Computational Sensitivity Analysis for State Constrained Optimal Control Problems

A theoretical sensitivity analysis for parametric optimal control problems subject to pure state constraints has recently been elaborated in [7,8]. The articles consider both first and higher order state constraints and develop conditions for solution differentiability of optimal solutions with respect to parameters. In this paper, we treat the numerical aspects of computing sensitivity differentials via appropriate boundary value problems. In particular, numerical methods are proposed that allow to verify all assumptions underlying solution differentiability. Three numerical examples with state constraints of order one, two and four are discussed in detail.     

Constrained variational refinement

A non-uniform, variational refinement scheme is presented for computing piecewise linear curves that minimize a certain discrete energy functional subject to convex constraints on the error from interpolation. Optimality conditions are derived for both the fixed and free-knot problems. These conditions are expressed in terms of jumps in certain (discrete) derivatives. A computational algorithm is given that applies to constraints whose boundaries are either piecewise linear or spherical. The results are applied to closed periodic curves, open curves with various boundary conditions, and (approximate) Hermite interpolation.     

Optimization Techniques for Solving Elliptic Control Problems with Control and State Constraints: Part 1. Boundary Control

We study optimal control problems for semilinear elliptic equations subject to control and state inequality constraints. In a first part we consider boundary control problems with either Dirichlet or Neumann conditions. By introducing suitable discretization schemes, the control problem is transcribed into a nonlinear programming problem. It is shown that a recently developed interior point method is able to solve these problems even for high discretizations. Several numerical examples with Dirichlet and Neumann boundary conditions are provided that illustrate the performance of the algorithm for different types of controls including bang-bang and singular controls. The necessary conditions of optimality are checked numerically in the presence of active control and state constraints.     

Level set based multi-scale methods for large deformation contact problems

We consider the numerical simulation of contact problems in elasticity with large deformations. The non-penetration condition is described by means of a signed distance function to the obstacle's boundary. Techniques from level set methods allow for an appropriate numerical approximation of the signed distance function preserving its non-smooth character. The emerging non-convex optimization problem subject to non-smooth inequality constraints is solved by a non-smooth multiscale SQP method in combination with a non-smooth multigrid method as interior solver. Several examples in three space dimensions including applications in biomechanics illustrate the capability of our methods.     

Locking constraints for elastic rods and a curvature bound for spatial curves

The paper presents necessary conditions for curves in R³ subject to the nonholonomic constraint of an upper bound for curvature and suitable boundary conditions. The proof essentially uses a reformulation of the problem by means of framed curves. The Euler–Lagrange equations for nonlinearly elastic Cosserat rods subject to a general class of locking constraints is derived by similar methods. Mathematics Subject Classification (2000) 49K15, 51M16, 74B20, 74K10     

Continuation and shooting methods for boundary value problems of Bernstein type

Fixed point continuation methods and shooting methods are combined to produce an effective numerical procedure for solving boundary value problems for nonlinear ordinary differential equations. Typical numerical solution schemes involve an iteration procedure. Continuation methods systematically generate good initial guesses and, when combined with a shooting method and an appropriate update procedure, give systematic means for the numerical solution of nonlinear boundary value problems. This paper concentrates on problems of Bernstein type, which arise naturally in the calculus of variations and in steady-state heat conduction.     

Solving elliptic control problems with interior point and SQP methods: control and state constraints

We study optimal control problems for semilinear elliptic equations subject to control and state inequality constraints. Both boundary control and distributed control problems are considered with boundary conditions of Dirichlet or Neumann type. By introducing suitable discretization schemes, the control problem is transcribed into a nonlinear programming problem. Necessary conditions of optimality are discussed both for the continuous and the discretized control problem. It is shown that the recently developed interior point method LOQO of [35] is capable of solving these problems even for high discretizations. Four numerical examples with Dirichlet and Neumann boundary conditions are provided that illustrate the performance of the algorithm for different types of controls including bang–bang controls.     

线性约束三次规划问题的全局最优性必要条件和最优化算法

讨论了带线性不等式约束三次规划问题的最优性条件和最优化算法. 首先, 讨论了带有线性不等式约束三次规划问题的 全局最优性必要条件. 然后, 利用全局最优性必要条件, 设计了解线性约束三次规划问题的一个新的局部最优化算法(强局部最优化算法). 再利用辅助函数和所给出的新的局部最优化算法, 设计了带有线性不等式约束三 规划问题的全局最优化算法. 最后, 数值算例说明给出的最优化算法是可行的、有效的.     

Numerical analysis of electric field formulations of the eddy current model

This paper deals with finite element methods for the numerical solution of the eddy current problem in a bounded conducting domain crossed by an electric current, subjected to boundary conditions involving only data easily available in applications. Two different cases are considered depending on the boundary data: input current intensities or differences of potential. Weak formulations in terms of the electric field are given in both cases. In the first one, the input current intensities are imposed by means of integrals on curves lying on the boundary of the domain and joining current entrances and exit. In the second one, the electric potentials are imposed by means of Lagrange multipliers, which are proved to represent the input current intensities. Optimal error estimates are proved in both cases and implementation issues are discussed. Finally, numerical tests confirming the theoretical results are reported. Partially supported by Xunta de Galicia research grant PGIDIT02PXIC20701PN (Spain). Partially supported by FONDAP in Applied Mathematics (Chile). Partially supported by MCYT (Spain) project DPI2003-01316.     

A boundary value approach to the numerical solution of initial value problems by multistep methos †

A boundary value appraoch to the numerical solution of initial value problems by means of linear multistep methods is presented. This theory is based on the study of linear difference equations when their general solution is computed by imposing boundary conditions. All the main stability and convergence properties of the obtained methods are investigated abd compared to those of the classical multistep methods. Then, as an example, new itegration formulas, called extended trapezoidal rules, are derived. For any order they have the same stability properties (in the sense of the definitions given in this paper) of the trapezoidal rule, which is the first method in this class. Some numerical examples are presented to confirm the theoretical expectations and to allow us to trust a future code based on boundary value methods.     

Incorporating boundary conditions in a stochastic volatility model for the numerical approximation of bond prices

In this paper, we consider a two-factor interest rate model with stochastic volatility, and we assume that the instantaneous interest rate follows a jump-diffusion process. In this kind of problems, a two-dimensional partial integro-differential equation is derived for the values of zero-coupon bonds. To apply standard numerical methods to this equation, it is customary to consider a bounded domain and incorporate suitable boundary conditions. However, for these two-dimensional interest rate models, there are not well-known boundary conditions, in general. Here, in order to approximate bond prices, we propose new boundary conditions, which maintain the discount function property of the zero-coupon bond price. Then, we illustrate the numerical approximation of the corresponding boundary value problem by means of an alternative direction implicit method, which has been already applied for pricing options. We test these boundary conditions with several interest rate pricing models.     

Numerical analysis of singularly perturbed delay differential turning point problem

In this paper, we describe a numerical method based on fitted operator finite difference scheme for the boundary value problems for singularly perturbed delay differential equations with turning point and mixed shifts. Similar boundary value problems are encountered while simulating several real life processes for instance, first exit time problem in the modelling of neuronal variability. A rigorous analysis is carried out to obtain priori estimates on the solution and its derivatives for the considered problem. In the development of numerical methods for constructing an approximation to the solution of the problem, a special type of mesh is generated to tackle the delay term along with the turning point. Then, to develop robust numerical scheme and deal with the singularity because of the small parameter multiplying the highest order derivative term, an exponential fitting parameter is used. Several numerical examples are presented to support the theory developed in the paper.     

Boundary value problems for differential-algebraic equations

This paper shows how certain regularization methods can be used to determine the solution structure of linear differential systems subject to linear constraints and boundary conditions. Attention is restricted to the relatively straightforward index-one problems and to certain index-two problems, where endpoint jumps are related to the underlying boundary layer structure of the corresponding singular perturbation problems.     

A new local and global optimization method for mixed integer quadratic programming problems

In this paper, a new local optimization method for mixed integer quadratic programming problems with box constraints is presented by using its necessary global optimality conditions. Then a new global optimization method by combining its sufficient global optimality conditions and an auxiliary function is proposed. Some numerical examples are also presented to show that the proposed optimization methods for mixed integer quadratic programming problems with box constraints are very efficient and stable.     

Potential transients for an electrochemical corrosion reaction under constant current conditions

Due to the electrochemical nature of almost all corrosion reactions, electrochemical methods are commonly used to measure the corrosion rate of a metal in the laboratory or in the field. In particular, steady state methods are the most widely used for corrosion rate measurements. Transient methods, which can be much more efficient, traditionally rely on an equivalent linear circuit representing the surface kinetics, with negligible mass transport effects. This has been reported to predict transients which are not observed experimentally in many practical situations. In this paper, we consider the galvanostatic method, wherein a constant current is applied across a corroding metal surface and the transient potential response is recorded. The resulting boundary value problems incorporating mixed kinetic and diffusion control involve highly nonlinear, coupled boundary conditions. We present numerical and approximate analytical solutions which can be incorporated into corrosion analysis routines in order to calculate corrosion parameters. The analytical expressions open the possibility of measuring corrosion parameters by merely fitting a class of elementary functions to experimental potential transients. This leads to a significant reduction in the number of computations required for the curve fitting, and hence increasing the overall efficiency of the measurement process compared to the conventional steady state methods.     

Boundary control of semilinear elliptic equations with discontinuous leading coefficients and unbounded controls

We consider optimal control problems governed by semilinear elliptic equations with pointwise constraints on the state variable. The main difference with previous papers is that we consider nonlinear boundary conditions, elliptic operators with discontinuous leading coefficients and unbounded controls. We can deal with problems with integral control constraints and the control may be a coefficient of order zero in the equation. We derive optimality conditions by means of a new Lagrange multiplier theorem in Banach spaces.     

Forward differential dynamic programming

The dynamic programming formulation of the forward principle of optimality in the solution of optimal control problems results in a partial differential equation with initial boundary condition whose solution is independent of terminal cost and terminal constraints. Based on this property, two computational algorithms are described. The first-order algorithm with minimum computer storage requirements uses only integration of a system of differential equations with specified initial conditions and numerical minimization in finite-dimensional space. The second-order algorithm is based on the differential dynamic programming approach. Either of the two algorithms may be used for problems with nondifferentiable terminal cost or terminal constraints, and the solution of problems with complicated terminal conditions (e.g., with free terminal time) is greatly simplified.     

An explicit series approximation to the optimal exercise boundary of American put options

This paper derives an explicit series approximation solution for the optimal exercise boundary of an American put option by means of a new analytical method for strongly nonlinear problems, namely the homotopy analysis method (HAM). The Black–Sholes equation subject to the moving boundary conditions for an American put option is transferred into an infinite number of linear sub-problems in a fixed domain through the deformation equations. Different from perturbation/asymptotic approximations, the HAM approximation can be applicable for options with much longer expiry. Accuracy tests are made in comparison with numerical solutions. It is found that the current approximation is as accurate as many numerical methods. Considering its explicit form of expression, it can bring great convenience to the market practitioners.     

New Direct Numerical Methods for Some Multidimensional Problems of the Calculus of Variations

In this paper, we develop a new approach to the design of direct numerical methods for multidimensional problems of the calculus of variations. The approach is based on a transformation of the problem with the use of a new class of Sobolev-like spaces that is studied in the article. This transformation allows one to analytically compute the direction of steepest descent of the main functional of the calculus of variations with respect to a certain inner product, and, in turn, to construct new direct numerical methods for multidimensional problems of the calculus of variations. In the end of the paper, we point out how the approach developed in the article can be extended to the case of problems with more general boundary conditions, problems for functionals depending on higher order derivatives, and problems with isoperimetric and/or pointwise constraints.     

初边值问题稳定性分析的一个注记

初边值问题稳定性的研究是在初值问题稳定性理论基础上开展的.1968年Kreiss得到了双曲型方程组二层显式差分近似初边值问题稳定性的充分条件,1972年Gustafs-son,Kreiss,Sundstr(?)m得到了多层隐式差分近似初边值问题稳定性的充分必要条件;一般称之为GKS理论.