Mesh Independence Principle Research Articles

Extending the Mesh Independence For Solving Nonlinear Equations Using Restricted Domains

The mesh independence principle states that, if Newton’s method is used to solve an equation on Banach spaces as well as finite dimensional discretizations of that equation, then the behaviour of the discretized process is essentially the same as that of the initial method. This principle was inagurated in Allgower et al. (SIAM J Numer Anal 23(1):160–169, 1986). Using our new Newton–Kantorovich-like theorem and under the same information we show how to extend the applicability of this principle in cases not possible before. The results can be used to provide more efficient programming methods.

Multigrid Optimization Methods for the Optimal Control of Convection–Diffusion Problems with Bilinear Control

Optimal control problems, governed by convection---diffusion equations with bilinear control, are studied. For the realization of the numerical solution, the multigrid for optimization method together with finite difference discretization is utilized and investigated. In addition, the extension to constrained optimal control problems with bilinear control is considered. Results of numerical experiments show the computational performance of the proposed multigrid scheme in solving optimal control problems subject to a convection---diffusion equation with bilinear control. We obtain that the proposed multigrid strategy accelerates classical one-grid optimization schemes and inherits the order of convergence of the finite difference discretization. Moreover, the mesh independence principle is obtained, which is a typical characterization of a multigrid strategy.

Newton’s method and a mesh-independence principle for certain semilinear boundary-value problems

We exhibit an algorithm which computes an ϵ-approximation of the positive solutions of a family of boundary-value problems with Neumann boundary conditions. Such solutions arise as the stationary solutions of a family of semilinear parabolic equations with Neumann boundary conditions. The algorithm is based on a finite-dimensional Newton iteration associated with a suitable discretized version of the problem under consideration. To determine the behavior of such a discrete iteration we establish an explicit mesh-independence principle. We apply a homotopy-continuation algorithm to compute a starting point of the discrete Newton iteration, and the discrete Newton iteration until an ϵ-approximation of the stationary solution is obtained. The algorithm performs roughly O((1/ϵ)1/2) flops and function evaluations.

Mesh-independent convergence of penalty methods applied to optimal control with partial differential equations

In this paper, optimal control problems with elliptic state equations and constraints on controls are considered. Also state constraints are briefly discussed. Barrier-penalty methods are applied to treat the occurring restrictions. In the case of finite-dimensional optimization problems, the considered methods have a linear rate of convergence in dependence of the penalty parameter. However, in the case of infinite-dimensional problems, as studied in this article, the direct application of finite-dimensional theory, as given in Grossmann and Zadlo [A general class of penalty/barrier path-following Newton methods for nonlinear programming, Optimization 54 (2005), pp. 161–190], would lead to mesh-dependent order one estimates that deteriorate if the discretization is refined. In this article a first rigorous proof is given for inequality constrained problems that in the case of quadratic penalties a mesh-independence principle holds, i.e. the first-order convergence estimate holds for the continuous problem as well as for discretized problems independently of the discretization step size. The penalty techniques rest upon the control approximate reduction as discussed, e.g. in Grossmann et al. [C. Grossmann, H. Kunz, and R. Meischner, Elliptic control by penalty techniques with control reduction, in System Modeling and Optimization, IFIP Advances in Information and Communication Technology, Vol. 312, Springer, Berlin, 2009, pp. 251–267; M. Hinze, A variational discretization concept in control constrained optimization: the linear-quadratic case, Comput. Optim. Appl. 30 (2005), pp. 45–61]. For the discretization conforming linear element discretization is applied. Some numerical examples illustrate and confirm the theoretical results.

A Mesh-Independence Principle for Quadratic Penalties Applied to Semilinear Elliptic Boundary Control

The quadratic loss penalty is a well known technique for opti- mization and control problems to treat constraints. In the present paper they are applied to handle control bounds in a boundary control problems with semilinear elliptic state equations. Unlike in the case of finite dimensional optimization for infinite dimensional problems the order of convergence could only be roughly estimated, but numerical experiments revealed a clearly better convergence behavior with constants independent of the dimension of the used discretization. The main result in the present paper is the proof of sharp con- vergence bounds for both, the finite und infinite dimensional problem with a mesh-independence in case of the discretization. Further, to achieve an efficient realization of penalty methods the principle of control reduction is applied, i.e. the control variable is represented by the adjoint state variable by means of some nonlinear function. The resulting optimality system this way depends only on the state and adjoint state. This system is discretized by conforming linear finite elements. Numerical experiments show exactly the theoretically predicted behavior of the studied penalty technique.

Characterizing Mesh Independent Quadratic Convergence of Newton's Method for a Class of Elliptic Problems

The classical mesh independence principle (MIP) describes a desirable property for Newton's method, its main feature being that the discrete iterations exhibit the same quadratic convergence behavior for any mesh size, i.e., uniformly as the mesh is refined. We study the latter property for a general class of second order nonlinear elliptic boundary value problems solved by finite element discretization. For this, a more specific principle, the mesh independence principle for quadratic convergence (MIPQC), is introduced. It is proved that the MIPQC holds if and only if the elliptic equation is semilinear.

Mesh-independence of semismooth Newton methods for Lavrentiev-regularized state constrained nonlinear optimal control problems

A class of nonlinear elliptic optimal control problems with mixed control-state constraints arising, e.g., in Lavrentiev-type regularized state constrained optimal control is considered. Based on its first order necessary optimality conditions, a semismooth Newton method is proposed and its fast local convergence in function space as well as a mesh-independence principle for appropriate discretizations are proved. The paper ends by a numerical verification of the theoretical results including a study of the algorithm in the case of vanishing Lavrentiev-parameter. The latter process is realized numerically by a combination of a nested iteration concept and an extrapolation technique for the state with respect to the Lavrentiev-parameter.

On Finite Element Methods for Fully Nonlinear Elliptic Equations of Second Order

For the first time, we present for the general case of fully nonlinear elliptic differential equations of second order a nonstandard $C^1$ finite element method (FEM). We consider, throughout the paper, two cases in parallel: For convex, bounded, polyhedral domains in ${\bf R}^n$, or for $C^2$ bounded domains in ${\bf R}^2$, we prove stability and convergence for the corresponding conforming or nonconforming $C^1$ FEM, respectively. The results for equations and systems of orders 2 and $2m$ and quadrature approximations appear elsewhere. The classical theory of discretization methods is applied to the differential operator or the combined differential and the boundary operator. The consistency error for satisfied or violated boundary conditions on polyhedral or curved domains has to be estimated. The stability has to be proved in an unusual way. This is the hard core of the paper. Essential tools are linearization, a compactness argument, the interplay between the weak and strong form of the linearized operator, and a new regularity result for solutions of finite element equations. An essential basis for our proofs are Davydov's results for $C^1$ FEs on polyhedral domains in ${\bf R}^n$ or of local degree 5 for $C^2$ domains in ${\bf R}^2$. Better convergence and extensions to ${\bf R}^n$ for $C^2$ domains are to be expected from his forthcoming results on curved domains. Our proof for the second case in ${\bf R}^n$, includes the first essentially as a special case. The method applies to quasi-linear elliptic problems not in divergence form as well. A discrete Newton method is shown to converge locally quadratically, essentially independently of the actual grid size by the mesh independence principle.

A refined Newton’s mesh independence principle for a class of optimal shape design problems

Shape optimization is described by finding the geometry of a structure which is optimal in the sense of a minimized cost function with respect to certain constraints. A Newton’s mesh independence principle was very efficiently used to solve a certain class of optimal design problems in [6]. Here motivated by optimization considerations we show that under the same computational cost an even finer mesh independence principle can be given.

An error analysis and the mesh independence principle for a nonlinear collocation problem

AbstractA nonlinear Dirichlet boundary value problem is approximated by an orthogonal spline collocation scheme using piecewise Hermite bicubic functions. Existence, local uniqueness, and error analysis of the collocation solution and convergence of Newton's method are studied. The mesh independence principle for the collocation problem is proved and used to develop an efficient multilevel solution method. Simple techniques are applied for estimating certain discretization and iteration constants that are used in the formulation of a mesh refinement strategy and an efficient multilevel method. Several mesh refinement strategies for solving a test problem are compared numerically. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006

Mesh-Independence of Lagrange-SQP Methods with Lipschitz-Continuous Lagrange Multiplier Updates

An augmented Lagrange-SQP method with Lipschitz continuous Lagrange multiplier updates is analyzed. The approximation of this method is investigated, and convergence results are presented. A mesh-independence principle for the augmented Lagrange-SQP method is proved, which asserts that asymptotically the infinite-dimensional algorithm and finite-dimensional discretizations have the same convergence property. More precisely, for sufficiently small mesh-size there is at most a difference of one iteration step between the number of steps required by the infinite-dimensional method and its discretization to converge within a given tolerance l > 0. The theoretical results are demonstrated numerically by a parameter identification problem.

Mesh-Independence of the Lagrange–Newton Method for Nonlinear Optimal Control Problems and their Discretizations

In a recent paper we proved a mesh-independence principle for Newton's method applied to stable and consistent discretizations of generalized equations. In this paper we introduce a new consistency condition which is easier to check in applications. Using this new condition we show that the mesh-independence principle holds for the Lagrange–Newton method applied to nonlinear optimal control problems with mixed control-state constraints and their discretizations by Euler's method or Ritz type methods.

A mesh independence principle for perturbed Newton-like methods and their discretizations

In this manuscript we study perturbed Newton-like methods for the solution of nonlinear operator equations in a Banach space and their discretized versions in connection with the mesh independence principle. This principle asserts that the behavior of the discretized process is asymptotically the same as that for the original iteration and consequently, the number of steps required by the two processes to converge to within a given tolerance is essentially the same. So far this result has been proved by others using Newton’s method for certain classes of boundary value problems and even more generally by considering a Lipschitz uniform discretization. In some of our earlier papers we extend these results to include Newton-like methods under more general conditions. However, all previous results assume that the iterates can be computed exactly. This is not true in general. That is why we use perturbed Newton-like methods and even more general conditions. Our results, on the one hand, extend, and on the other hand, make more practical and applicable all previous results.

Mesh-Independence for an Augmented Lagrangian-SQP Method in Hilbert Spaces

An augmented Lagrangian-SQP algorithm for optimal control of differential equations in Hilbert spaces is analyzed. This algorithm has second-order convergence rate provided that a second-order sufficient optimality condition is satisfied. The internal approximation of this method is investigated, and convergence results are presented. A mesh-independence principle for the augmented Lagrangian-SQP method is proved, which asserts that asymptotically the infinite dimensional algorithm and finite dimensional discretizations have the same convergence property. More precisely, for sufficiently small mesh-size there is at most a difference of one iteration step between the number of steps required by the infinite dimensional method and its discretization to converge within a given tolerance $\varepsilon >0$. The theoretical results are demonstrated by two optimal control problems for the Burgers equation.

Newton's Mesh Independence Principle for a Class Of Optimal Shape Design Problems

Many optimal shape design problems can be stated as infinite dimensional minimization problems. For deriving an implementable algorithm it has to be decided either to discretize the problem and to use a finite algorithm to solve the discrete problem or to state an algorithm in function space and to discretize this algorithm. This issue has yet to be addressed in the field of optimal shape design research. One big advantage for the latter procedure is a mesh independence behavior as it has been proven by Allgower et al. [ SIAM J. Numer. Anal., 23 (1986), pp. 160--169]. Since their assertions are not directly applicable to these specific kinds of problems, a modified version of their mesh independence principle is given here in order to derive more efficient algorithms for the resulting large scale problems.

On a new Newton-Mysovskii-type theorem with applications to inexact Newton-like methods and their discretizations

In this manuscript we first introduce a new Newton-Mysovskii-type theorem to study the convergence of inexact Newton-like methods to a locally unique solution of a nonlinear operator equation with a nondifferentiable term. We then study their discretized versions in connection with the mesh independence principle. This principle asserts that the behaviour of the discretized process is asymptotically the same as that for the original iteration and consequently, the number of steps required by the two processes to converge to within a given tolerance is essentially the same. So far this result has been proved by others using Newton's method for certain classes of boundary value problems and even more generally by considering a Lipschitz uniform discretization. In some of our earlier papers we extended these results to include Newton-like methods under more general conditions. However, all previous results assume that the iterates can be computed exactly. This is not true in general. That is why we use perturbed Newton-like methods and even more general conditions. Our results, on the one hand, extend and on the other hand, make more practical and applicable all previous results. In particular, we solve a nonlinear integral equation appearing in radioactive transfer.

Mesh independence principle for nonlinear equations on hilbert spaces by preconditioning

In the paper [2] the author prove that, under certain assumptions, the Mesh Independence Principle is satisfied for a class of nonlinear operator equations (on Banach spaces) and their discretizations. In this paper we show that, under few natural assumptions, the same results can be obtained for a class of nonlinear operator equations on Hilbert spaces and a family of preconditioned finite element discretizations of them. We also obtain (using a different proof technique) a generalization of some similar results from the paper [1]. In the last section of the paper we present numerical examples which confirm our theoretical results from the previous ones.

A mesh independence principle for inexact Newton-like methods and their discretizations under generalized Lipschitz conditions

In this manuscript, we study inexact Newton-like methods for the solution of nonlinear operator equations in a Banach space involving a nondifferentiable term, and their discretized versions in connection with the mesh independence principle. This principle asserts that the behavior of the discretized process is asymptotically the same as that for the original iteration and consequently, the number of steps required by the two processes to converge to within a given tolerance is essentially the same. So far this result has been proved by others using Newton's method for certain classes of boundary value problems and even more generally by considering a Lipschitz uniform (or not) discretization. In some of our earlier papers we extended these results to include Newton-like methods under more general conditions. However, all previous results assume that the iterates can be computed exactly. This is not true in general. That is why we use inexact Newton-like methods and even more general conditions. Our results, on the one hand, extend, and on the other hand, make more practical and applicable previous results. Moreover, we can solve a wider range of problems and find sharper error bounds. Several applications of our results to two point boundary value problems are illustrated.

A mesh-independence principle for operators equations and the Steffensen method

In this study we prove the mesh-independence principle via Steffensen’s method. This principle asserts that when Steffensen’s method is applied to a nonlinear equation between some Banach spaces, as well as to some finite-dimensional discretization of that equation, then the behavior of the discretized process is asymptotically the same as that for the original iteration. Local and semilocal convergence results as well as an error analysis for Steffensen’s method are also provided.

An extension of the mesh independence principle for operator equations in Banach space

In this study, we extend the validity of the mesh-independence principle to include perturbed Newton-like methods in Banach space. We show that when a perturbed Newton-like method is applied to a nonlinear equation as well as to some finite-dimensional discretization of that equation, there is a difference of at most one between the number of steps required by the two processes to converge to within a given tolerance. Our results cover and extend earlier results.