Problem reduction, renormalization, and memory

TL;DR

This paper explores methods to simplify complex computational problems by connecting problem reduction techniques with concepts from renormalization and statistical mechanics, highlighting the importance of memory and forcing terms in time-dependent cases.

Contribution

It introduces new approaches for problem reduction linked to renormalization and discusses the role of memory and forcing in time-dependent problem analysis.

Findings

Averaging often fails for time-dependent problems.

Memory and random forcing terms are essential in averaged equations.

Examples demonstrate the effectiveness of the proposed methods.

Abstract

Methods for the reduction of the complexity of computational problems are presented, as well as their connections to renormalization, scaling, and irreversible statistical mechanics. Several statistically stationary cases are analyzed; for time dependent problems averaging usually fails, and averaged equations must be augmented by appropriate memory and random forcing terms. Approximations are described and examples are given.