# Multiscale II: analysis and efficient simulation of multiscale problems.

## Abstracts for the talks

### A new approach to variational problems with multiple scales

**Giovanni Alberti ** *(Pisa)*

Wednesday, October 08, 1997

Many problems in science involve the creation and interaction of microstructures on multiple spatial or temporal scales. Most mathematical approaches have addressed "infinitely fine" microstructures or the case of a single small scale. In this talk we propose a new idea to deal with multiple scales and illustrate it with some simple examples. This is joint work with S. Müller.

### Weak convergence methods and adiabatic results in classical and quantum mechanics

**F. Bornemann ** *(ZIB)*

Thursday, October 09, 1997

We will show that weak convergence methods are conveniently suited to explicitly access singular limits of a certain family of mechanical systems with multiple time scales. This family turns out to be characterized by the existence of sufficiently many adiabatic invariants.

The key step is the idenfication of the weak limits of all those quadratic quantities which carry important information for the limit system. This idenfication becomes possible by a weak convergence analogue of the Virial Theorem, resulting in certain matrix commutation relations.

We will address natural mechanical systems with strong constraining potentials and the adiabatic theorem of quantum mechanics. These examples will show that our approach considerably extends the possibility of passing through resonances.

### Numerical analysis of the (relaxed) double well problem

**Carsten Carstensen ** *(Kiel)*

Thursday, October 09, 1997

The mathematical modelling of phase transitions in alloys, or optimal design of composites, is concerned with non-convex variational problems, where a minimum may or may not be attained. Young's model example, the two-well or double-well problem (P), is considered in this talk. Convexification yields a relaxed problem (RP) related to (P). The loss of information we face in recasting (P) as (RP) is not substantial as long as we are interested in the global displacements, stress fields and Young measures (which describe oscillations creating microstructures); these variables defined by (P) can be computed from solutions of (RP).

Typically, the problem (RP) is convex, but not strictly convex. Hence error estimates for finite element methods are more difficult to prove than for simpler uniformly convex problems. A priori and a posteriori error estimates are presented and illustrated in numerical examples which indicate that it is more reasonable to perform a numerical analysis of (RP) rather than of (P).

### The approximation of complicated dynamical behavior

**M. Dellnitz ** *(Bayreuth)*

Wednesday, October 08, 1997

Multilevel techniques are presented for the efficient numerical approximation of complicated dynamical behavior. Concretely we develop (adaptive) methods which allow to extract statistical information on the underlying dynamical system. This is done by an approximation of natural invariant measures as well as (almost) cyclic dynamical components. We discuss issues concerning the implementation (e.g. parallelization strategies) and indicate potential applications of these methods (e.g. to the computation of Lyapunov exponents). The results are illustrated by several numerical examples.

### Essential molecular dynamics: Progress in a new algorithmic approach

**P. Deuflhard ** *(ZIB, )*

Wednesday, October 08, 1997

Recently, the author -- together with his co-authors Michael Dellnitz, Oliver Junge, and Christof Schütte -- developed a new algorithmic approach to molecular dynamics, which is based on the computation of almost invariant sets of Hamiltonian systems. In this approach, only well-conditioned short term subtrajectories in lieu of ill-posed long term trajectories (which are typically used in Monte Carlo simulations) are exploited. The aim is to directly compute averages of physical observables, comformations and conformational changes - informations that are actually desired by computational chemistry. Mathematically speaking, such informations come out of the computation of the invariant measures and sets (corresponding to the dominant eigenvalue 1 and infinite relaxation time) and almost invariant measures and sets (corresponding to eigenmodes for eigenvalues close to 1 and therefore finite, but "large" relaxation times). An adaptive multilevel box method or subdivision technique is presented, which helps to solve the arising stochastic eigenvalue problem "fast". The basic concepts of the new algorithm and recent speed-ups will be presented.

### Energy density for the wave equation on thin domains

**Gilles Francfort ** *(Paris)*

Wednesday, October 08, 1997

In this joint work with P. Gérard, 3-d/2-d asymptotic analysis of the wave equation is undertaken for cylindrical domains with vanishing thickness. The limit equation is a 2-d wave equation but its energy density is not the weak-limit of its 3-d analogue. The computation of that limit is performed microlocally through the computation of the semi-classical measure associated to the space-time derivatives of the field.

### Flow properties for Young-measure solutions of semilinear hyperbolic systems

**Alexander Mielke ** *(Hannover)*

Friday, October 10, 1997

We consider a semilinear hyperbolic system of *d* equations of first order.
We consider sequences of initial data which approximate a Young measure
and study the Young-measure limit of the associated sequence of solutions.
Thus we are able to define the notion of Cauchy problem and semigroups
for Young measure solutions, at least for by using compactness
through compensation. For the subclass of product measure solutions
is considered, where each Cauchy problem has a unique solution.
Continuous dependence of the associated semigroup on the initial
data can be established in the Wasserstein topology and, under additional
structural condition on the nonlinearity, also in the weak topology.
The reported research extends previous work of L. Tartar and
is partly joint work with Florian Theil.

### Adiabatic invariants and their preservation under symplectic discretizations

**S. Reich ** *(ZIB)*

Thursday, October 09, 1997

We will apply normal form theory to highly oscillatory Hamiltonian systems and discuss the preservation of adiabatic invariants over exponentially long periods of time. Next, the conservation of adiabatic invariants under a symplectic discretization will be considered. Backward error analysis will provide us the appropriate tool to show that, again, the invariants are preserved over exponentially long periods of time.

### The amplitude equation for degenerate subcritical bifurcations in pattern formating systems

**G. Schneider ** *(Hannover)*

Friday, October 10, 1997

We consider a model for a pattern formating system on the infinite line
which leads to a subcritical bifurcation. For the degenerate case
we use multiple scaling analysis to derive

as an amplitude equation for the envelope *A* of the bifurcating
pattern which is modulated slowly in time and in space.
We show exact estimates between the approximations obtained
via the amplitude equation and true solutions of the original system.
Moreover, we show that every small solution of the original system
develops in such a way that it can be described after a certain time
by the solutions of the amplitude equation.
The difficulty is to show the estimates on an
-time
scale in contrast to
for the classical Ginzburg-Landau equation, if
is the order of the amplitude. This theory allows the description of modulated
N-pulses in the original system.

### Multiple scales and nonadiabatic effects in molecular quantum dynamics

**Christof Schütte ** *(ZIB)*

Thursday, October 09, 1997

The full quantum dynamical (QD) description of molecules includes multiple
scales in particular if particles of essentially different masses are included.
This makes full QD simulations of larger molecules simply impossible.
In real life molecular dynamics one deals with this problem by introducing
simplified "mixed quantum-classical" models which describe the heavy
particles of the molecule by the means of classical mechanics and only a
small portion by the means of a wavefunction.
This talk will present two of these simplified models. By using the
techniques of the previous talk "Weak Convergence Methods and
Adiabatic Results in Classical and Quantum Mechanics", it will be discussed
under which circumstances these models gain a suitable approximation of the
true QD solution. The discussion will be concentrated on the understanding
of the nonadiabatic effects which may cause such simplified descriptions
to fail and on the alternatives which may prevent such failure.

### H-measures and their variants

**Luc Tartar ** *(CMU/Leipzig)*

Wednesday, October 08, 1997

When a sequence converges weakly to 0 in but not
strongly ( being a subset of ), one says that it contains
oscillations or concentration effects depending upon if the limit of its square
has a N-dimensional density or not. One quantitative way to analyze these
oscillations or concentration effects is to use H-measures (which I have
introduced for a few applications and which Patrick GÉRARD has introduced
independently for other ones); they are measures on which
have a quadratic microlocal character (and make more precise the quadratic
theorem of Compensated Compactness Theory).

For dealing with problems with one characteristic length, Patrick GÉRARD then
introduced a variant, called semi-classical measures, living on (quite similar to the idea which I had proposed independently to use
H-measures after adding one dimension). It was then shown by Pierre-Louis LIONS
and Thierry PAUL that the semi-classical measures could be introduced using
WIGNER transform. I then found with Patrick GÉRARD how to use correlations
instead, and why these objects are not good enough when at least two scales are
present.

Some other variants of H-measures can be defined, adapted to different
questions, and I will describe the advantages and defects of some of these
variants.

## Date and Location

**October 08 - 10, 1997**

Max Planck Institute for Mathematics in the Sciences

Inselstraße 22

04103 Leipzig

Germany

see travel instructions

## Scientific Organizers

**P. Deuflhard**

**P. Dolzmann**

**Stefan Müller**