On the estimation of the Mori-Zwanzig memory integral

Abstract

We develop rigorous estimates and provably convergent approximations for the memory integral in the Mori-Zwanzig (MZ) formulation. The new theory is built upon rigorous mathematical foundations and is presented for both state-space and probability density function space formulations of the MZ equation. In particular, we derive errors bounds and sufficient convergence conditions for short-memory approximations, the $t$-model, and hierarchical (finite-memory) approximations. In addition, we derive computable upper bounds for the MZ memory integral, which allow us to estimate (a priori) the contribution of the MZ memory to the dynamics. Numerical examples demonstrating convergence of the proposed algorithms are presented for linear and nonlinear dynamical systems evolving from random initial states.

arXiv comment42 pages, 7 figures

References in corpus (7)

• Extensive Chaos in the Lorenz-96 Model
• Data-driven parameterization of the generalized Langevin equation
• Data-driven closures for stochastic dynamical systems
• A comparative study of two stochastic mode reduction methods
• The Numerical Approximation of Nonlinear Functionals and Functional Differential Equations
• Optimal Prediction for Hamiltonian partial differential equations
• Duality and Conditional Expectations in the Nakajima-Mori-Zwanzig Formulation

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