Some places to start are the wikipedia entry and the scholarpedia entry. Go more in-depth with the book "Principles of Multiscale Modeling". Also this review (pdf) (From 2000, but still a good overview of an array of fields and problems).
There are some other 'multi's that may be encountered in modeling:
- Multiphysics - combining multiple physical models in a single simulation. For instance, a thermal model and a mechanical model where the mechanical properties are temperature-dependent. Usually the models occur on the same length scale (vs. multiscale)
- Multigrid - method of accelerating a solution and bridging the largest and smallest lengths in a single model. Typically using grids of different resolution.
- Multiscale - combining models at different length (or time, energy) scales
What defines a 'scale'?
Each scale consists of a self-contained theory (or model), approximations for solution (often these approximations come to define variants of each model), and validation by experiments (carefully designed to probe at this scale, and isolate the influence of other scales).Transitioning between scales often involves a qualitative change in the type of theory (particle-based, continuum modeled with PDE, network of coupled ODE's, etc.) The "self-contained" part usually implies a small number of input parameters. Some of this is practical - if the scale had too many inputs it would be hard to understand, hard to reason about, and hard to compute.
The other type is concurrent modeling (in some cases called 'on the fly'), where simulations at different scales are done at the same time. In the example above, the atomistic simulation would call down to the electronic structure simulation as needed to get necessary parameters. Alternately, if it is not possible to parameterize the potential, the atomistic simulation would get forces and energies directly from the electronic structure simulation.
Multiscale in Materials
This whole process seems most developed for engineering certain metals and alloys, where it's called Integrated Computational Materials Engineering.The book "Integrated Computational Materials Engineering for Metals: Using Multiscale Modeling to Invigorate Engineering Design with Science" has some case studies, including one on the design of a Cadillac control arm, covering the scales from electronic structure (DFT) to the final mechanical design of arm. I found it helpful to have a specific example running the entire range.
Scales encountered for materials problems include
- electronic structure
- atomic motion
- dislocations
- voids and other microstructures
- continuum description
- engineering design (using FEA - Finite Element Analysis)
The Minerals, Metals and Materials Society (TMS) produced a report that covers the current state and challenges for the field: "Modeling Across Scales: A Roadmapping Study for connecting materials models across length and time scales"
Multiscale in Biology
There are a lot of directions to go when building upward from electronic structure. Another possibility is biology. This article is a good overview:"Multiscale computational models of complex biological systems".
Scales encountered include
- genetic
- protein
- pathway and signaling
- whole cell
- cell network and tissue
- whole organ
- multisystem and organism
(And in some cases, this could go higher to populations and ecosystems)
Multiscale in Computing
It may be instructive to compare with various scales in computing- materials (doped silicon, etc)
- transistors
- gates
- larger logic assemblies (shift registers, adders, etc)
- processors (CPU)
- assembly language
- low level programming language
- high level programming language
- operating system
- applications
- large-scale distributed systems
Now perhaps not every hierarchical description of abstractions should be considered 'multiscale', but it may be worth considering the similarities and differences.
One of those differences is where engineering (control) can occur. For computing, the entire stack is engineered. For materials, points of control are composition, manufacturing process, and component design. For drug design in biology, the only point of control/design is the drug molecule. All the effects above that level are no longer under direct control - understanding how these effects propagate through the scales is very difficult problem.
General Approaches
One of the limiting factors is how much computer time it takes to solve the model at each scale. Approaches for speeding up existing models (via multigrid or reducing the degrees of freedom) start to merge with methods for generating models for higher scales. One approach is systematic upscaling, based on multigrid and renormalization ideas. See Principles of Systematic Upscaling (pdf).Another approach is called Heterogeneous Multiscale Modeling (HMM) See this overview page and a review article (pdf)
Connections to QMC
Given this blog is focused on QMC, how these multiscale ideas might relate to QMC. Solving the electronic structure problem forms the lowest level of the hierarchy (for practical engineering problems, anyway). Can we speed up QMC calculations, possibly through creating reduced-order models? Also, can existing information (previous runs on the same system, or information from higher levels) be used to speed the calculation?