Stochastic inverse heat conduction using a spectral approach

An adjoint‐based functional optimization technique in conjunction with the spectral stochastic finite element method is proposed for the solution of an inverse heat conduction problem in the presence of uncertainties in material data, process conditions and measurement noise. The ill‐posed stochastic inverse problem is restated as a conditionally well‐posed L₂ optimization problem. The gradient of the objective function is obtained in a distributional sense by defining an appropriate stochastic adjoint field. The L₂ optimization problem is solved using a conjugate‐gradient approach. Accuracy and effectiveness of the proposed approach is appraised with the solution of several stochastic inverse heat conduction problems. Copyright © 2004 John Wiley & Sons, Ltd.     

A new numerical method for the boundary optimal control problems of the heat conduction equation

A new numerical method is developed for the boundary optimal control problems of the heat conduction equation in the present paper. When the boundary optimal control problem is solved by minimizing the objective function employing a conjugate‐gradient method, the most crucial step is the determination of the gradient of objective function usually employing either the direct differentiation method or the adjoint variable method. The direct differentiation method is simple to implement and always yields accurate results, but consumes a large amount of computational time. Although the adjoint variable method is computationally very efficient, the adjoint variable does not have sufficient regularity at the boundary for the boundary optimal control problems. As a result, a large numerical error is incurred in the evaluation of the gradient function, resulting in premature termination of the conjugate gradient iteration. In the present investigation, a new method is developed that circumvents this difficulty with the adjoint variable method by introducing a partial differential equation that describes the temporal and spatial dynamics of the control variable at the boundary. The present method is applied to the Neumann and Dirichlet boundary optimal control problems, respectively, and is found to solve the problems efficiently with sufficient accuracy. Copyright © 2001 John Wiley & Sons, Ltd.     

一个充分下降的修正 PRP 型谱共轭梯度法

谱共轭梯度法是共轭梯度法的一种重要延拓,可以通过共轭参数和谱参数二维度调整,使得所设计算法的搜索方向满足某一预设条件,比如充分下降条件或共轭条件等。谱参数和共轭参数的设计是谱共轭梯度法的两大核心工作,决定方法的收敛性和数值效果。基于 PRP 方法,构造了一个修正的 PRP 型共轭参数,该共轭参数不仅保持了 PRP 公式的结构和性能,而且具有 FR 方法的收敛性质。利用充分下降条件取定一个谱参数,与修正的 PRP 型共轭参数结合,建立一个新的谱共轭梯度算法。该算法不依赖于任何线搜索就可以满足充分下降条件。常规假设条件下,采用强 Wolfe 线搜索准则产生步长,证明了新算法的全局敛性。通过 100 个算例对该算法进行数值测试并与其他五个算法进行比较,同时采用性能图对数值结果进行直观展示,结果表明该算法是有效的。     

Constrained optimization using a multipoint type chaotic Lagrangian method with a coupling structure

This article proposes a new constrained optimization method using a multipoint type chaotic Lagrangian method that utilizes chaotic search trajectories generated by Lagrangian gradient dynamics with a coupling structure. In the proposed method, multiple search points autonomously implement global search using the chaotic search trajectory generated by the coupled Lagrangian gradient dynamics. These points are advected to elite points (which are chosen by considering their objective function values and their feasibility) by the coupling in order to explore promising regions intensively. In this way, the proposed method successfully provides diversification and intensification for constrained optimization problems. The effectiveness of the proposed method is confirmed through application to various types of benchmark problem, including the coil spring design problem, the benchmark problems used in the special session on constrained real parameter optimization in CEC2006, and a high-dimensional and multi-peaked constrained optimization problem.     

Conjugate gradient method for the Robin inverse problem associated with the Laplace equation

This paper studies a non-linear inverse problem associated with the Laplace equation of identifying the Robin coefficient from boundary measurements. A variational formulation of the problem is suggested, thereby transforming it into an optimization problem. Mathematical properties relevant to its numerical computation are established. The optimization problem is solved using the conjugate gradient method in conjunction with the discrepancy principle, and the algorithm is implemented using the boundary element method. Numerical results are presented for several benchmark problems with both exact and noisy data, and the convergence of the algorithm with respect to mesh refinement and decreasing the amount of noise in the data is investigated. Copyright © 2006 John Wiley & Sons, Ltd.     

The identification of a Robin coefficient by a conjugate gradient method

This paper investigates a non‐linear inverse problem associated with the heat conduction problem of identifying a Robin coefficient from boundary temperature measurement. The variational formulation of the problem is given. The conjugate gradient method combining with the discrepancy principle for choosing the suitable stop step are proposed for solving the optimization problem, in which the finite difference method is used to solve the direct problems. The performance of the method is verified by simulating four examples. The convergence with respect to the grid refinement and the amount of noise in the data is also investigated. Copyright © 2008 John Wiley & Sons, Ltd.     

Using Krylov‐Padé model order reduction for accelerating design optimization of structures and vibrations in the frequency domain

In many engineering problems, the behavior of dynamical systems depends on physical parameters. In design optimization, these parameters are determined so that an objective function is minimized. For applications in vibrations and structures, the objective function depends on the frequency response function over a given frequency range, and we optimize it in the parameter space. Because of the large size of the system, numerical optimization is expensive. In this paper, we propose the combination of Quasi‐Newton type line search optimization methods and Krylov‐Padé type algebraic model order reduction techniques to speed up numerical optimization of dynamical systems. We prove that Krylov‐Padé type model order reduction allows for fast evaluation of the objective function and its gradient, thanks to the moment matching property for both the objective function and the derivatives towards the parameters. We show that reduced models for the frequency alone lead to significant speed ups. In addition, we show that reduced models valid for both the frequency range and a line in the parameter space can further reduce the optimization time. Copyright © 2012 John Wiley & Sons, Ltd.     

Monte Carlo gradient estimation in high dimensions

A Monte Carlo procedure to estimate efficiently the gradient of a generic function in high dimensions is presented. It is shown that, adopting an orthogonal linear transformation, it is possible to identify a new coordinate system where a relatively small subset of the variables causes most of the variation of the gradient. This property is exploited further in gradient‐based algorithms to reduce the computational effort for the gradient evaluation in higher dimensions. Working in this transformed space, only few function evaluations, i.e. considerably less than the dimensionality of the problem, are required. The procedure is simple and can be applied by any gradient‐based method. A number of different examples are presented to show the accuracy and the efficiency of the proposed approach and the applicability of this procedure for the optimization problem using well‐known gradient‐based optimization algorithms such as the descent gradient, quasi‐Newton methods and Sequential Quadratic Programming. Copyright © 2009 John Wiley & Sons, Ltd.     

Multiresolution subspace-based optimization method for inverse scattering problems

This paper investigates an approach to inverse scattering problems based on the integration of the subspace-based optimization method (SOM) within a multifocusing scheme in the framework of the contrast source formulation. The scattering equations are solved by a nested three-step procedure composed of (a) an outer multiresolution loop dealing with the identification of the regions of interest within the investigation domain through an iterative information-acquisition process, (b) a spectrum analysis step devoted to the reconstruction of the deterministic components of the contrast sources, and (c) an inner optimization loop aimed at retrieving the ambiguous components of the contrast sources through a conjugate gradient minimization of a suitable objective function. A set of representative reconstruction results is discussed to provide numerical evidence of the effectiveness of the proposed algorithmic approach as well as to assess the features and potentialities of the multifocusing integration in comparison with the state-of-the-art SOM implementation.     

共轭梯度法的全局收敛性

探讨了在强Wolfe搜索规则下，与βk^PR相关的算法的收敛性，在不需要假设目标函数为凸的情况下，证明了充分下降及算法的全局收敛性。     

求解广义Lyapunov方程的非单调谱投影梯度法

本文研究双线性控制系统中的一类广义Lyapunov方程的半正定解．基于凸函数的局部极小解就是全局极小解这一良好性质,首先将广义Lyapunov方程的半正定解问题等价转化为凸优化问题．利用非单调线搜索技术确定步长,构造了非单调谱投影梯度方法求解这一等价问题．最后用数值例子验证了新方法的可行性和有效性．     

A human‐like numerical technique for design of engineering systems

Because of the necessity for considering various creative and engineering design criteria, optimal design of an engineering system results in a highly‐constrained multi‐objective optimization problem. Major numerical approaches to such optimal design are to force the problem into a single objective function by introducing unjustifiable additional parameters and solve it using a single‐objective optimization method. Due to its difference from human design in process, the resulting design often becomes completely different from that by a human designer. This paper presents a novel numerical design approach, which resembles the human design process. Similar to the human design process, the approach consists of two steps: (1) search for the solution space of the highly‐constrained multi‐objective optimization problem and (2) derivation of a final design solution from the solution space. Multi‐objective gradient‐based method with Lagrangian multipliers (MOGM‐LM) and centre‐of‐gravity method (CoGM) are further proposed as numerical methods for each step. The proposed approach was first applied to problems with test functions where the exact solutions are known, and results demonstrate that the proposed approach can find robust solutions, which cannot be found by conventional numerical design approaches. The approach was then applied to two practical design problems. Successful design in both the examples concludes that the proposed approach can be used for various design problems that involve both the creative and engineering design criteria. Copyright © 2005 John Wiley & Sons, Ltd.     

Determination of particle size distribution by light extinction method using improved pattern search algorithm with Tikhonov smoothing functional

An inversion technique which combines the pattern search algorithm with the Tikhonov smoothing functional for retrieval of particle size distribution (PSD) by light extinction method is proposed. In the unparameterized shape-independent model, we first transform the PSD inversion problem into an optimization problem, with the Tikhonov smoothing functional employed to model the objective function. The optimization problem is then solved by the pattern search algorithm. To ensure good convergence rate and accuracy of the whole retrieval, a competitive strategy for determining the initial point of the pattern search algorithm is also designed. The accuracy and limitations of the proposed technique are tested by the inversion results of synthetic and real standard polystyrene particles immersed in water. In addition, the issues about the objective function and computation time are further discussed. Both simulation and experimental results show that the technique can be successfully applied to retrieve the PSD with high reliability and stability in the presence of random noise. Compared with the Phillips–Twomey method and genetic algorithm, the proposed technique has certain advantages in terms of reaching a more accurate and steady optimal solution with less computational effort, thus making this technique more suitable for quick and accurate measurement of PSD.     

结合广义Armijo步长搜索的一类记忆梯度算法及其收敛特征

本文对求解无约束规划的超记忆梯度算法中线搜索方向中的参数,给了一个假设条件,从而确定了它的一个新的取值范围,保证了搜索方向是目标函数的充分下降方向,由此提出了一类新的记忆梯度算法.并在去掉迭代点列有界和广义Armijo步长搜索下,讨论了算法的全局收敛性,且给出了结合形如共轭梯度法FR,PR,HS的记忆梯度法的修正形式,数值实验表明,新算法比Armijo线搜索下的FR,PR,HS共轭梯度法和超记忆梯度法更稳定、更有效.     

Structural shape and topology optimization of cast parts using level set method

This paper presents a level set‐based shape and topology optimization method for conceptual design of cast parts. In order to be successfully manufactured by the casting process, the geometry of cast parts should satisfy certain moldability conditions, which poses additional constraints in the shape and topology optimization of cast parts. Instead of using the originally point‐wise constraint statement, we propose a casting constraint in the form of domain integration over a narrowband near the material boundaries. This constraint is expressed in terms of the gradient of the level set function defining the structural shape and topology. Its explicit and analytical form facilitates the sensitivity analysis and numerical implementation. As compared with the standard implementation of the level set method based on the steepest descent algorithm, the proposed method uses velocity field design variables and combines the level set method with the gradient‐based mathematical programming algorithm on the basis of the derived sensitivity scheme of the objective function and the constraints. This approach is able to simultaneously account for the casting constraint and the conventional material volume constraint in a convenient way. In this method, the optimization process can be started from an arbitrary initial design, without the need for an initial design satisfying the cast constraint. Numerical examples in both 2D and 3D design domain are given to demonstrate the validity and effectiveness of the proposed method. Copyright © 2017 John Wiley & Sons, Ltd.     

Study on semi‐conjugate direction methods for non‐symmetric systems

Some theoretical problems and implementation problems are studied here for the semi‐conjugate direction method established by Yuan, Golub, Plemmons and Cecilio (2002). The existence of semi‐conjugate directions is proved for almost all matrices except skew‐symmetric matrices. A new technique is proposed to overcome the breakdown problem appeared in the semi‐conjugate direction method. In the implementation of the semi‐conjugate direction method, the generation of the semi‐conjugate direction is very important and necessary, but very expensive. The technique of limited‐memory is introduced to economize the cost of the generation of the semi‐conjugate direction in the Yuan–Golub– Plemmons–Cecilio algorithm. Finally, some numerical experiments are given to confirm our theoretical results. Our results illustrate that the semi‐conjugate direction method is very nice alternative for solving non‐symmetric systems, and the limited‐memory left conjugate direction method is a good improvement of the left conjugate direction method. Copyright © 2004 John Wiley & Sons, Ltd.     

MISO: mixed-integer surrogate optimization framework

We introduce MISO, the mixed-integer surrogate optimization framework. MISO aims at solving computationally expensive black-box optimization problems with mixed-integer variables. This type of optimization problem is encountered in many applications for which time consuming simulation codes must be run in order to obtain an objective function value. Examples include optimal reliability design and structural optimization. A single objective function evaluation may take from several minutes to hours or even days. Thus, only very few objective function evaluations are allowable during the optimization. The development of algorithms for this type of optimization problems has, however, rarely been addressed in the literature. Because the objective function is black-box, derivatives are not available and numerically approximating the derivatives requires a prohibitively large number of function evaluations. Therefore, we use computationally cheap surrogate models to approximate the expensive objective function and to decide at which points in the variable domain the expensive objective function should be evaluated. We develop a general surrogate model framework and show how sampling strategies of well-known surrogate model algorithms for continuous optimization can be modified for mixed-integer variables. We introduce two new algorithms that combine different sampling strategies and local search to obtain high-accuracy solutions. We compare MISO in numerical experiments to a genetic algorithm, NOMAD version 3.6.2, and SO-MI. The results show that MISO is in general more efficient than NOMAD and the genetic algorithm with respect to finding improved solutions within a limited budget of allowable evaluations. The performance of MISO depends on the chosen sampling strategy. The MISO algorithm that combines a coordinate perturbation search with a target value strategy and a local search performs best among all algorithms.     

Adaptive range particle swarm optimization

This paper proposes a new technique for particle swarm optimization called adaptive range particle swarm optimization (ARPSO). In this technique an active search domain range is determined by utilizing the mean and standard deviation of each design variable. In the initial search stage, the search domain is explored widely. Then the search domain is shrunk so that it is restricted to a small domain while the search continues. To achieve these search processes, new parameters to determine the active search domain range are introduced. These parameters gradually increase as the search continues. Through these processes, it is possible to shrink the active search domain range. Moreover, by using the proposed method, an optimum solution is attained with high accuracy and a small number of function evaluations. Through numerical examples, the effectiveness and validity of ARPSO are examined.     

Level‐set topology optimization with many linear buckling constraints using an efficient and robust eigensolver

Linear buckling constraints are important in structural topology optimization for obtaining designs that can support the required loads without failure. During the optimization process, the critical buckling eigenmode can change; this poses a challenge to gradient‐based optimization and can require the computation of a large number of linear buckling eigenmodes. This is potentially both computationally difficult to achieve and prohibitively expensive. In this paper, we motivate the need for a large number of linear buckling modes and show how several features of the block Jacobi conjugate gradient (BJCG) eigenvalue method, including optimal shift estimates, the reuse of eigenvectors, adaptive eigenvector tolerances and multiple shifts, can be used to efficiently and robustly compute a large number of buckling eigenmodes. This paper also introduces linear buckling constraints for level‐set topology optimization. In our approach, the velocity function is defined as a weighted sum of the shape sensitivities for the objective and constraint functions. The weights are found by solving an optimization sub‐problem to reduce the mass while maintaining feasibility of the buckling constraints. The effectiveness of this approach in combination with the BJCG method is demonstrated using a 3D optimization problem. Copyright © 2016 John Wiley & Sons, Ltd.     

Calibration by optimization without using derivatives

Applications in engineering frequently require the adjustment of certain parameters. While the mathematical laws that determine these parameters often are well understood, due to time limitations in every day industrial life, it is typically not feasible to derive an explicit computational procedure for adjusting the parameters based on some given measurement data. This paper aims at showing that in such situations, direct optimization offers a very simple approach that can be of great help. More precisely, we present a numerical implementation for the local minimization of a smooth function \(f:{\mathbb R}^n\rightarrow {\mathbb R}\) subject to upper and lower bounds without relying on the knowledge of the derivative of f. In contrast to other direct optimization approaches the algorithm assumes that the function evaluations are fairly cheap and that the rounding errors associated with the function evaluations are small. As an illustration, this algorithm is applied to approximate the solution of a calibration problem arising from an engineering application. The algorithm uses a Quasi-Newton trust region approach adjusting the trust region radius with a line search. The line search is based on a spline function which minimizes a weighted least squares sum of the jumps in its third derivative. The approximate gradients used in the Quasi-Newton approach are computed by central finite differences. A new randomized basis approach is considered to generate finite difference approximations of the gradient which also allow for a curvature correction of the Hessian in addition to the Quasi-Newton update. These concepts are combined with an active set strategy. The implementation is public domain; numerical experiments indicate that the algorithm is well suitable for the calibration problem of measuring instruments that prompted this research. Further preliminary numerical results suggest that an approximate local minimizer of a smooth non-convex function f depending on \(n\le 300 \) variables can be computed with a number of iterations that grows moderately with n.