[Paper Review] A boundary value problem of a generalised linear discrete time system with no solutions and infinitely many solutions

Nicholas Apostolopoulos, Fernando Ortega+1 more|arXiv (Cornell University)|Oct 26, 2016

Differential Equations and Numerical Methods25 references3 citations

TL;DR

This paper proposes explicit, optimally regularized solutions for non-consistent boundary value problems (BVPs) in generalized linear discrete time systems with singular leading coefficient matrices. Using matrix pencil theory and Tikhonov-type regularization, it derives closed-form formulas for optimal solutions when the BVP has no solution or infinitely many solutions, with numerical examples validating the approach under rank-deficient or inconsistent systems.

ABSTRACT

In this article we study a class of generalised linear systems of difference equations with given boundary conditions and assume that the boundary value problem is non-consistent, i.e. it has infinite many or no solutions. We take into consideration the case that the coefficients are square constant matrices with the leading coefficient singular and provide optimal solutions. Numerical examples are given to justify our theory.

Motivation & Objective

• To address boundary value problems (BVPs) in generalized linear discrete time systems where the leading coefficient matrix is singular and the system may have no solution or infinitely many solutions.
• To provide a systematic method for computing optimal solutions in non-consistent BVPs, where traditional solution methods fail.
• To extend existing theory on generalized linear systems by incorporating regularization techniques for ill-posed BVPs.
• To derive explicit, computationally testable formulas for optimal solutions under various consistency conditions.
• To validate the theoretical framework with numerical examples demonstrating the method's robustness under rank-deficient or inconsistent systems.

Proposed method

• Utilizes matrix pencil theory to analyze the regularity of the system $ sF - G $, decomposing it into Weierstrass canonical form via non-singular transformations $ P $ and $ Q $.
• Employs the Weierstrass form to express the general solution of the system as $ Y_k = Q_p J_p^k C $, where $ C $ is a constant vector to be determined from boundary conditions.
• Defines the matrix $ K $ and vector $ L $ from boundary conditions to form the linear system $ K C = L $, which determines solution existence and consistency.
• Applies Tikhonov regularization via $ (K^*K + E^*E)^{-1}K^*L $ when $ K $ is rank-deficient and $ L \notin \text{col}(K) $, ensuring a unique, stable solution.
• Uses the Moore-Penrose inverse-like formulation $ (K^*K)^{-1}K^*L $ when $ K $ is full rank and $ L \notin \text{col}(K) $, minimizing the residual $ \|L - K\hat{C}\|_2^2 $.
• Introduces a small regularization parameter $ \theta $ via matrix $ E $ with $ \|E\|_2 = \theta \ll 1 $ to stabilize the inverse in rank-deficient cases.

Experimental results

Research questions

• RQ1 Under what conditions does a generalized linear discrete time system with a singular leading coefficient matrix admit no solution or infinitely many solutions?
• RQ2 How can an optimal solution be systematically derived when the boundary value problem is non-consistent?
• RQ3 What regularization strategy ensures a unique, stable, and computationally feasible solution in rank-deficient or inconsistent BVPs?
• RQ4 How can the solution be expressed in a closed-form formula that is both explicit and easily testable?
• RQ5 What role does matrix pencil theory play in characterizing the structure of solutions in such generalized systems?

Key findings

• For a non-consistent BVP with $ p < r_1 + r_2 $, $ \text{rank}(K) = p $, and $ L \notin \text{col}(K) $, the optimal solution is given by $ \hat{Y}_k = Q_p J_p^k (K^*K)^{-1}K^*L $.
• When $ K $ is rank-deficient and $ L \notin \text{col}(K) $, the optimal solution is $ \hat{Y}_k = Q_p J_p^k (K^*K + E^*E)^{-1}K^*L $, with $ \|E\|_2 = \theta \ll 1 $, ensuring invertibility.
• In Example 4.1, with $ \theta = 0.00001 $, the optimal solution converges to $ \hat{Y}_k = \left[0, 0, 0, \frac{11}{14 \cdot 4^k}, \frac{11}{14 \cdot 4^k}\right]^T $, demonstrating stability under regularization.
• In Example 4.2, with full-rank $ K $, the optimal solution is $ \hat{Y}_k = \left[0.0349 \cdot 2^{-k}, 0, 0, 0.5006 \cdot 4^{-k}, 0.5006 \cdot 4^{-k}\right]^T $, showing accurate residual minimization.
• The method successfully handles both cases of non-consistency: no solution and infinite solutions, providing a unified regularization framework.
• Theoretical results are validated numerically, confirming that the derived formulas yield stable, optimal solutions even when the original system is ill-posed.