Method In Function Space Research Articles

Flexible Continuous-Time Modeling for Multi-Objective Day-Ahead Scheduling of CHP Units

Increasing applications of CHP units have turned the problem of finding the best optimization model into a significant subject for scholars. In this respect, this paper is aimed at driving a novel formulation to the multi-objective day-ahead scheduling of CHP units using Bernstein polynomials, which more optimally schedules power and heat generations as well as ramping trajectories. This procedure includes yielding an affine function that closely approximates real-time net-load and generation trajectories, which is demonstrated to have a superior performance to the conventional hourly day-ahead scheduling of CHP units based on discrete-time approximation. The problem of how to handle various objective functions by function space method is also addressed. The simulations conducted on the sample test systems, which consist of CHP systems, thermal and heat-only units, as well as thermal and electrical loads, show that the suggested multi-objective model can perfectly cover the total heat and electrical loads in terms of economic and environmental criteria. More importantly, the results indicate that the accuracy of the proposed approach renders cost saving of 1.67% and emission saving of 1.46% in comparison with the conventional hourly-based model, apart from leading to fewer ramping scarcities in real-time operations.

A projection method based on self-adaptive rules for Stokes equations with nonlinear slip boundary conditions

In this work we propose and study a self-adaptive projection method for the numerical solution of Stokes equations under slip boundary conditions. We introduce the projection operator to formulate the slip constraint as a fixed point equation. To solve this problem we suggest a projection method by reformulating the nonlinear slip boundary conditions into an iterative form. To make the method more efficient, we find a self-adaptive rule that uses iterative functions to adjust the penalty parameter automatically. We show the convergence of the method in function space and give its application in detail. Finally, the numerical results are given to support our theoretical results.

Self-adaptive projection and boundary element methods for contact problems with Tresca friction

A self-adaptive projection method is considered for the numerical simulation of contact problems with Tresca friction. The contact constraints are imposed in the fixed point formulation using projection operators. It leads to a projection method where each iterative step consists of a linear elasticity problem with Robin conditions on the contact boundary. To ensure the efficiency of the method, a self-adaptive rule is given to adjust the parameter automatically. We prove the convergence of the method in function space and demonstrate its application with the boundary element method. The numerical experiments are presented to show the performance of the proposed method.

Operator-theoretic and regularization approaches to ill-posed problems

A general framework of regularization and approximation methods for ill-posed problems is developed. Three levels in the resolution processes are distinguished and emphasized in this expository-research paper: philosophy of resolution, regularization–approximation schema, and regularization algorithms. Dilemmas and methodologies of resolution of ill-posed problems and their numerical implementations are examined with particular reference to the problem of finding numerically minimum weighted-norm least squares solutions of first kind integral equations (and more generally of linear operator equations with non-closed range). An emphasis is placed on the role of constraints, function space methods, the role of generalized inverses, and reproducing kernels in the regularization and stable computational resolution of these problems. The thrust of the contribution is devoted to the interdisciplinary character of operator-theoretic and regularization methods for ill-posed problems, in particular in mathematical geoscience.

A novel computational method for solving Troesch's problem with high-sensitivity parameter

ABSTRACTTroesch's problem is a nonlinear boundary value problem arising in the confinement of a plasma column by radiation pressure, and also in the theory of gas porous electrodes. It is well known that finding numerical solutions to this problem is challenging, especially when the sensitivity parameter is large. In this article, we present an efficient and accurate numerical method for solving Troesch's problem. The method presented in this work is capable of computing the solution, even for extremely high-sensitivity parameter. The method is based on the the Newton–Raphson–Kantorovich approximation method in function space combined with the standard finite difference method. Although, available numerical solvers fail to provide accurate numerical solutions when the sensitivity parameter λ becomes large (λ exceeds 100) [1–5], the method proposed here is able to provide accurate numerical solutions for extremely large values of this sensitivity parameter, up to λ = 500. Numerical experiments are provided to show the accuracy of the method compared to existing solvers, as well as its capability to compute the solution for high values of the sensitivity parameter λ.

A convex analysis approach to optimal controls with switching structure for partial differential equations

Optimal control problems involving hybrid binary-continuous control costs are challenging due to their lack of convexity and weak lower semicontinuity. Replacing such costs with their convex relaxation leads to a primal-dual optimality system that allows an explicit pointwise characterization and whose Moreau–Yosida regularization is amenable to a semismooth Newton method in function space. This approach is especially suited for computing switching controls for partial differential equations. In this case, the optimality gap between the original functional and its relaxation can be estimated and shown to be zero for controls with switching structure. Numerical examples illustrate the effectiveness of this approach.

An iterative finite difference method for solving Bratu’s problem

In this paper we propose a new iterative finite difference (IFD) scheme based on the Newton–Raphson–Kantorovich approximation method in function space to solve the classical one-dimensional Bratu’s problem. This new numerical method produces accurate solutions with low computational cost. The effectiveness and accuracy of the IFD method are confirmed through several numerical examples and compared to some existing numerical solvers.

An interior point method designed for solving linear quadratic optimal control problems with hp finite elements

We consider linear quadratic optimal control problems with elliptic partial differential equations. The problem is solved with an interior point method in the control variable. We prove convergence of this method in function space by employing a suitable smoothing operator. As discretization we choose hp-finite element method based on local estimates on the smoothness of functions. A fully adaptive algorithm is implemented and a-posteriori error estimators are derived for the central path and the Newton system. The theoretical results are complemented by numerical examples.

An interior point method in function space for the efficient solution of state constrained optimal control problems

We propose and analyze an interior point path-following method in function space for state constrained optimal control. Our emphasis is on proving convergence in function space and on constructing a practical path-following algorithm. In particular, the introduction of a pointwise damping step leads to a very efficient method, as verified by numerical experiments.

Automatic Fréchet Differentiation for the Numerical Solution of Boundary-Value Problems

A new solver for nonlinear boundary-value problems (BVPs) in Matlab is presented, based on the Chebfun software system for representing functions and operators automatically as numerical objects. The solver implements Newton’s method in function space, where instead of the usual Jacobian matrices, the derivatives involved are Fréchet derivatives. A major novelty of this approach is the application of automatic differentiation (AD) techniques to compute the operator-valued Fréchet derivatives in the continuous context. Other novelties include the use of anonymous functions and numbering of each variable to enable a recursive, delayed evaluation of derivatives with forward mode AD . The AD techniques are applied within a new Chebfun class called which allows users to set up and solve nonlinear BVPs, both scalar and systems of coupled equations, in a few lines of code, using the “nonlinear backslash” operator (\). This framework enables one to study the behaviour of Newton’s method in function space.

An interior point algorithm with inexact step computation in function space for state constrained optimal control

We consider an interior point method in function space for PDE constrained optimal control problems with state constraints. Our emphasis is on the construction and analysis of an algorithm that integrates a Newton path-following method with adaptive grid refinement. This is done in the framework of inexact Newton methods in function space, where the discretization error of each Newton step is controlled by adaptive grid refinement in the innermost loop. This allows to perform most of the required Newton steps on coarse grids, such that the overall computational time is dominated by the last few steps. For this purpose we propose an a-posteriori error estimator for a problem suited norm.

Superlinear convergence of the control reduced interior point method for PDE constrained optimization

A thorough convergence analysis of the Control Reduced Interior Point Method in function space is performed. This recently proposed method is a primal interior point pathfollowing scheme with the special feature that the control variable is eliminated from the optimality system. Apart from global linear convergence we show that this method converges locally superlinearly, if the optimal solution satisfies a certain non-degeneracy condition. In numerical experiments we observe that a prototype implementation of our method behaves as predicted by our theoretical results.

Mesh independence and fast local convergence of a primal-dual active-set method for mixed control-state constrained elliptic control problems

A class of mixed control-state constrained optimal control problems for elliptic partial differential equations arising, for example, in Lavrentiev-type regularized state constrained optimal control is considered. Its numerical solution is obtained via a primal-dual activeset method, which is equivalent to a class of semi-smooth Newton methods. The locally superlinear convergence of the active-set method in function space is established, and its mesh independence is proved. The paper contains a report on numerical test runs including a comparison with a short-step path-following interior-point method and a coarse-to-fine mesh sweep, that is, a nested iteration technique, for accelerating the overall solution process. Finally, convergence and regularity properties of the regularized problems with respect to a vanishing Lavrentiev parameter are considered. 2000 Mathematics subject classification: primary 65K05; secondary 90C33.

Some results in nonlinear QFT

AbstractNonlinear QFT (quantitative feedback theory) is a technique for solving the problem of robust control of an uncertain nonlinear plant by replacing the uncertain nonlinear plant with an ‘equivalent’ family of linear plants. The problem is then finding a linear QFT controller for this family of linear plants. While this approach is clearly limited, it follows in a long tradition of linearization approaches to nonlinear control (describing functions, extended linearization, etc.) which have been found to be quite effective in a wide range of applications. In recent work, the authors have developed an alternative function space method for the derivation and validation of nonlinear QFT that has clarified and simplified several important features of this approach. In particular, single validation conditions are identified for evaluating the linear equivalent family, and as a result, the nonlinear QFT problem is reduced to a linear equivalent problem decoupled from the linear QFT formalism. In this paper, we review this earlier work and use it in the development of (1) new results on the existence of nonlinear QFT solutions to robust control problems, and (2) new techniques for the circumvention of problems encountered in the application of this approach. Copyright © 2001 John Wiley & Sons, Ltd.

Optimal design of elastic rods under axial gravitational load using the maximum principle

Optimal design problems for structural elements in equilibrium often come in pairs in which the cost function for one problem becomes a constraint for the second. In particular, the problem of minimizing structural volume or weight under size and stress constraints has a dual in which potential energy is minimized for fixed volume of material. These dual problems are solved here for a linearly elastic rod hanging from a rigid support. The design variable is the cross-sectional area of the rod. The method used is the Maximum Principle of Pontryagin and Hestenes, rather than the function space methods used by others for similar problems.

Newton's Method for a Class of Optimal Shape Design Problems

A class of optimal shape design problems is considered where a part of the boundary of the domain represents the free parameter. The variable domain is parametrized by a class of functions in such a way that the optimal design problem results in an optimal control problem on a fixed domain. The functions for the parametrization of the domain are used as controls, and the corresponding states are then given by the solution of an elliptic boundary value problem on a fixed domain. Discretizing this control problem normally leads to a large-scale optimization problem, where the corresponding solution methods are characterized by the requirement of solving many boundary value problems. In spite of this interesting numerical challenge, until now little work has been done to derive more efficient algorithms by taking advantage of the specific structure of this kind of problem. In this report, Newton's method in function space is derived, resulting in an efficient algorithm for the discretized optimization problems. By using the specific structure of these optimal shape design problems, an efficient implementation of the numerical algorithm is introduced. The properties of this algorithm are compared with those of the gradient method using illustrative numerical examples.

An adaptive Rothe method for the wave equation

The adaptive Rothe method approaches a time-dependent PDE as an ODE in function space. This ODE is solved virtually using an adaptive state-of-the-art integrator. The actual realization of each time-step requires the numerical solution of an elliptic boundary value problem, thus perturbing the virtual function space method. The admissible size of that perturbation can be computed a priori and is prescribed as a tolerance to an adaptive multilevel finite element code, which provides each time-step with an individually adapted spatial mesh. In this way, the method avoids the well-known difficulties of the method of lines in higher space dimensions. During the last few years the adaptive Rothe method has been applied successfully to various problems with infinite speed of propagation of information. The present study concerns the adaptive Rothe method for hyperbolic equations in the model situation of the wave equation. All steps of the construction are given in detail and a numerical example (diffraction at a corner) is provided for the 2D wave equation. This example clearly indicates that the adaptive Rothe method is appropriate for problems which can generally benefit from mesh adaptation. This should be even more pronounced in the 3D case because of the strong Huygens' principle.

A few remarks concerning countable unions of finite-dimensional spaces

We use Nagata's universal spaces and function space methods to give an alternative proof of recent theorems of Engelking concerning strongly countable-dimensional and locally finite-dimensional spaces.

Optimal controlcn of a voltage-driven stepping motor

The single-step optimal control of a voltage-driven variable-reluctance stepping motor, with an inertia load, is considered. A solution for the optimal control policy is achieved by a conjugate gradient method in function space, optimising the response of a well established nonlinear motor model. The open-loop bang-bang policy which results is refined using a switching-times algorithm which is more economical in computer requirements. The technique is implemented as a microprocessor controller and is shown, by experimental results on a typical motor, to be effective over a wide range of load inertia.

Operator-theoretic and computational approaches to Ill-posed problems with applications to antenna theory

A general framework for regularization and approximation methods for ill-posed problems is developed. Three levels in the resolution processes are distinguished and emphasized: philosophy of resolution, regularization-approximation schema, and regularization algorithms. Dilemmas and methodologies of resolution of ill-posed problems and their numerical implementations are examined in this framework with particular reference to the problem of finding numerically minimum weighted-norm least-squares solutions of first kind integral equations (and more generally of linear operator equations with nonclosed range). A common problem in all these methods is delineated: each method reduces the problem of resolution to a nonstandard minimization problem involving an unknown critical parameter whose optimal value is crucial to the numerical realization and amenability of the method. The nonstandardness results from the fact that one does not have explicitly, or a priori, the function to be minimized; it has to built up using additional information, convergence rate estimates, and robustness conditions, etc. Several results are developed that complement recent advances in numerical analysis and regularization of inverse and ill-posed (identification and pattern synthesis) problems. An emphasis is placed on the role of constraints, function space methods, the role of generalized inverses, and reproducing kernels in the regularization and stable computational resolution of these problems. The results will be applied specifically to problems of antenna synthesis and identification. However the thrust of the paper is devoted to the interdisciplinary character of operator-theoretic and numerical methods for ill-posed problems.