A fourth-order cubic spline method for linear second-order two-point boundary-value problems

A cubic spline method for linear second-order two-point boundary-valueproblems is analysed. The method is a Petrov-Galerkin methodusing a cubic spline trial space, a piecewise-linear test space,and a simple quadrature rule for the integration, and may beconsidered a discrete version of the H¹-Galerkin method. Themethod is fully discrete, allows an arbitrary mesh, yields alinear system with bandwidth five, and under suitable conditionsis shown to have an 0(h^4– rate of convergence in the W_p¹norm for i = 0, 1, 2, 1p. The H¹-Galerkin method and orthogonalspline collocation with Hermite cubics are also discussed.     

ROBUST MATHEMATICAL PROGRAMMING FOR NATURAL RESOURCE MODELING UNDER PARAMETRIC UNCERTAINTY

Abstract Inaccurate specification of model coefficients can lead to false or distorted findings in modeling investigations of natural resource management. Hence, this paper outlines a decision framework for optimization problems in which only the bounded set of outcomes for uncertain parameters is known. These models can be solved with standard mathematical programming software and are no larger than their deterministic equivalent. The robust approach is contrasted against deterministic analysis and is demonstrated for two applications regarding the management of natural resources. Deterministic plans are infeasible in at least 40% of cases when parameters vary from their point estimates. Inclusion of robust constraints immunizes against this infeasibility, thereby removing errors arising from false certainty. Additionally, incorporation of bounded parameters in the objective function yields interval‐valued sets containing potential outcomes. However, this increase in the general relevance of model output introduces some degree of suboptimality as deterministic plans are buffered to proactively account for potential variability. The cost of robustness increases with the simulated spread of uncertain coefficients but may be reduced through accounting for the uncertainty aversion of decision makers.