Mathematical Pathways
From Simple to Complex and Back
Peter Oswald (Mathematical Sciences)
Dynamics
Multilevel Solvers
Subdivision Surfaces
The Complex World of Math(ematicians)
?? Seldomly exercised
Ever
Changing
Real
World
Sciences
Qualitative
and
Quantitative
Models
Mathematics
Exact and
Abstract
Models and
Language
Complicated and dirty
Clean and beautiful
Full of life
Looks dead (to some)
Dynamic: Truth is questioned
from time to time
Absolute truth within
stated assumptions
Success remains interdisciplinary challenge
Complex Numbers
Counting: Naturals, Integers
1,2,3,...,n,... ; 0 (zero) ;
-1(=1-2),-2,-3,... ;
Comparing in size: Rationals
1/2, -2/3, ... , p/q, ...
Still: Cannot solve simple quadratic
equations such as
x 2+ x + 1 = 0
y
z=x+iy
Filling the gaps: Real line
Sqrt(2), pi, e, ...
(World of Calculus: Limits,
Derivatives, Integrals, ... )
i = sqrt(-1)
x
Complex is also used for sets with structure such as the set of points,
line segments, polygons, polyhedra, ... , with incidence relationships
(polyhedra have vertices (=points), edges (=line segments), faces
(=polygons))
Algebraic Topology
All you need to know about derivatives
Functions of one variable x = x(t) :
dx
x(t  Δt)  x(t)
x' (t) 
 lim Δt 0
dt
Δt
The ordinary derivative x‘(t) describes
• Instantaneous rate of change of dependent
variable (x) with respect to change in the
independent variable (t)
• Geometry: Slope of tangent, x’’(t) related to
curvature
• Mechanics: Velocity, x’’(t) is acceleration
Functions of several variables u = u(t,x,y,z) :
u
u(t  Δt, x,...)  u(t, x,...)
ut 
 lim Δt 0
,
t
Δt
ux ,

Partial derivatives and differential operators such as gradient or
Laplacian
Complex Behavior in Dynamical Systems
State
Transition rule
„Next“ State
F
Discrete time: Recursively defined sequences
Initial state
x n 1  F(x n ), n  0,1,..., x 0 given
Example: Conway‘s Game of Life (
Cellular Automata)
Continuous time: Differential equations such as Selkov model
x'  f(x, y),
y'  g(x, y)
2D
State space
3D
x'  f(x, y, z),
y'  g(x, y, z),
Examples: Van der Pol and Lorenz systems
z'  h(x, y, z)
The Game of Life (Conway 1970)
State space: Infinite grid of square cells, each cell „live“ or „dead“
State is pattern of marked live cells (or doubly-indexed 0-1 sequence)
Transition rules: Look at the 8 neighbors of a cell
1. A live cell with < 2 live neighbors dies (loneliness)
2. A live cell with > 3 live neighbors dies (overcrowding)
3. A live cell with 2 or 3 live neighbors survives
4. A dead cell with = 3 live neighbors becomes alive (birth process)
Initial state: Finite pattern of marked live cells
constant pattern
period 4 pattern
moving
period 2 pattern
no moving
zero pattern
n=0
n=1
n=2
n=3
n=4
A Nonlinear Oscillator (Van der Pol 1920-27)
x' 'b(1  x )  x'  x  a  cos(ct),
x  x(t)
2
|x|>1 damping
|x|<1 excitation
Periodic force
State variable
or rewritten as system with state variable (x,y)
x'  b(x  x /3)  y, y'  x  a  cos(ct)
3
b=0
b=3
b=30
Limit cycles, quasi-periodic solutions, frequency locking ...
Lorenz system (Lorenz 1963)
Smallest system (3 equations) that shows a completely
new behavior:
Bounded solution trajectories approach a special region
(called strange attractor) in a fractal, non-periodic fashion
(some kind of deterministic chaos)
x'  10 ( y  x),
y'  r  x - y - x  z,
z'  x  y  8z/3,
x  x(t)
y  y(t)
z  z(t)
Rayleigh coefficient: r > 0
Application area: Weather forecasting, reduced model for
so-called convection rolls in the lower atmosphere
What have we seen so far?
• Complex (dynamic) behavior can be described by simple setups,
both in discrete time (recursive algorithms) and continuous
time (systems of ordinary differential equations).
• Similar systems can behave quite differently. In particular,
types of nonlinearity, dimension, parameter changes influence
the behavior. Understanding requires deep Mathematics.
• Math and simulation is not all. We don’t know whether our
simple models explain any real mechanism of nature.
This needs good experimentation, data collection and analysis.
• Can use the knowledge gained from simple models to analyse
and simulate larger and more complex systems, a common
approach is multi-scale or hierarchical modelling.
Some toy versions of multi-scale algorithms
Solving PDE problems (large
small or fine-to-coarse)
Creating shapes
(small
large or coarse-to-fine)
will be discussed next.
Terrain Triangulation (Multiple Spatial Scales)
Courtesy of Prof. Griebel (Scientific Computing, Uni Bonn)
Terrain Triangulation ( Zoom-In Application)
Courtesy of Prof. Griebel (Scientific Computing, Uni Bonn)
Multiple Scales in Physical Systems
Example: Flow problems
Micro/mesoscopic scales:
Statistical thermodynamics
turbulence
particle simulations
Macroscopic fields:
Engineering formulas:
velocity, pressure,
Drag, peak velocity
density, temperature,...
as functions of space
and time (PDEs)
Example: Integrated circuits
Nanoscale:
Quantum effects, particle
models, MC simulations
Macroscopic fields:
Electron/hole densities,
Potential fields (PDEs)
I/O models:
Simplified circuit
description (ODEs)
Example: Flow simulations
Courtesy of Prof. Griebel (Scientific Computing, Uni Bonn)
Re Reynolds number
u t  (Re) Δu  u  u  p  f
-1
0  u
f external
forces
u=(u1,u2,u3)
velocity vector
p pressure function
Most famous system of partial differential equations (PDEs).
Describes incompressible fluid motion at constant temperature,
and is trusted by 99.999% of scientists (as macroscopic model).
PDEs are everywhere (e.g., spatio-temporal effects in bio-systems).
Example: Electrostatics of ICs
V S G S V S
V
S
G
V
S
S
q unknown charge density
on conductor surface
1
q(Q)
da Q  U(P)

4π Γ | P - Q |
U given electrostatic
potential („voltages
attached to surface“)
Example of an integral equation (IE), also heavily used in engineering
simulations. PDEs and IEs of this kind do not possess simple
analytic solutions
computational methods
computer simulations
Numerical Discretization
• Unknown functions
Vectors of function values on grid
x(t)
(x1 , x 2 ,, x n )T
• Derivatives
x' (t 2 )
• Integrals
 x(t) dt
• PDEs
• IEs
Differences
x3  x2
t3  t2
or
x 3  x1
t 3  t1
Sums
c1x1  c2 x 2  cn x n
Sparse systems of (linear) equations
Dense systems of (linear) equations
Toy Problem: Temperature Distribution
Txx  Tyy  0,
T  T(x, y)
T describes stationary temperature
field in a square layer (insulated
from both sides), with temperature
fixed at the edges.
x
T = 0 (cool surface)
Use a (m+1)x(m+1) square grid,
each interior grid point carries a
discrete temperature value.
T = 20
(Air)
Corner
singularity
T = 20
The PDE (Laplace‘s equation)
is discretized by finite differences.
T = 200 (hot surface)
y
This leads altogether to an
unknown vector of length n=m*m,
and a very sparse, nicely structured
linear system of dimension n.
Discrete solution
m = 20
m = 40
Solving: Gaussian elimination kills
The classical solution method (Gaussian elimination) leads to fill-in
and becomes impractical for m>100:
mxm fill-in block
Before elimination
After 3 steps
After m=6 steps
Due to the mxm fill-in block each of the remaining elimination steps
Needs roughly ~ m*m = n CPU cycles, i.e., the overall method needs
Roughly ~ n*n cycles:
8
Accuracy 0.0001
m~100
~10 flops (doable but slow)
14
Single precision
m~3000
~10 flops (kills you)
Need new ideas: Reordering, FFT, Multi-scale (multi-grid) approach
Multiscale Solver: Basic Idea
Two ideas:
1) Solve the dicretized problem approximately, not exactly
(only within the accuracy of the numerical model which
obviously depends on the chosen grid size, i.e., on m)
2) Solve it not only on the given grid but also on a whole
hierarchy of coarser grids, and combine cleverly!
Size n
Size n/4
# flops
~n
Size n/16
Multigrid idea (1960-80) for IEs and PDEs: Fedorenko, Bachvalov, Brandt, Hackbusch,...
Linear Algebra Interpretation
badly conditioned
nicely conditioned
~ n non-zero entries
~ n log n non-zero entries
fast matrix-vector multiplication
Subdivision schemes: Coarse to Fine
• Have evolved from the recursive evaluation of
spline curves and surfaces into a tool for
hierarchical surface generation
• Surfaces are created by local topology
refinement and geometric rules for inserting new
and moving old points in space. They are
rendered (displayed) as triangulated surfaces.
• Process similar to creating fractal objects
• Twist: Result should look smooth (since most
shapes consist of smooth pieces)
Dyadic or 2-refinement
Insert edge midpoints
Quadrisect triangles
Geometric rules (2-subdivision)
Interpolating Schemes
(old points not moved)
Approximating Schemes
(old points slightly moved)
3
1/2
(Linear interpolation
-w
1
1/2
2w
polyhedra)
1
3
-w
1
1/2
-w
1/2
2w
-w
(Butterfly scheme, Dyn et al. 1990)
1
w(6)=10
1
w(k)
1
1
(Quartic boxspline, Loop 1987)
Sqrt(3)-refinement (slower topology refinement)
Insert face midpoints
Create rotated triangulation by joining new and old vertices
How it works (sqrt(3) scheme by Guskov)
Step 1
Step 2
Initial configuration
Pumpkin?
Step 6
Art?
To me: Not so great,
artifacts that are not
coming from initial
shape!
More Pumpkins: Other Geometric Rules
Butterfly: Interpolating
Approximating Geometric Rules
Slightly different initial configuration
Sqrt(7)-refinement (faster topology refinement)
Insert 3 points per triangle
(each such point
has its “closest”
old edge resp.
vertex)
Connect new points with each other and old vertices
in a consistent way (requires orientable triangulation)
Triangular versus Hexagonal Refinement
Example: Composite Schemes
1. Trivial upsampling: oV
nV
3. F2F:
1/3
2. V2F:
7*oV,
a
1-3a
1/3
1/k
1/k
a
1/3
4. F2V:
(5. V2V)
1/k
0
a
c/k
c/k
c/k
1-c
1/k
1/k
c/k
Repeat 2. – 4.(5.) another (n-1) times
c/k
Irregular vertices: Combined scheme for n=2
Tetrahedron:
1
k = 3 (not C )
Double pyramid:
Double pyramid:
1
k = 3 (not C )
k = 12 (C1 but too flat)
k = 4 (barely C1 )
k = 4 (already “treated”)
V2V-modification: Tetrahedron
before
after
V2V-modification:
before
k=3,4
after
V2V-modification: k=12
before
after
Thank you for your attention!
Acknowledgements go (in no specific order) to:
Wikipedia (Game of Life, Demonstration for Van der Pol
equation and Lorenz system, etc.)
Matlab
(Numerical simulation and visualization support)
Prof. Michael Griebel, Dr. Alex Schweitzer, and Group,
Institute of Scientific Computing, Uni Bonn (PDE simulations)
MS Powerpoint (Hacking it together)
Detailed references on request!
Useful URLs:
http://www.ibiblio.org/lifepatterns/
(Game of Life implementations)
http://www.cmp.caltech.edu/~mcc/Chaos_Course/
(many demos, in particular forced nonlinear oscillator)
http://to-campos.planetaclix.pt/fractal/lorenz_eng.html
(simulations for Lorenz and Rössler systems)
http://wissrech.iam.uni-bonn.de/main/index.html
(homepage of Griebel‘s Scientific Computing Group
at Uni Bonn, with project descriptions and software
for flow problems and many other simulations)
http://www.faculty.iu-bremen.de/poswald/teaching/teaching.html
(temporary download of today‘s USC talk)