Geometric Versus Arithmetic Difference and $H^1$-Seminorm Versus $L^2$-Norm in Function Approximation

Abstract

In solving function approximation problems, sometimes, it is necessary to ensure non-negativity of the result and approximately uniform relative deviation from the target. In these cases, it is appropriate to use the geometric difference to define distance. On the other hand, sometimes, it is more important that the result looks like the target, rather than being close to it in Euclidean sense. Then, to define distance, it is more appropriate to use the $H^1$-seminorm rather than the $L^2$-norm. This paper considers four possible metrics (distance functions): (i) $L^2$-norm of the arithmetic difference, (ii) $L^2$-norm of the geometric difference, (iii) $H^1$-seminorm of the arithmetic difference, and (iv) $H^1$-seminorm of the geometric difference. Using the four different metrics, we solve the following function approximation problem: given a target function and a set of linear equality constraints, find the function that satisfies the constraints and minimizes the distance, in the considered metric, to the target function. To solve the problem, we convert it to a finite dimensional constrained optimization problem by discretizing it. Then, the method of Lagrange multipliers is applied. The obtained systems for case (i) and (iii) are linear and are solved exactly. The systems for case (ii) and (iv), however, are nonlinear. To solve them, we propose a self-consistent iterative procedure which is a combination of fixed-point iteration and Newton method. Particular examples are solved and discussed.