A hybrid method for inversion of 3D DC resistivity logging measurements

This paper focuses on the application of hp hierarchic genetic strategy (hp–HGS) for solution of a challenging problem, the inversion of 3D direct current (DC) resistivity logging measurements. The problem under consideration has been formulated as the global optimization one, for which the objective function (misfit between computed and reference data) exhibits multiple minima. In this paper, we consider the extension of the hp–HGS strategy, namely we couple the hp–HGS algorithm with a gradient based optimization method for a local search. Forward simulations are performed with a self-adaptive hp finite element method, hp–FEM. The computational cost of misfit evaluation by hp–FEM depends strongly on the assumed accuracy. This accuracy is adapted to the tree of populations generated by the hp–HGS algorithm, which makes the global phase significantly cheaper. Moreover, tree structure of demes as well as branch reduction and conditional sprouting mechanism reduces the number of expensive local searches up to the number of minima to be recognized. The common (direct and inverse) accuracy control, crucial for the hp–HGS efficiency, has been motivated by precise mathematical considerations. Numerical results demonstrate the suitability of the proposed method for the inversion of 3D DC resistivity logging measurements.     

Arbitrary-level hanging nodes and automatic adaptivity in the hp-FEM

In this paper we present a new automatic adaptivity algorithm for the hp-FEM which is based on arbitrary-level hanging nodes and local element projections. The method is very simple to implement compared to other existing hp-adaptive strategies, while its performance is comparable or superior. This is demonstrated on several numerical examples which include the L-shape domain problem, a problem with internal layer, and the Girkmann problem of linear elasticity. With appropriate simplifications, the proposed technique can be applied to standard lower-order and spectral finite element methods.     

An hp finite element adaptive scheme to solve the Laplace model for fluid–solid vibrations

In this paper we introduce an hp finite element method to solve a two-dimensional fluid–structure spectral problem. This problem arises from the computation of the vibration modes of a bundle of parallel tubes immersed in an incompressible fluid. We prove the convergence of the method and a priori error estimates for the eigenfunctions and the eigenvalues. We define an a posteriori error estimator of the residual type which can be computed locally from the approximate eigenpair. We show its reliability and efficiency by proving that the estimator is equivalent to the energy norm of the error up to higher order terms, the equivalence constant of the efficiency estimate being suboptimal in that it depends on the polynomial degree. We present an hp adaptive algorithm and several numerical tests which show the performance of the scheme, including some numerical evidence of exponential convergence.     

基于遗传算法的动态MC2运输问题求解

动态MC²运输问题是描述多阶段供求波动的运输问题,其模型框架可以应用到很多领域。目前对动态MC²运输模型的求解主要采用传统的单纯形法,针对该问题的特殊性采用具有全局搜索能力的遗传算法进行求解。通过三维数组编码,设计有效的交叉、变异算子和适应度函数,克服了单纯形法求解该问题出现的并行性差、求解整数规划困难的不足。用Matlab7.0编程对算法进行检验,结果表明经过特殊设计的遗传算法能够很好地解决动态MC²运输问题。     

Two-Grid hp-Version Discontinuous Galerkin Finite Element Methods for Second-Order Quasilinear Elliptic PDEs

In this article we propose a class of so-called two-grid hp-version discontinuous Galerkin finite element methods for the numerical solution of a second-order quasilinear elliptic boundary value problem of monotone type. The key idea in this setting is to first discretise the underlying nonlinear problem on a coarse finite element space $V({{\mathcal {T}_{H}}},\boldsymbol {P})$ . The resulting ‘coarse’ numerical solution is then exploited to provide the necessary data needed to linearise the underlying discretisation on the finer space $V({{\mathcal {T}_{h}}},\boldsymbol {p})$ ; thereby, only a linear system of equations is solved on the richer space $V({{\mathcal {T}_{h}}},\boldsymbol {p})$ . In this article both the a priori and a posteriori error analysis of the two-grid hp-version discontinuous Galerkin finite element method is developed. Moreover, we propose and implement an hp-adaptive two-grid algorithm, which is capable of designing both the coarse and fine finite element spaces $V({{\mathcal {T}_{H}}},\boldsymbol {P})$ and $V({{\mathcal {T}_{h}}},\boldsymbol {p})$ , respectively, in an automatic fashion. Numerical experiments are presented for both two- and three-dimensional problems; in each case, we demonstrate that the CPU time required to compute the numerical solution to a given accuracy is typically less when the two-grid approach is exploited, when compared to the standard discontinuous Galerkin method.     

货架分配问题的一个有效混合遗传算法

针对货架分配问题提出了一个遗传算法与模拟退火算法及一个局部搜索算法混合的算法。首先,设计了一种比较直观的编码方法,用一个矩阵作为一种货架分配方案。第二,设计了与编码相应的杂交和变异算子,并且杂交、变异都能生成可行解,不需要对解进行修正。第三,为了能够生成好的初始种群,定义了一个阀值,这个阀值不仅反映了解的适应值的信息,而且还反映解的结构的信息。第四,为了增加算法的局部搜索能力,同时又尽量不增加计算的复杂度,让模拟退火算法和一种局部搜索算法并行作用于相应的子群。通过大量的数据模拟实验及与其他的几种算法模拟结果进行比较,实验显示,该算法不论是计算结果还是算法的稳定性都优于其他算法。     

基于遗传算法的动态MC2运输问题求解

动态MC²运输问题是描述多阶段供求波动的运输问题,其模型框架可以应用到很多领域。目前对动态MC²运输模型的求解主要采用传统的单纯形法,针对该问题的特殊性采用具有全局搜索能力的遗传算法进行求解。通过三维数组编码,设计有效的交叉、变异算子和适应度函数,克服了单纯形法求解该问题出现的并行性差、求解整数规划困难的不足。用Matlab7.0编程对算法进行检验,结果表明经过特殊设计的遗传算法能够很好地解决动态MC²运输问题。     

D-optimality of non-regular design spaces by using a Bayesian modification and a hybrid method

In this work a hybrid method of a genetic algorithm and sequential linear programming is suggested to obtain a D-optimal design of experiments. Regular as well as non-regular design spaces are considered. A D-optimal design of experiments maximizes the determinant of the information matrix, which appears in the normal equation. It is known that D-optimal design of experiments sometimes include duplicate design points. This is, of course, not preferable since duplicates do not add any new information to the response surface approximation and the computational effort is therefore wasted. In this work a Bayesian modification, where higher order terms are added to the response surface approximation, is used in case of duplicates in the design of experiments. In such manner, the draw-back with duplicates might be eliminated. The D-optimal problem, which is obtained by using the Bayesian modification, is then solved by a hybrid method. A hybrid method of a genetic algorithm that generates a starting point for sequential linear programming is developed. The genetic algorithm performs genetic operators such as cross-over and mutation on a binary version of the design of experiments, while the real valued version is used to evaluate the fitness. Next, by taking the gradient of the objective, a LP-problem is formulated which is solved by an interior point method that is available in Matlab. This is repeated in a sequence until convergence is reached. The hybrid method is tested for five numerical examples. Results from the numerical examples show a very robust convergence to a global optimum. Furthermore, the results show that the problem with duplicates is eliminated by using the Bayesian modification.     

非线性最小二乘全局解的混合计算智能算法

通过在遗传算法中嵌入牛顿算子，并定义适当的适应度和数据结构，从而得到可结合遗传算法和牛顿法两者长处，既有较快收敛性，又能以较大概率求得非线性最小二乘全局解的混合计算智能算法．数值结果表明了该方法显著优于遗传算法和牛顿法．     

Multiresolution genetic clustering algorithm for texture segmentation

This work plans to approach the texture segmentation problem by incorporating genetic algorithm and K-means clustering method within a multiresolution structure. As the algorithm descends the multiresolution structure, the coarse segmentation results are propagated down to the lower levels so as to reduce the inherent class–position uncertainty and to improve the segmentation accuracy. The procedure is described as follows. In the first step, a quad-tree structure of multiple resolutions is constructed. Sampling windows of different sizes are utilized to partition the underlying image into blocks at different resolution levels and texture features are extracted from each block. Based on the texture features, a hybrid genetic algorithm is employed to perform the segmentation. While the select and mutate operators of the traditional genetic algorithm are adopted in this work, the crossover operator is replaced with K-means clustering method. In the final step, the boundaries and the segmentation result of the current resolution level are propagated down to the next level to act as contextual constraints and the initial configuration of the next level, respectively.     

GA-based H2/H∞ static output feedback design with average performance concept and techniques of family of polynomials

For continuous systems with polytopic uncertainties, the mixed H₂/H_∞ robust output control problem is considered in this paper. The design objective is to seek a static output feedback controller that is able to stabilize the uncertain system and to minimize the mixed H₂/H_∞ norm. To this end, a hybrid algorithm mixed by the technique of Hurwitz testing matrix of family of polynomial, average performance constraint and genetic algorithm (GA) is proposed. Due to the fact that the algorithm is based on GAs, we use a hierarchical structure to merge the conditions of Hurwitz testing matrix technique and average performance constraint into a fitness function which is called Hierarchical Fitness Function Structure (HFFS). By replacing fitness function with HFFS, any GAs can be applied to constitute the algorithm. Also, the proposed approach does not need any assumptions, so it is much more relaxed than that of Lyapunov-based design methods.     

Finite element analysis of the Girkmann problem using the modern hp-version and the classical h-version

We perform finite element analysis of the so called Girkmann problem in structural mechanics. The problem involves an axially symmetric spherical shell stiffened with a foot ring and is approached (1) by using the axisymmetric formulation of linear elasticity theory and (2) by using a dimensionally reduced shell-ring model. In the first approach the problem is solved with a fully automatic hp-adaptive finite element solver whereas the classical h-version of the finite element method is used in the second approach. We study the convergence behaviour of the different numerical models and show that accurate stress resultants can be obtained with both models by using effective post-processing formulas.     

Applying particle swarm optimization algorithm to roundness measurement

Measuring the roundness of a circular workpiece is a common problem of quality control and inspection. In this area, maximum inscribed circle (MIC) and maximum circumscribing circle (MCC), minimum zone circle (MZC) and least square circle (LSC) are four commonly used methods. In particular, MIC, MCC, and MZC, which are nonlinear constrained optimization problems, have not been thoroughly discussed lately. This study proposes a machine vision-based roundness measuring method that applies the particle swarm optimization algorithm (PSO) to compute MIC, MCC and MZC. To facilitate the PSO process, five different PSO’s were encoded using a radius (R) and circle center (x, y) and extensively evaluated using an experimental design, in which the impact of inertia weight, maximum velocity and the number of particles on the performance of the particle swarm optimizer was analyzed. The proposed method was verified with a set of testing images and benchmarked with the GA-based (genetic algorithm) method [Chen, M. C. (2000). Roundness inspection strategies for machine visions using non-linear programs and genetic algorithms. International Journal of Production Research, 38, 2967–2988]. The experimental results reveal that the PSO-based method effectively solved the MIC, MCC, and MZC problems and outperforms GA-based method in both accuracy and the efficiency. As a finals, several industrial applications are presented to explore the effectiveness and efficiency of the proposed method.     

Discontinuous Petrov–Galerkin method with optimal test functions for thin-body problems in solid mechanics

We study the applicability of the discontinuous Petrov–Galerkin (DPG) variational framework for thin-body problems in structural mechanics. Our numerical approach is based on discontinuous piecewise polynomial finite element spaces for the trial functions and approximate, local computation of the corresponding ‘optimal’ test functions. In the Timoshenko beam problem, the proposed method is shown to provide the best approximation in an energy-type norm which is equivalent to the L₂-norm for all the unknowns, uniformly with respect to the thickness parameter. The same formulation remains valid also for the asymptotic Euler–Bernoulli solution. As another one-dimensional model problem we consider the modelling of the so called basic edge effect in shell deformations. In particular, we derive a special norm for the test space which leads to a robust method in terms of the shell thickness. Finally, we demonstrate how a posteriori error estimator arising directly from the discontinuous variational framework can be utilized to generate an optimal hp-mesh for resolving the boundary layer.     

A memetic algorithm for cardinality-constrained portfolio optimization with transaction costs

A memetic approach that combines a genetic algorithm (GA) and quadratic programming is used to address the problem of optimal portfolio selection with cardinality constraints and piecewise linear transaction costs. The framework used is an extension of the standard Markowitz mean–variance model that incorporates realistic constraints, such as upper and lower bounds for investment in individual assets and/or groups of assets, and minimum trading restrictions. The inclusion of constraints that limit the number of assets in the final portfolio and piecewise linear transaction costs transforms the selection of optimal portfolios into a mixed-integer quadratic problem, which cannot be solved by standard optimization techniques. We propose to use a genetic algorithm in which the candidate portfolios are encoded using a set representation to handle the combinatorial aspect of the optimization problem. Besides specifying which assets are included in the portfolio, this representation includes attributes that encode the trading operation (sell/hold/buy) performed when the portfolio is rebalanced. The results of this hybrid method are benchmarked against a range of investment strategies (passive management, the equally weighted portfolio, the minimum variance portfolio, optimal portfolios without cardinality constraints, ignoring transaction costs or obtained with L₁ regularization) using publicly available data. The transaction costs and the cardinality constraints provide regularization mechanisms that generally improve the out-of-sample performance of the selected portfolios.     

Divergence-Free <Emphasis Type="Italic">H</Emphasis>(div)-FEM for Time-Dependent Incompressible Flows with Applications to High Reynolds Number Vortex Dynamics

In this article, we consider exactly divergence-free H(div)-conforming finite element methods for time-dependent incompressible viscous flow problems. This is an extension of previous research concerning divergence-free \(H^1\)-conforming methods. For the linearised Oseen case, the first semi-discrete numerical analysis for time-dependent flows is presented whereby special emphasis is put on pressure- and Reynolds-semi-robustness. For convection-dominated problems, the proposed method relies on a velocity jump upwind stabilisation which is not gradient-based. Complementing the theoretical results, H(div)-FEM are applied to the simulation of full nonlinear Navier–Stokes problems. Focussing on dynamic high Reynolds number examples with vortical structures, the proposed method proves to be capable of reliably handling the planar lattice flow problem, Kelvin–Helmholtz instabilities and freely decaying two-dimensional turbulence.     

基于混沌搜索策略进行空洞探测的进化算法

结合混沌搜索策略和进化算法提出了一种混合优化方法，解决了平板内的空洞探测问题。该算法相较于此前提出的各种方法具有更好的拟合精度和更灵活的计算尺度，且能够有效地避免常规进化算法中容易出现的早熟现象。另外，在解空间的搜索过程中不受初值的影响，从任意点出发均能收敛到全局最优解。仿真结果证明了该算法的有效性。     

To CG or to HDG: A Comparative Study

Hybridization through the border of the elements (hybrid unknowns) combined with a Schur complement procedure (often called static condensation in the context of continuous Galerkin linear elasticity computations) has in various forms been advocated in the mathematical and engineering literature as a means of accomplishing domain decomposition, of obtaining increased accuracy and convergence results, and of algorithm optimization. Recent work on the hybridization of mixed methods, and in particular of the discontinuous Galerkin (DG) method, holds the promise of capitalizing on the three aforementioned properties; in particular, of generating a numerical scheme that is discontinuous in both the primary and flux variables, is locally conservative, and is computationally competitive with traditional continuous Galerkin (CG) approaches. In this paper we present both implementation and optimization strategies for the Hybridizable Discontinuous Galerkin (HDG) method applied to two dimensional elliptic operators. We implement our HDG approach within a spectral/hp element framework so that comparisons can be done between HDG and the traditional CG approach.     

Stochastic sampling design using a multi-objective genetic algorithm and adaptive neural networks

This paper presents a novel multi-objective genetic algorithm (MOGA) based on the NSGA-II algorithm, which uses metamodels to determine optimal sampling locations for installing pressure loggers in a water distribution system (WDS) when parameter uncertainty is considered. The new algorithm combines the multi-objective genetic algorithm with adaptive neural networks (MOGA–ANN) to locate pressure loggers. The purpose of pressure logger installation is to collect data for hydraulic model calibration. Sampling design is formulated as a two-objective optimization problem in this study. The objectives are to maximize the calibrated model accuracy and to minimize the number of sampling devices as a surrogate of sampling design cost. Calibrated model accuracy is defined as the average of normalized traces of model prediction covariance matrices, each of which is constructed from a randomly generated sampling set of calibration parameter values. This method of calculating model accuracy is called the ‘full’ fitness model. Within the genetic algorithm search process, the full fitness model is progressively replaced with the periodically (re)trained adaptive neural network metamodel where (re)training is done using the data collected by calling the full model. The methodology was first tested on a hypothetical (benchmark) problem to configure the setting requirement. Then the model was applied to a real case study. The results show that significant computational savings can be achieved by using the MOGA–ANN when compared to the approach where MOGA is linked to the full fitness model. When applied to the real case study, optimal solutions identified by MOGA–ANN are obtained 25 times faster than those identified by the full model without significant decrease in the accuracy of the final solution.     

Numerical solution of nonlinear delay differential equations of fractional variable-order using a novel shifted Jacobi operational matrix

This paper presents the generalized nonlinear delay differential equations of fractional variable-order. In this article, a novel shifted Jacobi operational matrix technique is introduced for solving a class of multi-terms variable-order fractional delay differential equations via reducing the main problem to an algebraic system of equations that can be solved numerically. The suggested technique is successfully developed for the aforementioned problem. Comprehensive numerical experiments are presented to demonstrate the efficiency, generality, accuracy of proposed scheme and the flexibility of this method. The numerical results compared it with other existing methods such as fractional Adams method (FAM), new predictor–corrector method (NPCM), a new approach, Adams–Bashforth–Moulton algorithm and L1 predictor–corrector method (L1-PCM). Comparing the results of these methods as well as comparing the current method (NSJOM) with the exact solution, indicating the efficiency and validity of this method. Note that the procedure is easy to implement and this technique will be considered as a generalization of many numerical schemes. Furthermore, the error and its bound are estimated.