Comments on Truncation Errors for Polynomial Chaos Expansions

TL;DR

This paper analyzes the errors introduced by truncating polynomial chaos expansions, providing bounds and conditions for zero error, with applications in polynomial mappings and predictive control.

Contribution

It derives explicit error bounds for polynomial chaos truncation and investigates conditions for zero approximation errors, including for convex quadratic programs.

Findings

Error bounds for polynomial chaos approximations established

Conditions for achieving zero truncation error identified

Simulation examples demonstrate theoretical results

Abstract

Methods based on polynomial chaos expansion allow to approximate the behavior of systems with uncertain parameters by deterministic dynamics. These methods are used in a wide range of applications, spanning from simulation of uncertain systems to estimation and control. For practical purposes the exploited spectral series expansion is typically truncated to allow for efficient computation, which leads to approximation errors. Despite the Hilbert space nature of polynomial chaos, there are only a few results in the literature that explicitly discuss and quantify these approximation errors. This work derives error bounds for polynomial chaos approximations of polynomial and non-polynomial mappings. Sufficient conditions are established, which allow investigating the question whether zero truncation errors can be achieved and which series order is required to achieve this. Furthermore, convex quadratic programs, whose argmin operator is a special case of a piecewise polynomial mapping, are studied due to their relevance in predictive control. Several simulation examples illustrate our findings.