Defect correction and a posteriori error estimation of Petrov-Galerkin methods for nonlinear Volterra integro-differential equations

We present two defect correction schemes to accelerate the Petrov-Galerkin finite element methods [19] for nonlinear Volterra integro-differential equations. Using asymptotic expansions of the errors, we show that the defect correction schemes can yield higher order approximations to either the exact solution or its derivative. One of these schemes even does not impose any extra regularity requirement on the exact solution. As by-products, all of these higher order numerical methods can also be used to form a posteriori error estimators for accessing actual errors of the Petrov-Galerkin finite element solutions. Numerical examples are also provided to illustrate the theoretical results obtained in this paper.     

Estimation of a Bivariate Extreme Value Distribution

Several threshold methods have been proposed for the purpose of estimating a bivariate extreme value distribution from a sample of data whose distribution is only in its domain of attraction. An integrated view of these methods is presented which leads to the introduction of a new asymptotically consistent estimator of the dependence function characterizing the extreme dependence structure. Through Monte Carlo simulations, the new estimator is also shown to do as well as its competitors and to outperform them in cases of weak dependence. To the authors' knowledge, this is the first time that the small-sample behavior of nonparametric bivariate threshold methods has ever been investigated.     

On the iterative solution of the algebraic equations in fully implicit Runge-Kutta methods

This paper deals with the iterative solution of stage equations which arise when some fully implicit Runge-Kutta methods, in particular those based on Gauss, Radau and Lobatto points, are applied to stiff ordinary differential equations. The error behaviour in the iterates generated by Newton-type and, particularly, by single-Newton schemes which are proposed for the solution of stage equations is studied. We consider stiff systems y'(t) = f(t,y(t)) which are dissipative with respect to a scalar product and satisfy a condition on the relative variation of the Jacobian of f(t,y) with respect to y, similar to the condition considered by van Dorsselaer and Spijker in [7] and [17]. We prove new convergence results for the single-Newton iteration and derive estimates of the iteration error that are independent of the stiffness. Finally, some numerical experiments which confirm the theoretical results are presented. This revised version was published online in June 2006 with corrections to the Cover Date.     

Diagonally implicit Runge–Kutta methods for 3D shallow water applications

We construct A‐stable and L‐stable diagonally implicit Runge–Kutta methods of which the diagonal vector in the Butcher matrix has a minimal maximum norm. If the implicit Runge–Kutta relations are iteratively solved by means of the approximately factorized Newton process, then such iterated Runge–Kutta methods are suitable methods for integrating shallow water problems in the sense that the stability boundary is relatively large and that the usually quite fine vertical resolution of the discretized spatial domain is not involved in the stability condition. This revised version was published online in June 2006 with corrections to the Cover Date.     

SQP-methods for solving optimal control problems with control and state constraints: adjoint variables, sensitivity analysis and real-time control

Parametric nonlinear optimal control problems subject to control and state constraints are studied. Two discretization methods are discussed that transcribe optimal control problems into nonlinear programming problems for which SQP-methods provide efficient solution methods. It is shown that SQP-methods can be used also for a check of second-order sufficient conditions and for a postoptimal calculation of adjoint variables. In addition, SQP-methods lead to a robust computation of sensitivity differentials of optimal solutions with respect to perturbation parameters. Numerical sensitivity analysis is the basis for real-time control approximations of perturbed solutions which are obtained by evaluating a first-order Taylor expansion with respect to the parameter. The proposed numerical methods are illustrated by the optimal control of a low-thrust satellite transfer to geosynchronous orbit and a complex control problem from aquanautics. The examples illustrate the robustness, accuracy and efficiency of the proposed numerical algorithms.     

Symmetric Projection Methods for Differential Equations on Manifolds

Projection methods are a standard approach for the numerical solution of differential equations on manifolds. It is known that geometric properties (such as symplecticity or reversibility) are usually destroyed by such a discretization, even when the basic method is symplectic or symmetric. In this article, we introduce a new kind of projection methods, which allows us to recover the time-reversibility, an important property for long-time integrations.     

Kernel polynomials for the solution of indefinite and ill-posed problems

We introduce a new family of semiiterative schemes for the solution of ill-posed linear equations with selfadjoint and indefinite operators. These schemes avoid the normal equation system and thus benefit directly from the structure of the problem. As input our method requires an enclosing interval of the spectrum of the indefinite operator, based on some a priori knowledge. In particular, for positive operators the schemes are mathematically equivalent to the so-called -methods of Brakhage. In a way, they can therefore be seen as appropriate extensions of the -methods to the indefinite case. This extension is achieved by substituting the orthogonal polynomials employed by Brakhage in the definition of the -methods by appropriate kernel polynomials. We determine the rate of convergence of the new methods and establish their regularizing properties.     

Second-Derivative-Free Variant of the Chebyshev Method for Nonlinear Equations

In this paper, we introduce a numerical method for nonlinear equations, based on the Chebyshev third-order method, in which the second-derivative operator is replaced by a finite difference between first derivatives. We prove a semilocal convergence theorem which guarantees local convergence with R-order three under conditions similar to those of the Newton-Kantorovich theorem, assuming the Lipschitz continuity of the second derivative. In a subsequent theorem, the latter condition is replaced by the weaker assumption of Lipschitz continuity of the first derivative.     

A Quasi-Newton Quadratic Penalty Method for Minimization Subject to Nonlinear Equality Constraints

We present a modified quadratic penalty function method for equality constrained optimization problems. The pivotal feature of our algorithm is that at every iterate we invoke a special change of variables to improve the ability of the algorithm to follow the constraint level sets. This change of variables gives rise to a suitable block diagonal approximation to the Hessian which is then used to construct a quasi-Newton method. We show that the complete algorithm is globally convergent. Preliminary computational results are reported.     

Exploiting additional structure in equality constrained optimization by structured SQP secant algorithms

In 1988, Tapia (Ref. 1) developed and analyzed SQP secant methods in equality constrained optimization taking explicitly the additive structure of the problem setting into account. In this paper, we extend Tapia's augmented scale Lagrangian secant method to the case where additional structure coming from the objective function is available. Using the example of nonlinear least squares with equality constraints, we demonstrate these ideas and develop a convergence theory proving local and q-superlinear convergence for this kind of structured SQP-algorithms.This research was supported by the Studienstiftung des Deutschen Volkes.