Constrained data smoothing via optimal control

The article considers a problem of best smoothing in a strip, where the objective is to find a function f : [0, 1] -> R that satisfies bilateral constraints on its values, d(t) <= f (t) <= e(t) for all 0 <= t <= 1 and minimizes a weighted sum of the L-2-norm of the second derivative and squared deviations from specified values, y(i), at discrete points 0 = t(1) < t(2) < ... < t(N+2). We assume that constraints d(t) and e(t) are continuous functions that are linear in each interval [t(i), t(i+1)], i = 1, ... , N + 1. We connect this problem to a state-constrained optimal control problem for the double integrator, and give conditions for the existence and uniqueness of the solution under which we also show that the solution is a cubic spline with knots at ti and no more than two additional knots in each interval (t(i), t(i+1)). We propose a numerical algorithm for solving this problem based on a two stage minimization, where the outer loop optimization problem is finite-dimensional and convex, while the inner loop optimization problem admits a solution which is easy to compute. Numerical results that show the efficacy of the proposed approach are reported.

Automated summary

This article considers the problem of best smoothing in a strip and proposes a numerical algorithm for solving it. Numerical results are reported to demonstrate the effectiveness of the proposed approach.