Vortex control of instationary channel flows using translation invariant cost functionals

The use of translation invariant cost functionals for the reduction of vortices in the context of shape optimization of fluid flow domains is investigated. Analytical expressions for the shape design sensitivity involving different cost functionals are derived. Instationary channel flow problems with a bump and an obstacle as possible control boundaries are taken as test examples. Numerical results are provided in various graphical forms for relatively low Reynolds numbers. Striking differences are found for the optimal shapes corresponding to the different cost functionals, which constitute different quantification of a vortex.     

Shape optimization for Stokes flow

This paper is concerned with a shape sensitivity analysis of a viscous incompressible fluid driven by Stokes equations. The structures of continuous shape gradients with respect to the shape of the variable domain for some given cost functionals are established by introducing the Piola transformation and then deriving the state derivative and its associated adjoint state. Finally we give the finite element approximation of the problem and a gradient type algorithm is effectively used for our problem.     

Local Stabilization of the Navier–Stokes Equations with a Feedback Controller Localized in an Open Subset of the Domain

We study the local exponential stabilization, near a given steady-state flow, of solutions of the 2-D and 3-D Navier–Stokes equations in a bounded domain. The control is effectuated through a distributed control localized in an open subset of the domain. We apply a linear feedback controller, obtained by the solution of a control problem. Lyapunov functionals are given for a large class of cost functionals.     

Optimal shape design for systems governed by variational inequalities,part 2: Existence theory for the evolution case

Some general existence results for optimal shape design problems for systems governed by parabolic variational inequalities are established by the mapping method and variational convergence theory. Then, an existence theorem is given for the optimal shape for an electrochemical machining problem, in which the cost functional is not lower semicontinuous, by extending the general results to this case. Furthermore, this problem is approximated by a set of optimal shape design problems which have more smooth cost functionals and are easier to handle computationally.The authors wish to express their sincere thanks to the reviewers for supplying additional references and for their valuable comments, which made the paper more readable.     

Domain dependence of solutions to the boundary value problem for equations of mixtures of compressible viscous fluids

Under study is the dependence of solutions of an inhomogeneous boundary value problem for a system of equations of mixtures of compressible viscous fluids on the shape of the flow domain. The obtained results can be used to proof the differentiability of solutions and functionals of the solutions (e.g., the drag functional) with respect to a parameter that characterizes the shape variations of an obstacle in the flow.     

Optimal shape design for systems governed by variational inequalities,part 1: Existence theory for the elliptic case

Some general existence results for optimal shape design problems for systems governed by elliptic variational inequalities are established by the mapping method and variational convergence theory. Then, an existence theorem is given for the optimal shape for an electrochemical machining problem, in which the cost functional is not lower semicontinuous, by extending the general results to this case. Furthermore, this problem is approximated by a set of optimal shape design problems which have more smooth cost functionals and are easier to handle computationally.The authors with to express their sincere thanks to the reviewers for supplying additional references and for their valuable comments, which made the paper more readable.     

Risk averse elastic shape optimization with parametrized fine scale geometry

Shape optimization of the fine scale geometry of elastic objects is investigated under stochastic loading. Thus, the object geometry is described via parametrized geometric details placed on a regular lattice. Here, in a two dimensional set up we focus on ellipsoidal holes as the fine scale geometric details described by the semiaxes and their orientation. Optimization of a deterministic cost functional as well as stochastic loading with risk neutral and risk averse stochastic cost functionals are discussed. Under the assumption of linear elasticity and quadratic objective functions the computational cost scales linearly in the number of basis loads spanning the possibly large set of all realizations of the stochastic loading. The resulting shape optimization algorithm consists of a finite dimensional, constraint optimization scheme where the cost functional and its gradient are evaluated applying a boundary element method on the fine scale geometry. Various numerical results show the spatial variation of the geometric domain structures and the appearance of strongly anisotropic patterns.     

Coefficient inverse extremum problems for stationary heat and mass transfer equations

A technique is developed for analyzing coefficient inverse extremum problems for a stationary model of heat and mass transfer. The model consists of the Navier-Stokes equations and the convection-diffusion equations for temperature and the pollutant concentration that are nonlinearly related via buoyancy in the Boussinesq approximation and via convective heat and mass transfer. The inverse problems are stated as the minimization of certain cost functionals at weak solutions to the original boundary value problem. Their solvability is proved, and optimality systems describing the necessary optimality conditions are derived. An analysis of the latter is used to establish sufficient conditions ensuring the local uniqueness and stability of solutions to the inverse extremum problems for particular cost functionals.     

Stopping Problems of Certain Multiplicative Functionals and Optimal Investment with Transaction Costs

Optimal stopping and impulse control problems with certain multiplicative functionals are considered. The stopping problems are solved by showing the unique existence of the solutions of relevant variational inequalities. However, since functions defining the multiplicative costs change the signs, some difficulties arise in solving the variational inequalities. Through gauge transformation we rewrite the variational inequalities in different forms with the obstacles which grow exponentially fast but with positive killing rates. Through the analysis of such variational inequalities we construct optimal stopping times for the problems. Then optimal strategies for impulse control problems on the infinite time horizon with multiplicative cost functionals are constructed from the solutions of the risk-sensitive variational inequalities of "ergodic type" as well. Application to optimal investment with fixed ratio transaction costs is also considered.     

Uniqueness of unbounded viscosity solutions for impulse control problem

We study here the impulse control problem in infinite as well as finite horizon. We allow the cost functionals and dynamics to be unbounded and hence the value function can possibly be unbounded. We prove that the value function is the unique viscosity solution in a suitable subclass of continuous functions, of the associated quasivariational inequality. Our uniqueness proof for the infinite horizon problem uses stopping time problem and for the finite horizon problem, comparison method. However, we assume proper growth conditions on the cost functionals and the dynamics.     

Shape optimization for Stokes problem with threshold slip

We study the Stokes problems in a bounded planar domain Ω with a friction type boundary condition that switches between a slip and no-slip stage. Our main goal is to determine under which conditions concerning the smoothness of Ω solutions to the Stokes system with the slip boundary conditions depend continuously on variations of Ω. Having this result at our disposal, we easily prove the existence of a solution to optimal shape design problems for a large class of cost functionals. In order to release the impermeability condition, whose numerical treatment could be troublesome, we use a penalty approach. We introduce a family of shape optimization problems with the penalized state relations. Finally we establish convergence properties between solutions to the original and modified shape optimization problems when the penalty parameter tends to zero.     

部分信息下期望消费效用最大的优化问题

研究了部分信息下期望消费效用最大的优化问题.利用凸分析理论,非线性滤波和Malliavin导数技术,得到了最优投资-消费策略和代价泛函.对于对数效用函数情形,给出了一个估算信息价值的公式,它是完全信息下和部分信息下所对应的最优代价泛函的差值.     

Solution of Parabolic Equations by Means of Lyapunov Functionals

We propose a new approach to defining the notion of a solution to linear and nonlinear parabolic equations. The main idea consists in studying connections between solutions to dynamic problems in the variational shape and the properties of the corresponding Lyapunov functionals which are strictly decreasing along the trajectories of the above-mentioned dynamic equations except for the equilibrium points. It turns out that the families of Lyapunov functionals constructed by T. I. Zelenyak enable us to propose a new approach to defining solutions to both linear and nonlinear parabolic problems. All results are given in the case of smooth solutions. Note that the Lyapunov functionals can be used for studying solutions with unbounded gradients.     

Identification of distributed systems and the theory of the regularization

The identification problems, i.e., the problems of finding unknown parameters in distributed systems from the observations are very important in modern control theory. The solutions of these identification problems can be obtained by solving the equations of the first kind. However, the solutions are often unstable. In other words, they are not continuously dependent on the data. The regularization or Tihonov's regularization is known as one of the stabilizing algorithms to solve these non well-posed problems. In this paper is studied the regularization method for identification of distributed systems. Several approximation theorems are proved to solve the equations of the first kind. Then, identification problems are reduced to the minimization of quadratic cost functionals by virtue of these theorems. On the other hand, it is known that the statistical methods for identification such as the maximum likelihood lead to the minimization problems of certain quadratic functionals. Comparing these quadratic cost functionals, the relations between the regularization and the statistical methods are discussed. Further, numerical examples are given to show the effectiveness of this method.     

On the problem of optimizing contact force distributions

The problem of optimizing the distribution of contact forces between a rigid obstacle and a discretized linear elastic body is considered. The design variables are the initial gaps between the potential contact nodal points and the obstacle. Two different cost functionals are investigated: the first reflects the objective of minimizing the maximum contact force; the second is the equilibrium potential energy. Contrary to what has been claimed in the literature, it is shown that these cost functionals do not give, in general, the same optimal design. However, it is also shown that, if a certain frequently realized assumption is met by the system flexibility matrix, then this equality does hold.The min-max cost functional is nonconvex and nondifferentiable, and Clarke's theory of nonsmooth optimization is used to establish a sufficient optimality condition. Investigating its consequences, both necessary and sufficient optimality conditions can be given. The equilibrium potential energy cost functional, on the other hand, turns out to have the remarkable porperties of differentiability and convexity.This work was supported by The Center for Industrial Information Technology (CENHT), Linköping Institute of Technology, Linköping, Sweden.     

Statistical shape priors for level set segmentation

Starting in the early 1990's level set methods have become a popular mathematical framework for variational image segmentation. In many applications of segmentation, however, cost functionals which merely take into account the intensity information of the input image will not give rise to the desired segmentation results. To cope with missing or misleading image information, researchers have proposed to impose statistical shape priors into the segmentation process. Such shape priors favor the evolving embedding function to remain similar to embedding functions associated with a collection of training shapes. As a consequence, one can obtain shape-consistent segmentation despite large amounts of noise, background clutter and partial occlusion of the object of interest. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Cloaking via impedance boundary condition for the 2-D Helmholtz equation

We consider control problems for the 2-D Helmholtz equation in an unbounded domain with partially coated boundary. Dirichlet boundary condition is given on one part of the boundary and the impedance boundary condition is imposed on another its part. The role of control in control problem under study is played by boundary impedance. Quadratic tracking–type functionals for the field play the role of cost functionals. Solvability of control problems is proved. The uniqueness and stability of optimal solutions with respect to certain perturbations of both cost functional and incident field are established.     

类Hartree-Fock方程的数值方法

本文的主要目的是介绍近年来大基组下的类Hartree-Fock方程数值求解的一些进展.类Hartree-Fock方程出现在Hartree-Fock理论和含杂化泛函的Kohn-Sham密度泛函理论中,是电子结构理论中一类重要的方程.该方程在复杂的化学和材料体系的电子结构计算中有广泛地应用.由于计算代价的原因,类Hartree-Fock方程一般只被用在较小规模的量子体系（含几十到几百个电子）的计算.从数学角度上讲,类Hartree-Fock方程是一个非线性积分-微分方程组,其计算代价主要来自于积分算子的部分,也就是Fock交换算子.通过发展和结合自适应压缩交换算子方法（ACE）,投影的C-DⅡS方法（PC-DⅡS）方法,以及插值可分密度近似方法（ISDF）,我们大大降低了杂化泛函密度泛函理论的计算代价.以含1000个硅原子的体系为例,我们将平面波基组下的杂化泛函的计算代价降至接近不含Fock交换算子的半局域泛函计算的水平.同时,我们发现类Hartree-Fock方程的数学结构也为一类特征值问题的迭代求解提供了新的思路.     

On Topological Derivatives for Elastic Solids with Uncertain Input Data

In this paper, a new approach to the derivation of the worst scenario and the maximum range scenario methods is proposed. The derivation is based on the topological derivative concept for the boundary-value problems of elasticity in two and three spatial dimensions. It is shown that the topological derivatives can be applied to the shape and topology optimization problems within a certain range of input data including the Lamé coefficients and the boundary tractions. In other words, the topological derivatives are stable functions and the concept of topological sensitivity is robust with respect to the imperfections caused by uncertain data. Two classes of integral shape functionals are considered, the first for the displacement field and the second for the stresses. For such classes, the form of the topological derivatives is given and, for the second class, some restrictions on the shape functionals are introduced in order to assure the existence of topological derivatives. The results on topological derivatives are used for the mathematical analysis of the worst scenario and the maximum range scenario methods. The presented results can be extended to more realistic methods for some uncertain material parameters and with the optimality criteria including the shape and topological derivatives for a broad class of shape functionals. This research is partially supported by the Brazilian Agency CNPq under Grant 472182/2007-2, FAPERJ under Grant E-26/171.099/2006 (Rio de Janeiro) and Brazilian-French Research Program CAPES/COFECUB under Grant 604/08 between LNCC in Petrópolis and IECN in Nancy, and by the Research Grant CNRS-CSAV between Institut Elie Cartan in Nancy and the Institut of Mathematics in Prague. The support is gratefully acknowledged.