developed to solve the decomposed sub-problems and coordinate the coupling variables among them to find the optimal solution. However, both the synthesis of the decomposition structure and the coordination of the coupling variables require additional function evaluations, in terms of evaluating the functional dependency between each sub-problem and determining the proper weighting coefficients between each coupling functions respectively. In this paper, a new divide-and-conquer strategy, Gradient-based Transformation Method (GTM), is proposed to overcome the challenges in structure synthesis and variable coordination. The proposed method first decomposes the MDO problem into several sub-systems and distributes one constraint from the original problem to each sub-system without evaluating the dependency between each sub-system. Each sub-system is then transformed to the single-variate coordinate along the gradient direction of the constraint. The total function evaluations equal the number of constraints times the number of variables plus one in every iteration. Due to the monotonicity characteristics of the transformed sub-problems, they are efficiently solved by Monotonicity Analyses without any additional function evaluations. Two coordination principles are proposed to determine the significances of the responses based on the feasibility and activity conditions of every sub-problem and to find the new design point at the average point of the most significant responses. The coordination principles are capable of finding the optimal solution in the convex feasible space bounded by the linearized sub-system constraints without additional function evaluations. The optimization processes continue until the convergence criterion is satisfied. The numerical examples show that the proposed methodology is capable of effectively and efficiently finding the optimal solutions of MDO problems.

Stochastic Response Surface Method with Enhanced Weighting Strategy

The weighted stochastic response surface method (WSRSM) has been demonstrated to be effective in improving the accuracy of the estimation of statistical moments and probability of failure (PoF) upon the stochastic response surface method (SRSM). However, it has been noticed that the weighting method in WSRSM may have little and sometimes negative impact on PoF estimation especially in the cases of low PoF. To address this issue, an enhanced weighting strategy is proposed that the weights of sample points are determined based on their importance not only to regression but also to PoF estimation. Specifically, relatively larger weights are assigned to points closer to the failure surface, which significantly accounts for the accuracy of PoF estimation. Comparative studies show that SRSM with the proposed weighting method outperforms WSRSM producing more accurate PoF estimation without incurring additional function evaluations.

On an efficient family of derivative free three-point methods for solving nonlinear equations

New three-step derivative free families of three-point methods for solving nonlinear equations are presented. First, a new family without memory of optimal order eight, consuming four function evaluations per iteration, is proposed by using two weight functions. The improvement of the convergence rate of this basic family, even up to 50%, is obtained without any additional function evaluation using a self-accelerating parameter. This varying parameter is calculated in each iterative step employing only information from the current and the previous iteration, defining in this way a family with memory. The self-accelerating parameter is calculated applying Newton’s interpolating polynomials of degree scaling from 1 to 4. The corresponding R-orders of convergence are increased from 8 to 10, 11, 6+42≈11.66 and 12, providing very high computational efficiency of the proposed methods with memory. Another convenient fact is that these methods do not use derivatives. Numerical examples and comparison with the existing three-point methods are included to confirm theoretical results.

A double weighted stochastic response surface method for reliability analysis

The weighted stochastic response surface method (WSRSM) has been demonstrated to be effective in improving the accuracy of the estimation of statistical moments and probability of failure (PoF) upon the stochastic response surface method (SRSM). However, it has been noticed that the weighting method in WSRSM may have little and sometimes negative impact on PoF estimation especially in the cases of low PoF. To mitigate this issue, a new double weighted SRSM (DWSRSM) is proposed that the weights of sample points are determined based on their importance not only to regression but also to PoF estimation. Specifically, relatively larger weights are assigned to points closer to the failure surface, which significantly accounts for the accuracy of PoF estimation. Comparative studies show that DWSRSM outperforms WSRSM producing more accurate PoF estimation without incurring additional function evaluations. An application of DWSRSM to the rocket design further demonstrates its effectiveness for PoF estimation.

Some efficient derivative free methods with memory for solving nonlinear equations

Based on the family of three-point derivative free methods without memory of eighth order convergence proposed by Džunić et al. [J. Džunić, M.S. Petković, L.D. Petković, Three-point methods with and without memory for solving nonlinear equations, Appl. Math. Comput. 218 (2012) 4917–4927], we present three methods with memory by suitable variation of a free parameter in each iterative step. This free parameter is calculated using Newton’s interpolatory polynomial of the third degree in two ways and Newton’s interpolatory polynomial of the fourth degree. Consequently, the R-order of convergence is increased from 8 to 11+1372≈11.352, 6+42≈11.657 and 12. The increase in the convergence order is achieved without any additional function evaluations and therefore, the proposed methods possess a very high computational efficiency. Numerical examples are presented and the performance is compared with the existing three-point methods with and without memory of the basic family. Moreover, theoretical order of convergence is verified on the examples.

Several improvements of the 2-point third order midpoint iterative method using weight functions

In this paper, we improve the order of the Midpoint iterative method from three to four with same number of function evaluations using weight functions to obtain a class of 2-point fourth order Jarratt-type methods. Then we prove a general result to further improve the order of the old methods through an additional function evaluation using weight functions. In this way, we develop two 3-point sixth order midpoint methods using different weights. A family of n-point I.F.s of order 2n is proposed. Using weight functions, we further improve the methods to obtain a 5-point sixteenth order midpoint method with same efficiency index as the optimal 2-point fourth order Jarratt-type methods. Two numerical examples using different starting are provided to test the robustness of the methods.

On generalized multipoint root-solvers with memory

The improved versions of the Kung–Traub family and the Zheng–Li–Huang family of n-point derivative free methods for solving nonlinear equations are proposed. The convergence speed of the modified families is considerably accelerated by employing a self-correcting parameter. This parameter is calculated in each iteration using information from the current and previous iteration so that the proposed families can be regarded as the families with memory. The increase of convergence order is attained without any additional function evaluations meaning that these families with memory possess high computational efficiency. Numerical examples are included to confirm theoretical results and demonstrate convergence behaviour of the proposed methods.

Complications vasculaires de la transplantation rénale : conduite diagnostique

Vascular complications after renal transplantation are the most frequent type of complication following urological complications. They may affect the function of the transplant. Early vascular complications include renal artery or vein thrombosis, lesions to the iliac vessels and cortical necrosis. Delayed complications mainly include renal artery stenosis, arteriovenous fistula, and rarely false aneurysm. Doppler sonography, sometimes with the use of intravenous contrast, is the imaging modality of choice in the acute setting or routine follow-up. MRI may be performed for additional morphological and functional evaluation while CT may provide additional evaluation of the arterial supply. Angiography is performed prior to endovascular treatment.

Efficient simulation of unsaturated flow using exponential time integration

We assess the performance of an exponential integrator for advancing stiff, semidiscrete formulations of the unsaturated Richards equation in time. The scheme is of second order and explicit in nature but requires the action of the matrix function φ( A) = A −1( e A − I) on a suitability defined vector v at each time step. When the matrix A is large and sparse, φ( A) v can be approximated by Krylov subspace methods that require only matrix–vector products with A. We prove that despite the use of this approximation the scheme remains second order. Furthermore, we provide a practical variable-stepsize implementation of the integrator by deriving an estimate of the local error that requires only a single additional function evaluation. Numerical experiments performed on two-dimensional test problems demonstrate that this implementation outperforms second-order, variable-stepsize implementations of the backward differentiation formulae.

Efficient structural reliability methods considering incomplete knowledge of random variable distributions

The determination of an exact distribution function of a random phenomena is not possible using a limited number of observations. Therefore, in the present paper the stochastic properties of a random variable are assumed as uncertain quantities and instead of predefined distribution types the maximum entropy distribution is used. Efficient methods for a reliability analysis considering these uncertain stochastic parameters are presented. Based on approximation strategies this extended analysis requires no additional limit state function evaluations. Later, variance based sensitivity measures are used to evaluate the contribution of the uncertainty of each stochastic parameter to the total variation of the failure probability.

A prototype and demonstrator of Akogrimo’s architecture: An approach of merging grids, SOA, and the mobile Internet

The trend of merging telecommunication infrastructures with traditional Information Technology (IT) infrastructures is ongoing and important for commercial service providers. The driver behind this development is, on one hand, the strong need for enhanced services and on the other hand, the need of telecommunication operators aiming at value-added service provisioning to a wide variety of customers. In the telecommunications sector, the IP Multimedia Subsystem (IMS) is a promising service platform, which may become a “standard” for supporting added-value services on top of the next generation network infrastructure. However, since its range of applicability is bound to SIP- enabled services, IMS extensions are being proposed by “SIPifying” applications. In parallel to these developments within the traditional IT sector, the notion of Virtual Organizations (VO) enabling collaborative businesses across organizational boundaries is addressed in the framework of Web Services (WS) standards implementing a Service-oriented Architecture (SOA). Here, concepts for controlled resource and service sharing based on WS and Semantic Technologies have been consolidated. Since the telecommunications sector has become, in the meantime “mobile”, all concepts brought into this infrastructure must cope with the dynamics mobility brings in. Therefore, within the Akogrimo project the VO concept has been extended towards a Mobile Dynamic Virtual Organization (MDVO) concept, additionally considering key requirements of mobile users and resources. Especial attention is given to ensure the duality of the merge of both, SOA and IMS approaches to holistically support SOA-enabled mobile added-value services and their users. This work describes major results of the Akogrimo project, paying special attention to the overall Akogrimo architecture, the prototype implemented, and the key scenario in which the instantiated Akogrimo architecture shows a very clear picture of applicability, use, and an additional functional evaluation.

Some variants of the Chebyshev–Halley family of methods with fifth order of convergence

In this paper we present some techniques for constructing high-order iterative methods in order to approximate the zeros of a non-linear equation f(x)=0, starting from a well-known family of cubic iterative processes. The first technique is based on an additional functional evaluation that allows us to increase the order of convergence from three to five. With the second technique, we make some changes aimed at minimizing the calculus of inverses. Finally, looking for a better efficiency, we eliminate terms that contribute to the error equation from sixth order onwards. The paper contains a comparative study of the asymptotic error constants of the methods and some theoretical and numerical examples that illustrate the given results. We also analyse the efficiency of the aforementioned methods, by showing some numerical examples with a set of test functions and by using adaptive multi-precision arithmetic in the computation.

More life for a “terminal” cell

More life for a “terminal” cell

Note on a Cubically Convergent Newton-Type Method Under Weak Conditions

In Fang et al. (J. Comput. Appl. Math. 220:409–412, 2008), a new two-step predictor–corrector type iterative method free from second derivatives for solving nonlinear equations has been proposed. But if λ k ≠0, k=1,2,… , then their main theorem does not hold. In this note, we first show that the iterative method given by authors has second-order convergence in the case λ k ≠0, k=1,2,… , not the third-order one as claimed in their work. Then, we present a modified scheme to increase the order, and it does not need additional function and derivatives evaluations per iteration. Several examples are given to demonstrate the efficiency and performance of a new modified method. Finally, we further discuss and propose several new modified families of iterative root-finding methods by using a new proposed technique.

Special Issue Devoted to Professor Ian Gladwell

Special Issue Devoted to Professor Ian Gladwell

Vectorized Solution of ODEs in Matlab

Vectorization is very important to the efficiency of computation in the popular problem-solving environment Matlab. It is shown that a class of Runge-Kutta methods investigated by Milne and Rosser that compute a block of new values at each step are well-suited to vectorization. Local error estimates and continuous extensions that require no additional function evaluations are derived. A (7,8) pair is derived and implemented in a program BV78 that is shown to perform quite well when compared to the well-known Matlab ODE solver ode45 which is based on a (4,5) pair.

Some modifications of Newton’s method with higher-order convergence for solving nonlinear equations

In [YoonMee Ham etal., Some higher-order modifications of Newton’s method for solving nonlinear equations, J. Comput. Appl. Math., 222 (2008) 477–486], some higher-order modifications of Newton’s method for solving nonlinear equations are constructed. But if p = 2 , then their main theorem did not hold. In this paper, we first give an example to show that YoonMee Ham etal.’s methods are not always correct in the case p = 2 . Then, we present the condition that H ( x , y ) should satisfy such that the order of convergence increases three or four or five units. Per iteration they only need two additional function evaluations to increase the order. Based on this and multi-step Newton’s scheme, we give further modifications of the method to obtain higher-order convergent iterative methods. Finally, several examples are given to demonstrate the efficiency and performance of our modified methods and compare them with some other methods.

From high oscillation to rapid approximation III: multivariate expansions

In this paper, we expand upon the theme of modified Fourier expansions and extend the theory to a multivariate setting and to expansions in eigenfunctions of the Laplace-Neumann operator. We pay detailed attention to expansions in a d-dimensional cube and to an effective derivation of expansion coefficients there by means of quadratures of highly oscillatory integrals. Thus, we present asymptotic and Filon-type formulae for an effective derivation of expansion coefficients and discuss their design and relative advantages. Such methods are effective only for large indices; hence, we introduce and analyse alternative quadrature schemes that require a relatively modest number of additional function evaluations.

A Method for Handling Uncertainty in Evolutionary Optimization With an Application to Feedback Control of Combustion

We present a novel method for handling uncertainty in evolutionary optimization. The method entails quantification and treatment of uncertainty and relies on the rank based selection operator of evolutionary algorithms. The proposed uncertainty handling is implemented in the context of the covariance matrix adaptation evolution strategy (CMA-ES) and verified on test functions. The present method is independent of the uncertainty distribution, prevents premature convergence of the evolution strategy and is well suited for online optimization as it requires only a small number of additional function evaluations. The algorithm is applied in an experimental setup to the online optimization of feedback controllers of thermoacoustic instabilities of gas turbine combustors. In order to mitigate these instabilities, gain-delay or model-based H infin controllers sense the pressure and command secondary fuel injectors. The parameters of these controllers are usually specified via a trial and error procedure. We demonstrate that their online optimization with the proposed methodology enhances, in an automated fashion, the online performance of the controllers, even under highly unsteady operating conditions, and it also compensates for uncertainties in the model-building and design process.