Equation-free modeling

Equation-free modeling is a method for multiscale computation and computer-aided analysis. It is designed for a class of complicated systems in which one observes evolution at a macroscopic, coarse scale of interest, while accurate models are only given at a finely detailed, microscopic, level of description. The framework empowers one to perform macroscopic computational tasks (over large space-time scales) using only appropriately initialized microscopic simulation on short time and small length scales. The methodology eliminates the derivation of explicit macroscopic evolution equations when these equations conceptually exist but are not available in closed form; hence the term equation-free.

Introduction

In a wide range of chemical, physical and biological systems, coherent macroscopic behavior emerges from interactions between microscopic entities themselves (molecules, cells, grains, animals in a population, agents) and with their environment. Sometimes, remarkably, a coarse-scale differential equation model (such as the Navier-Stokes equations for fluid flow, or a reaction-diffusion system) can accurately describe macroscopic behavior. Such macroscale modeling makes use of general principles of conservation (atoms, particles, mass, momentum, energy), and closed into a well-posed system through phenomenological constitutive equations or equations of state. However, one increasingly encounters complex systems that only have known microscopic, fine scale, models. In such cases, although we observe the emergence of coarse-scale, macroscopic behavior, modeling it through explicit closure relations may be impossible or impractical. Non-Newtonian fluid flow, chemotaxis, porous media transport, epidemiology, brain modeling and neuronal systems are some typical examples. Equation-free modeling aims to use such microscale models to predict coarse macroscale emergent phenomena.

Performing coarse-scale computational tasks directly with fine-scale models is often infeasible: direct simulation over the full space-time domain of interest is often computationally prohibitive. Moreover, modeling tasks, such as numerical bifurcation analysis, are often impossible to perform on the fine-scale model directly: a coarse-scale steady state may not imply a steady state for the fine-scale system, since individual molecules or particles do not stop moving when the gas density or pressure become stationary. Equation-free modeling circumvents such problems by using short bursts of appropriately initialized fine-scale simulation.

The coarse time-stepper

Dynamic problems invoke the coarse time-stepper. In essence, short bursts of computational experiments with the fine-scale simulator estimate local time derivatives. Given an initial condition for the coarse variables Failed to parse (syntax error): {\displaystyle U(t_k)\} at time $t_{k}$, the coarse time-stepper involves four steps:

• Lifting, creates microscale initial conditions $u(t_{k})$, consistent with the macrostate $U(t_{k})$;
• Simulation, uses the microscale simulator to compute the microscale state $u(t)$ at over a short interval $t_{k}\leq t\leq t_{k}+\delta t$;
• Restriction, obtains the macrostate $U(t_{k}+\delta t)$ from the fine-scale state $u(t)$;
• the Time-step, extrapolation of macrostate $U$ from $t_{k}$ to $t_{k+1}=t_{k}+\Delta t$ predicts the state a macrotime in the future.

Multiple time steps simulates the system into the macro-future. If the microscale model is stochastic, then an ensemble of microscale simulations may be needed to obtain sufficiently good extrapolation in the time step. Such a coarse time-stepper may be used in many algorithms of traditional continuum numerical analysis, such as numerical bifurcation analysis, optimization, control, and even accelerated coarse-scale simulation.

Traditionally, algebraic formulae determine time derivatives of the coarse model. In our approach, the macroscale derivative is estimated by the inner microscale simulator, in effect performing a closure on demand. A reason for the name equation-free is by analogy with matrix-free numerical linear algebra ; the name emphasizes that macro-level equations are never constructed explicitly in closed form.

Restriction

The restriction operator often follows directly from the specific choice of the macroscale variables. For example, when the microscale model evolves an ensemble of many particles, the restriction typically computes the first few moments of the particle distribution (the density, momentum, and energy).

Lifting

The lifting operator is usually much more involved. For example, consider a particle model: we need to define a mapping from a few low order moments of the particle distribution to initial conditions for each particle. The assumption that a relation exists that closes in these low order, coarse, moments, implies that the detailed microscale configurations are functionals of the moments (sometimes referred to as slaving ). We assume this relationship is established/emerges on time scales that are fast compared to the overall system evolution (see slow manifold theory and applications ). Unfortunately, the closure (slaving relations) are algebraically unknown (as otherwise the coarse evolution law would be known).

Initializing the unknown microscale modes randomly introduces a lifting error: we rely on the separation of macro and micro time scales to ensure a quick relaxation to functionals of the coarse macrostates (healing). A preparatory step may be required, possibly involving microscale simulations constrained to keep the macrostates fixed . When the system has a unique fixed point for the unknown microscale details conditioned upon the coarse macrostates, a constrained runs algorithm may perform this preparatory step using only the microscale time-stepper .

An illustrative example

A toy problem illustrates the basic concepts. For example, consider the differential equation system for two variables $(U(t),u(t))$:

${\frac {dU}{dt}}=-U-u+2\,,\quad {\frac {du}{dt}}=100(U^{3}-u).$

Capital $U$ denotes the presumed macroscale variable, and lowercase $u$ the microscale variable. equation. This classification means that we assume a coarse model of the form $dU/dt=G(U)$ exists, but do not necessarily know what its is. Arbitrarily define the lifting from any given macrosate $U$ as $(U,u)=(U,0.5)$. A simulation using this lifting and the coarse time-stepper is shown in Figure 2.

Equation-free coarse time-stepper applied to the illustrative example differential equation system using Failed to parse (syntax error): {\displaystyle \delta t=0.1</math and <math>U(0)=−1} .

The solution of the differential equation rapidly moves to the slow manifold $u\approx U^{3}$ for any initial data. The coarse time-stepper solution would agree better with the full solution when the 100 factor is increased. Figure 1 shows the lifted solution (blue solid line) $(U,u)$. At times $t=n\delta t$, the solution is restricted and then lifted again, which here is simply setting $u(n\delta t)=0.5$. The slow manifold is shown as a red line. The right figure shows the time derivative of the restricted solution as a function of time (blue solid line), as well as the time derivative $dU/dt$ (the coarse time derivative), as observed from a full simulation (red line).

External links

• Yannis Kevrekidis (ed.). "Equation-free modeling". Scholarpedia.