We develop a new effective approximation of the Mori-Zwanzig equation based on operator series expansions of the orthogonal dynamics propagator. In particular, we study the Faber series, which yields asymptotically optimal approximations converging at least R-superlinearly with the polynomial order for linear dynamical systems. We provide a through theoretical analysis of the new method and present numerical applications to random wave propagation and harmonic chains of oscillators interacting on the Bethe lattice and on graphs with arbitrary topology.

We present a new method to approximate the Mori-Zwanzig (MZ) memory integral in generalized Langevin equations (GLEs) describing the evolution of observables in high-dimensional nonlinear systems with local interactions. Building upon the Faber operator series we recently developed for the orthogonal dynamics propagator, and an exact combinatorial algorithm that allows us to compute memory kernels to any desired accuracy from first principles, we demonstrate that the proposed method is effective in computing auto-correlation functions, intermediate scattering functions and other important statistical properties of the observables. We also develop a new stochastic process representation that combines MZ memory kernels and Karhunen-Loève (KL) series expansions to build reduced-order models of observables in statistical equilibrium. Numerical applications are presented for the Fermi-Pasta-Ulam β-model, and for random wave propagation in homegeneous media.

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.

Built upon the hypoelliptic analysis of the effective Mori-Zwanzig (EMZ) equation for observables of stochastic dynamical systems, we show that the obtained semigroup estimates for the EMZ equation can be used to drive prior estimates of the observable statistics for system in the equilibrium and nonequilibrium state. In addition, we introduce both first-principle and data-driven methods to approximate the EMZ memory kernel, and prove the convergence of the data-driven parametrization schemes using the regularity estimate of the memory kernel. The analysis results are validated numerically via the Monte-Carlo simulation of the Langevin dynamics for a Fermi-Pasta-Ulam chain model. With the same example, we also show the effectiveness of the proposed memory kernel approximation methods.