Optimal Control for Two-Dimensional Stochastic Navier-Stokes Equations

Loeb space methods are used to prove the existence of an optimal control for the two-dimensional stochastic Navier--Stokes equations in a variety of settings—including that of control based on digital observations of the evolution of the solution.     

Convergence of Legendre Methods for Navier-Stokes Equations

1.IntroductionInthispaPerwestudyspectraltyPeofmethodsbasedonLegendrepo1ynomials.Weprovestabilityandconvergenceresultsforthesemethodsusingenergyestimates-TheconvergenceresultsweobtainedaxenearlyoptimalinthesensethattheerrorestimatesforthenumericalsolutionisofthesameorderastheerrorestimatesinaPproalmationtheoryl7,14].Atrivialconsequenceisthatthesemethodsareindeedspectrallyaccurate.Spectralmethodshavebeenusedqulteextensivelyinthepasttwodecades-Be-causeoftheirhighresolutionpower,thesemethodsrec…     

Navier-Stokes方程最优控制问题的一种非协调有限元局部稳定化方法

基于局部Gauss积分和梯形外推公式,速度/压力空间采用最低等阶非协调元NCP1-P1逼近,针对非定常Navier-Stokes方程最优控制问题,建立了一种全离散的非协调有限元局部稳定化格式.该格式绕开了inf-sup条件的束缚,且在每一时间步上,只需要做线性计算,减少了计算量.证明了该格式是无条件稳定的,给出了详细的误差分析.误差结果表明,该线性格式在时间上具有二阶精度.     

定常Navier—Stokes方程的Petrov—Galerkin方法

研究了定常Navier－Stokes方程的四种Petrov－Galerkin有限元方法：PG1，PG2，SD和GLS．它们都是稳定的，避免了经典混合方法中必要的Babuska－Brezzi条件．给出了各种方法有限元解的存在性、唯一性和唯一解的误差估计．     

Optimal Control in Heterogeneous Domain Decomposition Methods for Advection-Diffusion Equations

New domain decomposition methods (DDM) based on optimal control approach are introduced for the coupling of first and second order equations on overlapping subdomains. Several cost functionals and control functions are proposed. Uniqueness and existence results are proved for the coupled problem, and the convergence of iterative processes is analyzed. The work was supported by the Russian Foundation for Basic Research (04-01-00615) and it was partly carried out while the first author was visiting the IACS at EPFL.     

Reduced Finite Element Discretizations of the Stokes and Navier-Stokes Equations

ABSTRACT

If finite element spaces for the velocity and pressure do not satisfy the Babu?ka-Brezzi condition, a stable conforming discretization of the Stokes or Navier-Stokes equations can be obtained by enriching the velocity space by suitable functions. Writing any function from the enriched space as a sum of a function from the original space and a function from the supplementary space, the discretization will contain a number of additional terms compared with a conforming discretization for the original pair of spaces. We show that not all these terms are necessary for the solvability of the discrete problem and for optimal convergence properties of the discrete solutions, which is useful for saving computer memory and for establishing a connection to stabilized methods.     

On Optimal Control Using Particle Methods

Mesh-based methods like finite elements and finite volumes enjoy limited use in deforming domain and free surface applications due to exorbitant computational demands. Instead, particle methods are very suitable for simulating such problems. However, they have not been used for optimisation yet. We therefore derive optimisation strategies using this kind of methods. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mixed Finite Element Methods for Fourth Order Elliptic Optimal Control Problems

In this paper, a priori error estimates are derived for the mixed finite element discretization of optimal control problems governed by fourth order elliptic partial differential equations. The state and co-state are discretized by Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. The error estimates derived for the state variable as well as those for the control variable seem to be new. We illustrate with a numerical example to confirm our theoretical results.     

Optimal Control of Iterative Solution Methods for Linear Systems of Equations

Iterative solution methods for linear systems of equations can be regarded as discrete-time control systems, for which a stabilizing feedback control has to be found. Well known algorithms such as GMRES(m) may exhibit unstable dynamics or sensitive dependence on initial conditions, thus preventing the algorithm to converge to the desired solution. Based on linear system feedback design techniques a new algorithm is proposed that does not suffer under such shortcomings. Global convergence to the desired solution is shown for any initial state. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal Secant-Type Methods for Operator Equations

We prove an existence and uniqueness theorem for operator equations in Banach spaces with (generally non-differentiable) operators whose divided differences are Lipschitz continuous on operator's domain. The theorem makes possible to apply the concept of entropy optimality of iterative methods introduced in our earlier work to the class of secant-type methods. Using this concept, we show that it is feasible to get a method that needs the same information (one value of the operator) per iteration but exhibits a faster convergence than the secant and secant-update methods.     

Unconditionally Optimal Error Estimates of the Bilinear-Constant Scheme for Time-Dependent Navier-Stokes Equations

In this paper,the unconditional error estimates are presented for the time-dependent Navier-Stokes equations by the bilinear-constant scheme.The corresponding optimal error estimates for the velocity and the pressure are derived unconditionally,while the previous works require certain time-step restrictions.The analysis is based on an iterated time-discrete system,with which the error function is split into a temporal error and a spatial error.The τ-independent(τ is the time stepsize)error estimate between the numerical solution and the solution of the time-discrete system is proven by a rigorous analysis,which implies that the numerical solution in L^∞-norm is bounded.Thus optimal error estimates can be obtained in a traditional way.Numerical results are provided to confirm the theoretical analysis.     

Adjoint-Based Optimal Boundary Control of a Stefan Problem System Fully Coupled with Navier-Stokes Equations

Free boundary and moving boundary problems, that can be used to model crystal growth or the solidification and melting of pure materials, receive growing attention in science and technology. The optimal control of these problems appear even more interesting since certain desired shapes of the boundaries improve, e.g., the material quality in the case of crystal growth. We consider the so called two-phase Stefan problem that models a solid and a liquid phase separated by a moving interface. In the work presented, we take a sharp interface model approach and define a quadratic tracking-type cost functional that penalizes the deviation of the interface from the desired state at a final time as well as the control costs. Following the “optimize-then-discretize” paradigm, we formulate a first order optimality system using the formal Lagrange approach and derive the adjoint PDE system that provides the needed gradient of the cost functional. By means of an example setup of a container with an in- and outflow of water and a cooling unit at the bottom, we illustrate how the derived formulations can be used to achieve a desired interface between the solid and the fluid phase by controlling the flow at the inlet. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Decoupled Mixed Element Methods for Fourth Order Elliptic Optimal Control Problems with Control Constraints

In this paper, we study the finite element methods for distributed optimal control problems governed by the biharmonic operator. Motivated from reducing the regularity of solution space, we use the decoupled mixed element method which was used to approximate the solution of biharmonic equation to solve the fourth order optimal control problems. Two finite element schemes, i.e., Lagrange conforming element combined with full control discretization and the nonconforming Crouzeix-Raviart element combined with variational control discretization, are used to discretize the decoupled optimal control system. The corresponding a priori error estimates are derived under appropriate norms which are then verified by extensive numerical experiments.     

Nash Equilibria for the Multiobjective Control of Linear Partial Differential Equations

This article is concerned with the numerical solution of multiobjective control problems associated with linear partial differential equations. More precisely, for such problems, we look for the Nash equilibrium, which is the solution to a noncooperative game. First, we study the continuous case. Then, to compute the solution of the problem, we combine finite-difference methods for the time discretization, finite-element methods for the space discretization, and conjugate-gradient algorithms for the iterative solution of the discrete control problems. Finally, we apply the above methodology to the solution of several tests problems.     

A Second Order Nonconforming Triangular Mixed Finite Element Scheme for the Stationary Navier-Stokes Equations

In this paper,a nonconforming triangular mixed finite element scheme with second order convergence behavior is proposed for the stationary Navier-Stokes equations.The new nonconforming triangular element is taken as approximation space for the velocity and the linear element for the pressure.The convergence analysis is presented and optimal error estimates of both broken H1-norm and L2-norm for velocity as well as the L2-norm for the pressure are derived.     

Boundary Shape Control of the Navier-Stokes Equations and Applications

In this paper, the geometrical design for the blade's surface in an impeller or for the profile of an aircraft, is modeled from the mathematical point of view by a boundary shape control problem for the Navier-Stokes equations. The objective function is the sum of a global dissipative function and the power of the fluid. The control variables are the geometry of the boundary and the state equations are the Navier-Stokes equations. The Euler-Lagrange equations of the optimal control problem are derived, which are an elliptic boundary value system of fourth order, coupled with the Navier-Stokes equations. The authors also prove the existence of the solution of the optimal control problem, the existence of the solution of the Navier-Stokes equations with mixed boundary conditions, the weak continuity of the solution of the Navier-Stokes equations with respect to the geometry shape of the blade's surface and the existence of solutions of the equations for the Gateaux derivative of the solution of the Navier-Stokes equations with respect to the geometry of the boundary.     

Optimal Eighth Order Iterative Methods

We develop an eighth order family of methods, consisting of three steps and three parameters, for solving nonlinear equations. Per iteration the methods require four evaluations (three function evaluations and one evaluation of the first derivative). Convergence analysis shows that the family is eighth-order convergent which is also substantiated through the numerical work. Computational results ascertain that family of methods are efficient and demonstrate equal or better performance as compared with other well known methods.     

Navier-Stokes方程的脉动速度方程的最优动力系统建模和动力学分析

研究了基于Navier-Stokes方程的脉动速度方程的最优低维动力系统建模理论.最优目标泛函为脉动速度基函数的不可压缩性和正交性.数值计算了充分发展的并排双方柱绕流问题,并基于双尺度全局最优化方法,建立了它的脉动速度的最优动力系统模型.对其相空间轨道、Poincaré截面、分岔特性、功率谱和Lyapunov指数集等动力学特性进行了分析.随着Reynolds数的增加,双方柱绕流的脉动速度方程最优动力系统具有复杂的类倍周期分岔行为.     

On Error Estimates of the Projection Methods for the Navier-Stokes Equations: Second-order Schemes

We present in this paper a rigorous error analysis of several projection schemes for the approximation of the unsteady incompressible Navier-Stokes equations. The error analysis is accomplished by interpreting the respective projection schemes as second-order time discretizations of a perturbed system which approximates the Navier-Stokes equations. Numerical results in agreement with the error analysis are also presented.