O(h² ) Global Pointwise Algorithms for Delay Quadratic Problems in the Calculus of Variations

Abstract

The main purpose of this paper is to give numerical algorithms and the error analysis for delay quadratic problems in the calculus of variations. These methods are new, efficient, and accurate and have a global a priori error of O(h 2 ), where h is the distance between any two successive node points.We also derive the results for the general, numerical, delay problem, but we focus on proving our results in the simpler quadratic case since the extra technical numerical details have been given previously by the second author. In addition, the authors have previously shown how to reformulate general delay constrained problems in optimal control theory/constrained calculus of variations as unconstrained delay problems. Thus our numerical results and methods will hold for these general constrained problems also.Finally, we note that our algorithm, which solves the stationary condition(s) numerically, avoids the more difficult problems of piecing the solution of second order equations together and requires less smoothness in the solution. Thus we replace difficult second order boundary value problems with the easier task of approximating definite integrals involving first order derivatives.

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

“…However, this problem is, usually, too complicated to be solved analytically, and even in simple cases it is equivalent to the problem of solving a linear partial differential equation (see, e.g., [42], Chapter 9). In order to overcome this difficulty, let us transform problem (2). For the sake of simplicity, suppose that n = 2, ψ ≡ 0, and let Ω be an open box, i.e.…”

“…Since we know the direction of steepest descent for problem (5), (6), we can apply an obvious modification of almost any gradient-based algorithm of finite dimensional optimization to this problem and, in turn, to the initial problem (2).…”

“…Various types of reduction techniques and corresponding numerical methods in the one dimensional case (i.e. in the case when Ω = (a, b) ⊂ R) were proposed in [9,26,28,27,33,56,57,19,54,2,32,3]. In the multidimensional case, the range of choice of direct methods is much more narrow, and it includes (although is not exhausted by) the finite elements methods, the Galerkin method and the Ritz method [35,48,30,8,7,44,43,5].…”

New efficient numerical procedures for solving stochastic variational problems with a priori maximum pointwise error estimates

“…These results have also been partially applied to problems in partial differential equations and to achieve a "complete" solution for delay problems in the deterministic case [1,2].…”