A new stabilized finite element method for optimal control for a Ladyzhenskaya model for unsteady flows

This article considers the time‐dependent optimal control problem of tracking the velocity for the viscous incompressible flows which is governed by a Ladyzhenskaya equations with distributed control. The existence of the optimal solution is shown and the first‐order optimality condition is established. The semidiscrete‐in‐time approximation of the optimal control problem is also given. The spatial discretization of the optimal control problem is accomplished by using a new stabilized finite element method which does not need a stabilization parameter or calculation of high order derivatives. Finally a gradient algorithm for the fully discrete optimal control problem is effectively proposed and implemented with some numerical examples. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 263–287, 2012     

Computational method for optimal control of switched systems with input and state constraints

In this paper, we consider an optimal control problem of switched systems with input and state constraints. Since the complexity of such constraint and switching laws, it is difficult to solve the problem using standard optimization techniques. In addition, although conjugate gradient algorithms are very useful for solving nonlinear optimization problem, in practical implementations, the existing Wolfe condition may never be satisfied due to the existence of numerical errors. And the mode insertion technique only leads to suboptimal solutions, due to only certain mode insertions being considered. Thus, based on an improved conjugate gradient algorithm and a discrete filled function method, an improved bi-level algorithm is proposed to solve this optimization problem. Convergence results indicate that the proposed algorithm is globally convergent. Three numerical examples are solved to illustrate the proposed algorithm converges faster and yields a better cost function value than existing bi-level algorithms.     

Three-term conjugate gradient method for the convex optimization problem over the fixed point set of a nonexpansive mapping

Many constrained sets in problems such as signal processing and optimal control can be represented as a fixed point set of a certain nonexpansive mapping, and a number of iterative algorithms have been presented for solving a convex optimization problem over a fixed point set. This paper presents a novel gradient method with a three-term conjugate gradient direction that is used to accelerate conjugate gradient methods for solving unconstrained optimization problems. It is guaranteed that the algorithm strongly converges to the solution to the problem under the standard assumptions. Numerical comparisons with the existing gradient methods demonstrate the effectiveness and fast convergence of this algorithm.     

Conjugate gradient-boundary element solution for distributed elliptic optimal control problems

An optimality system of equations for the optimal control problem governed by Helmholtz-type equations is derived. By the associated first-order necessary optimality condition, we obtain the conjugate gradient method (CGM) in the continuous case. Introducing the sequence of higher-order fundamental solutions, we propose an iterative algorithm based on the conjugate gradient-boundary element method using the multiple reciprocity method (CGM+MRBEM) for solving the discrete control input. This algorithm has an advantage over that of the existing literatures because the main attribute (the reduced dimensionality) of the boundary element method is fully utilized. Finally, the local error estimates for this scheme are obtained, and a test problem is given to illustrate the efficiency of the proposed method.     

A COMPUTATIONAL APPROACH FOR OPTIMAL CONTROL SYSTEMS GOVERNED BY PARABOLIC VARIATIONAL INEQUALITIES

Iterative techniques for solving optimal control systems governed by parabolic variational inequalities are presented. The techniques we use are based on linear finite elements method to approximate the state equations and nonlinear conjugate gradient methods to solve the discrete optimal control problem. Convergence results and numerical experiments are presented.     

Spectral Optimization Methods for the Time Fractional Diffusion Inverse Problem

An inverse problem of reconstructing the initial condition for a time fractional diffusion equation is investigated. On the basis of the optimal control framework, the uniqueness and first order necessary optimality condition of the minimizer for the objective functional are established, and a time-space spectral method is proposed to numerically solve the resulting minimization problem. The contribution of the paper is threefold: 1) a priori error estimate for the spectral approximation is derived; 2) a conjugate gradient optimization algorithm is designed to efficiently solve the inverse problem; 3) some numerical experiments are carried out to show that the proposed method is capable to find out the optimal initial condition, and that the convergence rate of the method is exponential if the optimal initial condition is smooth.     

A control operator and some of its applications

The conventional function space algorithms for solving minimization of penalized cost functional for optimal control problem characterized by linear-system integral quadratic cost due to Di Pillo and others, though falling within the framework pertinent to the conjugate gradient method algorithm, is difficult to apply computationally. The difficulty arises principally because there exists in the algorithm a number of stringent requirements imposed on the minimization procedures to facilitate its convergence. Incidentally, such computations are very cumbersome to carry out numerically. To circumvent this major numerical draw back, we construct here a control operator associated with this class of problems and use our explicit knowledge of the operator to devise an extended conjugate gradient method algorithm for solving this family of problems. Furthermore, the establishment of some functional inequalities which are obtained using the knowledge of the control operator is discussed.     

特殊双变量矩阵方程组异类约束解的MCG算法

本文研究了约束矩阵方程问题中异类约束解的迭代算法.利用修正共轭梯度法,求得了特殊双变量线性矩阵方程组的异类约束解,选取特殊的初始矩阵,得到唯一极小范数异类约束解.理论证明和数值算例验证了该方法的有限步收敛性,推广了修正共轭梯度法在求约束矩阵方程问题中的应用范围.     

一般线性或非线性约束下的共轭投影变尺度方法

梯度投影法是一类有效的约束最优化算法，在最优化领域中占有重要的地位．但是，梯度投影法所采用的投影是正交投影，不包含目标函数和约束函数的二阶导数信息·因而；收敛速度不太令人满意．本文介绍一种共轭投影概念，利用共轭投影构造了一般线性或非线性约束下的共轭投影变尺度算法，并证明了算法在一定条件下具有全局收敛性．由于算法中的共轭投影恰当地包含了目标函数和约束函数的二阶导数信息，因而收敛速度有希望加快．数值试验的结果表明算法是有效的．     

Adjoint concepts for the optimal control of Burgers equation

Adjoint techniques are a common tool in the numerical treatment of optimal control problems. They are used for efficient evaluations of the gradient of the objective in gradient-based optimization algorithms. Different adjoint techniques for the optimal control of Burgers equation with Neumann boundary control are studied. The methods differ in the point in the numerical algorithm at which the adjoints are incorporated. Discretization methods for the continuous adjoint are discussed and compared with methods applying the application of the discrete adjoint. At the example of the implicit Euler method and the Crank Nicolson method it is shown that a discretization for the adjoint problem that is adjoint to the discretized optimal control problem avoids additional errors in gradient-based optimization algorithms. The approach of discrete adjoints coincides with that of automatic differentiation tools (AD) which provide exact gradient calculations on the discrete level.     

Gradient Descent Approach to Optimal Mode Scheduling in Hybrid Dynamical Systems

This paper concerns the problem of optimally scheduling the sequence of dynamic response functions in nonlinear switched-mode hybrid dynamical systems. The control parameter has a discrete component and a continuous component, namely the sequence of modes and the duration of each mode, while the performance criterion consists of a cost functional on the state trajectory. The problem is naturally cast in the framework of optimal control. This framework has established techniques sufficient to address the continuous part of the parameter, but lacks adequate tools to consider the discrete element. To get around this difficulty, the paper proposes a bilevel hierarchical algorithm. At the lower level, the algorithm considers a fixed mode sequence and minimizes the cost functional with respect to the mode durations; at the upper level, it updates the mode sequence by using a gradient technique that is tailored to the special structure of the discrete variable (mode sequencing). The resulting algorithm is not defined on a single parameter space, but rather on a sequence of Euclidean spaces of increasing dimensions, an unusual setting for which there is no established notion of convergence. The paper suggests first a suitable definition of convergence based on the concepts of optimality functions; then, it proves that the proposed algorithm converges in that sense. Research supported in part by the National Science Foundation under Grant #0509064.     

改进HS共轭梯度算法及其全局收敛性

１．引 言 １９５２年 Ｍ．Ｈｅｓｔｅｎｅｓ和Ｅ．Ｓｔｉｅｆｅｌ提出了求解正定线性方程组的共轭梯度法［１］．１９６４年Ｒ．Ｆｌｅｔｃｈｅｒ和Ｃ．Ｒｅｅｖｅｓ将该方法推广到求解下列无约束优化问题： ｍｉｎｆ（ｘ）,ｘ∈Ｒｎ,（１）其中ｆ：Ｒｎ→Ｒ１为连续可微函数,记ｇｋ＝ ｆ（ｘｋ）,ｘｋ∈ Ｒｎ． 若点列｛ｘｋ｝由如下算法产生：其中 βｋ＝［ｇＴｋ（ｇｋ－ｇｋ－１）］／［ｄＴｋ－１（ｇｋ－ｇｋ－１）］．（Ｈｅｓｔｅｎｅｓ－Ｓｔｉｅｆｅｌ）　　（４）则称该算法为 Ｈｅｓｔｅｎｅｓ—Ｓｔｉｅｆｅｌ共轭梯度算…     

A variational conjugate gradient method for determining the fluid velocity of a slow viscous flow

The problem considered is that of determining the fluid velocity for linear hydrostatics Stokes flow of slow viscous fluids from measured velocity and fluid stress force on a part of the boundary of a bounded domain. A variational conjugate gradient iterative procedure is proposed based on solving a series of mixed well-posed boundary value problems for the Stokes operator and its adjoint. In order to stabilize the Cauchy problem, the iterations are ceased according to an optimal order discrepancy principle stopping criterion. Numerical results obtained using the boundary element method confirm that the procedure produces a convergent and stable numerical solution.     

有界约束非线性优化问题的仿射共轭梯度路径法

本文提出仿射内点离散共轭梯度路径法解有界约束的非线性优化问题,通过构造预条件离散的共轭梯度路径解二次模型获得预选迭代方向,结合内点回代线搜索获得下一步的迭代,在合理的假设条件下,证明了算法的整体收敛性与局部超线性收敛速率,最后,数值结果表明了算法的有效性.     

Truncated-newtono algorithms for large-scale unconstrained optimization

We present an algorithm for large-scale unconstrained optimization based onNewton's method. In large-scale optimization, solving the Newton equations at each iteration can be expensive and may not be justified when far from a solution. Instead, an inaccurate solution to the Newton equations is computed using a conjugate gradient method. The resulting algorithm is shown to have strong convergence properties and has the unusual feature that the asymptotic convergence rate is a user specified parameter which can be set to anything between linear and quadratic convergence. Some numerical results on a 916 vriable test problem are given. Finally, we contrast the computational behavior of our algorithm with Newton's method and that of a nonlinear conjugate gradient algorithm. This research was supported in part by DOT Grant CT-06-0011, NSF Grant ENG-78-21615 and grants from the Norwegian Research Council for Sciences and the Humanities and the Norway-American Association. This paper was originally presented at the TIMS-ORSA Joint National Meeting, Washington, DC, May 1980.     

基于凸组合共轭梯度法的ARIMA模型参数估计

提出了一种凸组合共轭梯度算法,并将其算法应用到ARIMA模型参数估计中.新算法由改进的谱共轭梯度算法与共轭梯度算法作凸组合构造而成,具有下述特性:1)具备共轭性条件;2)自动满足充分下降性.证明了在标准Wolfe线搜索下新算法具备完全收敛性,最后数值实验表明通过调节凸组合参数,新算法更加快速有效,通过具体实例证实了模型的显著拟合效果.     

Chan-Vese模型的共轭梯度算法

随着图像采集设备的发展和对图像分辨率要求的提高,人们对图像处理算法在收敛速度和鲁棒性方面提出了更高的要求.从优化的角度对Chan-Vese模型进行算法上的改进,即将共轭梯度法应用到该模型中,使得新算法有更快的收敛速度.首先,简单介绍了Chan-Vese模型的变分水平集方法的理论框架;其次,将共轭梯度算法引入到该模型的求解,得到了模型的新的数值解方法;最后,将得到的算法与传统求解Chan-Vese模型的最速下降法进行了比较.数值实验表明,提出的共轭梯度算法在保持精度的前提下有更快的收敛速度.     

Gradient-type penalty method with inertial effects for solving constrained convex optimization problems with smooth data

We consider the problem of minimizing a smooth convex objective function subject to the set of minima of another differentiable convex function. In order to solve this problem, we propose an algorithm which combines the gradient method with a penalization technique. Moreover, we insert in our algorithm an inertial term, which is able to take advantage of the history of the iterates. We show weak convergence of the generated sequence of iterates to an optimal solution of the optimization problem, provided a condition expressed via the Fenchel conjugate of the constraint function is fulfilled. We also prove convergence for the objective function values to the optimal objective value. The convergence analysis carried out in this paper relies on the celebrated Opial Lemma and generalized Fejér monotonicity techniques. We illustrate the functionality of the method via a numerical experiment addressing image classification via support vector machines.     

A new,globally convergent Riemannian conjugate gradient method

This article deals with the conjugate gradient method on a Riemannian manifold with interest in global convergence analysis. The existing conjugate gradient algorithms on a manifold endowed with a vector transport need the assumption that the vector transport does not increase the norm of tangent vectors, in order to confirm that generated sequences have a global convergence property. In this article, the notion of a scaled vector transport is introduced to improve the algorithm so that the generated sequences may have a global convergence property under a relaxed assumption. In the proposed algorithm, the transported vector is rescaled in case its norm has increased during the transport. The global convergence is theoretically proved and numerically observed with examples. In fact, numerical experiments show that there exist minimization problems for which the existing algorithm generates divergent sequences, but the proposed algorithm generates convergent sequences.     

Dynamic responses of shear flows over a deformable porous surface layer in a cylindrical tube

This paper investigates dynamic responses of a viscous fluid flow introduced under a time dependent pressure gradient in a rigid cylindrical tube that is lined with a deformable porous surface layer. With the Darcy’s law and a linear elasticity assumption, we have solved the coupling effect of the fluid movement and the deformation of the porous medium in the Laplace transform space. Governing equations are deduced for the solid displacement and the fluid velocity in the porous layer. Analytical solutions in the transformed domain are derived and the time dependent variables are inverted numerically using Durbin’s algorithm. Interaction between the solid and the fluid phases in the porous layer and its effects on fluid flow in tube are investigated under steady and unsteady flow conditions when the solid phase is either rigid or deformable. Examples are presented for flows driven by a Heaviside or a sinusoid pressure gradient. Significant effects of the porous surface layer on the flow in the tube are observed. The analytical solutions can be used to test more complicated numerical schemes.