Small Errors Imply Large Instabilities

TL;DR

This paper demonstrates that for many numerical methods, improving accuracy inherently leads to increased instability, especially in function recovery and differential equation solving, highlighting a fundamental trade-off.

Contribution

It establishes a general trade-off principle between error and stability in numerical methods for function recovery and differential equations.

Findings

Unavoidable trade-off between error and stability in many methods

Symmetric collocation is more stable than unsymmetric collocation

Improving accuracy can cause increased instability in numerical techniques

Abstract

Numerical Analysts and scientists working in applications often observe that once they improve their techniques to get a better accuracy, some instability creeps in through the back door. This paper shows for a large class of numerical methods that such a Trade-off Principle between error and stability is unavoidable. The setting is confined to recovery of functions from data, but it includes solving differential equations by writing such methods as a recovery of functions under constraints imposed by differential operators and boundary values. It is shown in particular that Kansa's Unsymmetric Collocation Method sacrifices accuracy for stability, when compared to symmetric collocation.