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Markovian Embedding of Nonlinear Memory via Spectral Representation

Divya Jaganathan, Rahil N. Valani

TL;DR

The paper addresses non-Markovian dynamics with nonlinear memory by introducing an exact spectral representation to embed the memory into an extended, Markovian framework. It constructs an explicit local-in-time reformulation in an augmented space using a history function H(k,t) and demonstrates the method on two one-dimensional models: walking droplets and the Stefan problem. Numerical results validate that the embedded system reproduces known behaviors, including steady-state speeds and moving-front dynamics, while offering a time-independent computational cost after spectral truncation. The approach's efficiency depends on the model's spectral demands, with Stefan requiring many modes and walking droplets fewer, illustrating the method's versatility for memory-dependent systems.

Abstract

Differential equations containing memory terms that depend nonlinearly on past states model a variety of non-Markovian processes. In this study, we present a Markovian embedding procedure for such equations with distributed delay by utilising an exact spectral representation of the nonlinear memory function. This allows us to transform the nonlocal system to an equivalent local-in-time system in an abstract extended space. We demonstrate our embedding procedure for two one-dimensional physical models: (i) the walking droplet and (ii) the single-phase Stefan problem. In addition to providing an alternative representation of the underlying physical system, the local representation finds applications in designing efficient time-integrators with time-independent computational costs for memory-dependent systems which typically suffer from growing-in-time costs.

Markovian Embedding of Nonlinear Memory via Spectral Representation

TL;DR

The paper addresses non-Markovian dynamics with nonlinear memory by introducing an exact spectral representation to embed the memory into an extended, Markovian framework. It constructs an explicit local-in-time reformulation in an augmented space using a history function H(k,t) and demonstrates the method on two one-dimensional models: walking droplets and the Stefan problem. Numerical results validate that the embedded system reproduces known behaviors, including steady-state speeds and moving-front dynamics, while offering a time-independent computational cost after spectral truncation. The approach's efficiency depends on the model's spectral demands, with Stefan requiring many modes and walking droplets fewer, illustrating the method's versatility for memory-dependent systems.

Abstract

Differential equations containing memory terms that depend nonlinearly on past states model a variety of non-Markovian processes. In this study, we present a Markovian embedding procedure for such equations with distributed delay by utilising an exact spectral representation of the nonlinear memory function. This allows us to transform the nonlocal system to an equivalent local-in-time system in an abstract extended space. We demonstrate our embedding procedure for two one-dimensional physical models: (i) the walking droplet and (ii) the single-phase Stefan problem. In addition to providing an alternative representation of the underlying physical system, the local representation finds applications in designing efficient time-integrators with time-independent computational costs for memory-dependent systems which typically suffer from growing-in-time costs.
Paper Structure (8 sections, 22 equations, 4 figures)

This paper contains 8 sections, 22 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Spectral structure and Markovian embedding
  3. Prototypical physical models with nonlinear memory effects
  4. Walking droplets
  5. Markovian embedding for the stroboscopic model
  6. Single-phase Stefan problem
  7. Markovian embedding for Stefan problem
  8. Conclusions

Figures (4)

  • Figure 1: (a) Schematic of a 1D walking droplet. Typical known droplet states in the stroboscopic model obtained by solving the Markovian system (\ref{['eq:markovianSWD']}) for $(x_{d0}, \dot{x}_{d0})$ = $(1,1)$: (b) Non-walker$(C_1=0.01, C_2=0.1)$, (c) Steady walker $(C_1=C_2=0.1)$, (d) Chaotic walker $(C_1=1.5,C_2=0.01)$. Velocity in (c) is scaled by a factor of $20$ for visibility.
  • Figure 2: Evolution of real (red)/imaginary (blue) parts of the history function $H(k,t)$ in the spectral space at representative times for a steady walker $(C_1=C_2=0.1)$, with $H(k,0)=0$. The finite support of $H(k,t)$ in $[-1,1]$ and its smoothness demands a nominal, fixed (in time) requirement of as few as $M=30$ Chebyshev quadrature nodes to accurately compute the history integral.
  • Figure 3: (a) Schematic of 1D single-phase Stefan problem. (b) Numerical response of the melting front to a constant temperature forcing $\theta(0,t)=f(t)=1$ at the fixed end $x=0$ overlaid on the exact solution along with (c) the instantaneous pointwise error.
  • Figure 4: Evolution of real (red)/imaginary (blue) parts of the history function $H(k,t)$ in the truncated $k-$domain $k \in [0,500]$ at different time instances for the single-phase Stefan problem subject to constant temperature forcing $(f(t)=1)$ at the fixed end $x=0$. $M=2000$ Chebyshev nodes were used to accurately compute the integral of the highly oscillatory history function.