Improved Approximation Bounds for Moore-Penrose Inverses of Banded Matrices with Applications to Continuous-Time Linear Quadratic Control
Sungho Shin, Wallace Gian Yion Tan, Mihai Anitescu
TL;DR
This work introduces geometric, exponentially decaying bounds for approximating Moore-Penrose inverses of banded matrices indexed by finite metric spaces. By generalizing bandedness to arbitrary metrics and employing odd polynomial (via Chebyshev) approximations, the authors derive κ-banded approximations $ ilde{A}$ of $A^+$ with error $ rm{A^+- ilde{A}}_2$ that decays exponentially in the bandwidth ratio $ frac{ ext{κ}}{ar{arkappa}}$, with tighter bounds in the PSD case. They compare against prior results, showing improved decay rates and constants, and extend the theory to indefinite, rectangular cases. The results are then applied to continuous-time linear-quadratic control, establishing uniform, mesh-size-invariant perturbation bounds and proving that the solution mapping decays exponentially in time; the discretized problem yields a banded structure that remains well-behaved as the mesh refines, enabling a rigorous link between discrete and continuous domains. Overall, the paper provides a principled framework for exploiting exponential locality in large-scale, discretized optimization and control problems, with implications for PDE-constrained and network-structured systems.
Abstract
We present improved approximation bounds for the Moore-Penrose inverses of banded matrices, where the bandedness is induced by a metric on the index set. We show that the pseudoinverse of a banded matrix can be approximated by another banded matrix, and the error of approximation is exponentially small in the ratio of the bandwidth of the approximation to that of the original matrix. An intuitive corollary can be obtained: the off-diagonal blocks of the pseudoinverse decay exponentially with the distance between the node sets associated with row and column indices, on the given metric space. Our bounds are expressed in terms of the bound of singular values of the system. For saddle point systems, commonly encountered in optimization, we provide the bounds of singular values associated under standard regularity conditions. Remarkably, our bounds improve previously reported ones and allow us to establish a perturbation bound for continuous-domain optimal control problems by analyzing the asymptotic limit of their finite difference discretization, which has been challenging with previously reported bounds.