# Numerical Analysis of Multiscale Computations (09w5109)

Arriving Sunday, December 6 and departing Friday December 11, 2009

## Organizers

Bjorn Engquist (University of Texas at Austin)
Olof Runborg (Royal Institute of Technology, Sweden)
Steve Ruuth (Simon Fraser University)
Richard Tsai (University of Texas at Austin)

## Objectives

In early 2007, BIRS sponsored our workshop on Numerical Analysis of Multiscale Computations. This workshop successfully facilitated further communications and collaborations of the participant researchers. Following that workshop, there has been a series of further events with expanded frontier:
Cambridge University hosted a workshop on highly oscillatory problems. A two-day meeting took place
in Princeton University that concentrated on multiscale aspects of molecular simulations and analysis. Several plenary talks as well as minisymposia focussed on various aspects of multiscale computations in ICIAM this year.
We observed that new techniques were being introduced, some prevously brought-up problems being resolved, while some important common problems remain.

Based on our previous success and our aspiration to push research forward in
this important field, we propose to organize a new workshop at BIRS in 2009.
This workshop is designed to be both a follow-up workshop for the previous participants and the previously discussed topics, as well as a new one with an expanded horizon.
This time we will have a full size workshop with 40 persons.

There was a realization at the previous workshop that
multiscale problems are most often also multi-disciplinary,
and input from experts in other areas is necessary.
To understand where and how mathematics can help and improve
the existing computational techniques used in the applied fields,
it is important to be able to connect and relate to the physics and
the modeling behind the problems.
Thus, although the focus on the workshop will be on mathematics,
we therefore also plan to invite a
few leading scientists in physics, biology and chemistry to reflect
on our work and discuss problems where new mathematics is needed.

In order to solve multiscale problems of practical interests and to bring the impacts to
the fields, we thus propose to bring computational
scientists, analysts and scientists in applied fields together for a brain
storming workshop at Banff. The workshop will be mutually beneficial
for the participants.
On the one hand, the analysts will be informed
of new multiscale methodologies directly from the computational
scientists
and discover new mathematical problems that are crucial to the
applications from experts in the field;
on the other hand, the computational people will be exposed to new
mathematical tools and ideas that are developed by the analysts.
Most importantly, these groups of people will be able to have a
more direct interaction and communication during this workshop and
determine a set of important problems for building a theoretically
sound and computationally feasible multiscale approach.

We set the main themes to be:

1. Coarse models and coarse variables

In the typical situation, a fine scale model for the high-dimensional
microscopic variables is known, but it is too expensive to simulate.
To reduce the computational
cost we want instead to compute a number of coarse variables,
using a coarse model. Finding such variables and a corresponding
model are two major difficulties.

The fine and coarse models can be
of different types. PDEs or ODEs are commonly used, but
in many applications, stochastic modeling is more appropriate.
For instance, molecular dynamics (ODEs) can be a
fine scale model of fluids, while a coarse scale model
could be Navier--Stokes (a PDE). Similarly, Monte--Carlo
methods can be used either as a valid method for the coarse scale system
or used in a fine scale to evaluate coarse scale quantities,
while some other type of computational model is used in the complement
scale.

From a practical point of view, we would like to be able
to find reasonable coarse models and variables
in a systematic way. In the new class of numerical methods
it is often sufficient to find a tentative, incomplete, coarse
model. Numerical simulation of the fine scale model can then
be used to reconstruct the missing parts (see below), but can
numerical methods also be used to automate how we find the tentative
coarse model and the coarse variables?

Another non-trivial issue is whether a
coarse system is properly closed, i.e. whether a certain set of coarse
variables are enough to describe the coarse dynamics that is consistent
with the fine scale system, and if not,
what additional auxilliary coarse variables are needed to
close the coarse system.

We note that it is sometimes also possible to solve for
the coarse variables without any direct reference to
a coarse model at all. We are interested in developing
such direct numerical procedures that consistently
drive the coarse scale evolution by using snapshots of fine scale
solutions. As an example, the Fermi-Pasta-Ulam problem is to study
the adiabatic energy transfer in a system of interlaced linear stiff
springs and soft'' nonlinear ones. The adiabatic energy transfer
in the system is an important phenomenon that only becomes
observable when the energies of the right springs in the original
system is explictly computed.
Each spring in the system oscillates at similar fast time
scales, and it is not obvious what coarse model (and variables)
should be used to consistenly compute the energy transfer in the
system.

2. Convergence, accuracy and complexity

One of the main goals of developing multiscale methods is to significantly reduce the complexity of finding accurate numerical solutions for phenomena that involve
wide ranges of scales.
A challange in developing multiscale methods is the question about convergence.
Conventional convergence theory roughly says that when the mesh size goes to zero,
the computed approximations converge to the solution of the problem.
In the multiscale setting, this may have to be modified so that both the mesh size and
the parameter for the smallest scale in the problem goes to zero together in one limiting
process instead of two nested limits. Therefore, we would like to find out:
Under what conditions and in what sense do the multiscale
approaches converge, while still having a significantly smaller complexity than
that of direct simulation of the fine scale model? What is the accuracy
and the stability properties of the methods?

3. Coupling of models

At the heart of the new multiscale approaches
is the coupling of the coarse and the fine models. There are two
main coupling scenarios: top down'' and bottom up''.
In the first case, a tentative coarse model is known but (a) certain
data, such as constants or coefficients in the equations,
are not available; or (b) the coarse model is not valid in
parts of the domain.
In the second case, a fine scale model exists, but cannot be computed in the entire domain of interest. Certain "up-scaling" is needed but there
may not be a readily available, even tentative,
analytical form of the macroscale model.
In both these cases one needs to couple to the fine scale
model and use it to supplement the macroscale simulations.
The challenges of coupling between different models in a multiscale computation can be typically divided into three issues: (1) finding a suitable initial
and boundary condition for the fine scale domains,
that correctly reflects macroscale information;
(2) evaluating the macroscopic information from fine scale calculations; (3) identification of subdomains in which macroscale model are not sufficient and a fine scale model is required.
If certain notion of homogeneity in the macro domain exists, periodic boundary conditions for the fine scale simulations are appropriate choice. Generally speaking, microscale simulations feedback to the macro model through Dircihlet or Neumann type conditions. One of the goal of this workshop is to discuss if there is a systematic way
and theory for issues raised above or novel couplings from new applications.

4. New challenges

Here we invite scientists from physics, biology and chemistry to
discuss problems in their fields.
The aim is to see if the mathematics of multiscale problems developed
so far can have an impact on their problems, as well as if
novel mathematical modeling of the presented problems is needed.
Often, analyzing good model problems can be very effective in advancing the solution to the practical problems, as one can attack the essential difficulties without
being distracted by other less relevant factors. The development of shock capturing schemes for conservation laws is one such example. The model problem, here Burgers' equation, played an important role in the advancement of numerical computations for conservation laws.
Therefore, we set one of the aims of this workshop to
identify new challenges and their corresponding model problems.
In this effort, it is crucial to interact with scientists
who have deep insights into the physics of the underlying problem
and how it is modeled.

These aspects pose challenging computational and analytical problems.
by either applying empirical rules based on physics, or by rigorous
analytical approaches such as homogenization or methods from
mathematical physics on much simplified settings.
Some initial work has been done on Item 2 for some classes of methods,
but many open questions remain. Much numerical simulations has been
done along the lines of 3, but the analysis of the methods
is largely missing.

Despite considerable progress in the past decades, the mathematics of multiscale modeling
and computation is still not completely developed. The wide variety of applications and methods developed in different fields of science call for some generality, so that the
difficult questions raised above may be answered
(even if partially) through some unifying framework. Indeed, with the accumulation of knowledge and experience by researches, the field of multiscale modeling and simulation has matured enough to admit a systematic mathematical analysis. The first step towards a general framework goes through careful study and analysis of model problems.

On the other hand, the mathematics of multiscale models is important for the more pragmatic aspect of providing tools for the construction of efficient and robust methods applicable in other branches of science. In lay man's words, how does an engineer,
faced with a practical problem, develop and apply a reliable
multiscale scheme? Convergence, complexity and the choice of model play a crucial role in these issues. The workshop themes were chosen accordingly.

A full answer to the above questions as well as those posed as the objectives of the workshop is still far from our reach. The importance of the workshop is in facilitating an open discussion between experts in the field in order to clarify the important issues and point out possible directions where solutions can be sought.