[Paper Review] The minimax approach to the estimation of solutions to first order linear systems of ordinary differential periodic equations with inexact data

A. G. Nakonechny, Yuri Podlipenko|arXiv (Cornell University)|Oct 16, 2018

Statistical and numerical algorithms5 references4 citations

TL;DR

This paper develops a minimax estimation approach for first-order linear systems of periodic ordinary differential equations with inexact data, where right-hand sides are subject to quadratic constraints. By reducing the estimation problem to an optimal control problem, it derives explicit expressions for minimax estimates of solutions and linear functionals through the solution of uniquely solvable periodic ODE systems.

ABSTRACT

We consider first-order linear systems of ordinary differential equations with periodic coefficients. Supposing that right-hand sides of equations are not known and subjected to some quadratic restrictions, we obtain optimal, in certain sense, estimates of solutions to above-mentioned problems from indirect noisy observations of these solutions on a finite system of points and intervals.

Motivation & Objective

• Address the challenge of estimating solutions to first-order linear periodic ODEs when right-hand sides are uncertain and constrained by quadratic bounds.
• Develop guaranteed (minimax) estimates for solutions and linear functionals from indirect noisy observations on finite intervals and points.
• Establish conditions under which the estimation problem reduces to solving uniquely solvable systems of periodic ODEs.
• Provide explicit analytical expressions for minimax estimates and estimation errors using solutions of adjoint and primal periodic systems.

Proposed method

• Formulate the estimation problem as a minimax optimization: minimize the worst-case mean square error over all admissible perturbations within quadratic constraints.
• Reduce the minimax estimation problem to an optimal control problem via duality principles, leveraging adjoint systems.
• Derive the adjoint system of periodic ODEs with impulse conditions at observation points to characterize the optimal estimate.
• Express the minimax estimate of the solution as a linear functional of observations, with coefficients determined by solutions of a system of periodic ODEs.
• Construct a system of linear algebraic equations for the values of the costate function at observation points, enabling numerical computation of the estimate.
• Use fundamental matrix solutions and transition matrices (e.g., $X(t)$, $Z(t) = [X^*(t)]^{-1}$) to derive explicit formulas for the estimation error and the estimate itself.

Experimental results

Research questions

• RQ1 How can one obtain minimax estimates for solutions of periodic linear ODEs when the right-hand side is only known to satisfy quadratic constraints?
• RQ2 What is the structure of the optimal linear estimate under indirect noisy observations of the solution at discrete points and intervals?
• RQ3 How can the minimax estimation problem be transformed into a solvable optimal control problem?
• RQ4 What conditions ensure the unique solvability of the resulting system of periodic ODEs for the estimation process?
• RQ5 How are the estimation error and the optimal estimate explicitly expressed in terms of the system's fundamental solutions?

Key findings

• The minimax estimate of the solution is explicitly expressed as a linear combination of observations, with coefficients derived from solutions of a system of periodic ODEs.
• The estimation error is determined by the solution of a unique periodic ODE system, ensuring stability and optimality under worst-case perturbations.
• The values of the costate function at observation points satisfy a system of linear algebraic equations, solvable under the condition $\det(E_n - X(T)) \neq 0$.
• The optimal estimate and estimation error are expressed in closed form using the fundamental matrix $X(t)$ and its adjoint $Z(t) = [X^*(t)]^{-1}$.
• The method yields a minimax Kalman–Bucy-type filter for periodic systems under quadratic uncertainty constraints.
• The approach is validated through the derivation of explicit formulas for $p(t)$, $\hat{z}(t)$, and $\hat{x}(t)$, linking the estimate to the solution of a dual periodic system with impulse conditions.