A Rellich's result revisited and sensitivity of solutions of parametrized linear systems

TL;DR

This paper revisits Rellich's theorem on the smoothness of solutions to parametrized linear systems, deriving finer results and proposing an efficient adjoint algorithm for sensitivity analysis of nearly deficient systems.

Contribution

It provides new smoothness results for parametrized systems and introduces an efficient adjoint method for sensitivity computation in nearly deficient systems.

Findings

Finer smoothness conditions for solutions of parametrized linear systems.

An efficient adjoint algorithm for sensitivity analysis of (n-1)-deficient systems.

Enhanced understanding of solution behavior near system deficiencies.

Abstract

In this paper we revisit a result due to Franz Rellich on smoothness of solutions of parametrized linear systems. With this result as a starting point, we obtain finer smoothness results in an elementary fashion and propose an efficient adjoint algorithm for computing sensitivities of $(n-1)$-deficient systems, being $n$ the order of the system.