Equation-free modeling
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.
[1] 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.
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.
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.
In essence, short bursts of computational experiments with the fine-scale simulator estimate local time derivatives.
In this 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;[5] the name emphasizes that macro-level equations are never constructed explicitly in closed form.
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 [7]).
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.
The examples illustrate the various ways to construct and assemble the algorithmic building blocks.
[10] Applying the equation-free paradigm to a real problem requires considerable care, especially defining the lifting and restriction operators, and the appropriate outer solver.
The recursive projection method[14] enables the computation of bifurcation diagrams using legacy simulation code.
Consider the coarse time stepper in its effective form which includes explicit dependence upon one or more parameters
Bifurcation analysis computes equilibria or periodic orbits, their stability and dependence upon parameter
Additionally, for problems where the macroscale has continuous symmetries, one can use a template based approach [15] to compute coarse self-similar or travelling wave solutions as fixed points of a coarse time-stepper that also encodes appropriate rescaling and/or shifting of space-time and/or solution.
For example, self-similar diffusion solutions may be found as the probability density function of detailed molecular dynamics.
[19] Higher order schemes for systems where the microscale noise is still apparent on the macroscale time step are more problematic.
When the microscale simulator is computationally expensive the gap-tooth scheme empowers efficient large scale prediction.
[21][22][23] The Matlab/Octave toolbox provides support to users to implement simulations on an rectangular grid of patches in 1D or 2D space.
[4] The combination of the gap-tooth scheme with coarse projective integration is called patch dynamics.
[24] This answer is related to the coupling in holistic discretization and theoretical support provided by the theory of slow manifolds.
Then one generally has to form averages over a core of each tooth/patch, and apply the coupling condition over a finite action region on the edges of each tooth/patch.
[25] That is, for efficiency one makes the microscale tooth/patch as small as possible, but limited by the need to fit in action and core regions big enough to form accurate enough averages.
Just as for normal projective integration, at the start of each burst of microscale simulation, one has to create an initial condition for each patch that is consistent with the local macroscale variables, and the macroscale gradients from neighboring interpolated patches.
If the chosen macroscale length is too small then more coarse scale variables may be needed: for example, in fluid dynamics we conventionally close the PDEs for density, momentum and energy; yet in high speed flow especially at lower densities we need to resolve modes of molecular vibration because they have not equilibrated on the time scales of the fluid flow.
However, in complex situations there is a need to automatically detect the appropriate coarse variables, and then use them in the macroscale evolution.
In some problems it could be that as well as densities, the appropriate coarse variables also need to include spatial correlations, as in the so-called Brownian bugs.
