Multiscale and Multiresolution Methods

General Description

MrHyDE is designed to use heterogeneous computational architectures to enable concurrent multiscale modeling and simulation. The general form of a coupled multiscale system is:

where $u_c$ denotes the coarse-scale variables and $u_f$ denotes the fine scale variables. Either $F_c$ or $F_f$ may also depend on space and/or time, but we omit this explicit dependence for simplicity. If we assume that the fine scale variables can be eliminated (preferably locally), then we can solve an equation only involving the coarse scale variables:

where the notation $h(u_c)$ represents the local nonlinear elimination of the fine scale information, which depends on the current coarse scale state. As we will soon see, this is related to the Schur complement in the case of steady-state linear operators. At this point, there are two possible interpretations of the coupled system and the reduced system. If the objective is to compute an approximation to 𝑢𝑐 that is informed by the fine scale information, then one interpretation is that this can be obtained by solving the reduced system. However, if the objective is to approximate the fine scale solution that is informed by global information, then we can interpret the reduced system as providing the time evolution of the coarse scale global coupling between fine scale models. The appropriate interpretation is problem dependent.

For simplicity, we will ignore any spatial discretizations throughout this section and proceed as if we are solving coupled systems of ODEs, but in general, all of the use cases involve spatial discretization and are typically based on variational formulations.

To simplify the notation, we will rewrite the reduced system as:

where $g(u_c)$ represents the time derivative. The precise form of h(𝑢𝑐) depends on the multiscale formulation, e.g., variational multiscale, homogenization, FE2, etc. In some cases, it may simply be a numerical approximation of $u_f$ given $u_c$, and in other cases it may be a model for the subgrid effects. However, in all cases we refer to h as the Macro-Micro-Macro map, since macro-scale information is sent to the local fine scale (subgrid) models, fine scale information is computed, and a reduced (upscaled) version of this information gets returned to the coarse scale problem (see the following figure).

MrHyDE is designed to utilize any type of subgrid model, but the primary demonstration is a multiscale Dirichlet-to-Neumann map which is described in below. MrHyDE also requires the gradient of $h$ with respect to $u_c$ to enable accurate Jacobians for implicit time integration. This is often complicated since $h$ will can depend on $u_c$ either explicitly or implicitly through $u_f$, which depends on $u_c$. Thus, the subgrid models typically compute

Computing $\partial u_f / \partial u_c$ is the most challenging step. We first describe it for the steady-state case before moving on to the transient case.

Steady-state Case

Transient Case

Synchronous Time Integration

Asynchronous Time Integration