Mathematical Pathways From Simple to Complex and Back

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Mathematical PathwaysFrom Simple to Complex and
Back
Peter Oswald (Mathematical Sciences)

• Dynamics

Subdivision Surfaces
Multilevel Solvers
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The Complex World of Math(ematicians)
?? Seldomly exercised
Sciences Qualitative and Quantitative Models
Mathematics Exact and Abstract Models and Language
Ever ChangingReal World
Complicated and dirty Full of life Dynamic
Truth is questioned from time to time
Clean and beautiful Looks dead (to
some) Absolute truth within stated assumptions
Success remains interdisciplinary challenge
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Complex Numbers
Counting Naturals, Integers 1,2,3,...,n,...
0 (zero) -1(1-2),-2,-3,...
Still Cannot solve simple quadratic equations
such as x x 1 0
2
Comparing in size Rationals 1/2, -2/3, ... ,
p/q, ...
z x i y
y
Filling the gaps Real line Sqrt(2), pi, e, ...

i sqrt(-1)
(World of Calculus Limits, Derivatives,
Integrals, ... )
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
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All you need to know about derivatives

• Functions of one variable x x(t)
• 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)
Partial derivatives and differential
operators such as gradient or Laplacian
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Complex Behavior in Dynamical Systems
Transition rule
State
Next State
F
Initial state
Discrete time Recursively defined sequences
Example Conways Game of Life (
Cellular Automata)
Continuous time Differential equations such as
Selkov model
State space
2D
3D
Examples Van der Pol and Lorenz systems
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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

• A live cell with lt 2 live neighbors dies
  (loneliness)
• A live cell with gt 3 live neighbors dies
  (overcrowding)
• A live cell with 2 or 3 live neighbors survives
• 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
n0
n1
n2
n3
n4
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A Nonlinear Oscillator (Van der Pol 1920-27)
xgt1 damping xlt1 excitation
State variable
Periodic force
or rewritten as system with state variable (x,y)
b0
b30
b3
Limit cycles, quasi-periodic solutions, frequency
locking ...
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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)
Rayleigh coefficient r gt 0
Application area Weather forecasting, reduced
model for so-called convection rolls in the
lower atmosphere
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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 dont 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.

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Terrain Triangulation (Multiple Spatial Scales)
Courtesy of Prof. Griebel (Scientific Computing,
Uni Bonn)
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Terrain Triangulation ( Zoom-In Application)
Courtesy of Prof. Griebel (Scientific Computing,
Uni Bonn)
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Multiple Scales in Physical Systems
Example Flow problems
Macroscopic fields velocity, pressure, density,
temperature,... as functions of space and time
(PDEs)
Micro/mesoscopic scales Statistical
thermodynamics turbulence particle simulations
Engineering formulas Drag, peak velocity
Example Integrated circuits
Nanoscale Quantum effects, particle models, MC
simulations
I/O models Simplified circuit description (ODEs)
Macroscopic fields Electron/hole
densities, Potential fields (PDEs)
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Example Flow simulations
Courtesy of Prof. Griebel (Scientific Computing,
Uni Bonn)
Re Reynolds number
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).
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Example Electrostatics of ICs
q unknown charge density on conductor
surface
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
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Numerical Discretization

• Unknown functions Vectors of
  function values on grid
• Derivatives
  Differences
• Integrals
  Sums
• PDEs
  Sparse systems of (linear) equations
• IEs
  Dense systems of (linear) equations

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Toy Problem Temperature Distribution
T describes stationary temperature field in a
square layer (insulated from both sides), with
temperature fixed at the edges. Use a
(m1)x(m1) square grid, each interior grid point
carries a discrete temperature value. The PDE
(Laplaces equation) is discretized by finite
differences. This leads altogether to an
unknown vector of length nmm, and a very
sparse, nicely structured linear system of
dimension n.
x
T 0 (cool surface)
T 20 (Air)
T 20
Corner singularity
y
T 200 (hot surface)
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Discrete solution
m 20
m 40
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Solving Gaussian elimination kills
The classical solution method (Gaussian
elimination) leads to fill-in and becomes
impractical for mgt100
mxm fill-in block
Before elimination
After 3 steps
After m6 steps
Due to the mxm fill-in block each of the
remaining elimination steps Needs roughly mm
n CPU cycles, i.e., the overall method
needs Roughly nn cycles Accuracy 0.0001
m100 10 flops
(doable but slow) Single precision
m3000 10 flops (kills you)
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Need new ideas Reordering, FFT, Multi-scale
(multi-grid) approach
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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!
flops n
Size n
Size n/4
Size n/16
Multigrid idea (1960-80) for IEs and PDEs
Fedorenko, Bachvalov, Brandt, Hackbusch,...
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Linear Algebra Interpretation
badly conditioned
nicely conditioned n non-zero entries
n log n non-zero entries
fast matrix-vector
multiplication
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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)

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Dyadic or 2-refinement
Insert edge midpoints
Quadrisect triangles
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Geometric rules (2-subdivision)
Interpolating Schemes (old points not moved)
Approximating Schemes (old points slightly moved)
3
1
1/2
1/2
(Linear interpolation polyhedra)
1
3
-w
2w
-w
w(6)10
1
1
1
1/2
1/2
w(k)
1
1
-w
-w
2w
(Butterfly scheme, Dyn et al. 1990)
(Quartic boxspline, Loop 1987)
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Sqrt(3)-refinement (slower topology refinement)
Insert face midpoints
Create rotated triangulation by joining new and
old vertices
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How it works (sqrt(3) scheme by Guskov)
Step 1
Step 2
Initial configuration
Pumpkin? Art? To me Not so great, artifacts
that are not coming from initial shape!
Step 6
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More Pumpkins Other Geometric Rules
Butterfly Interpolating
Approximating Geometric Rules
Slightly different initial configuration
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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)
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Triangular versus Hexagonal Refinement
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Example Composite Schemes
1. Trivial upsampling oV 7oV,
nV 0
3. F2F
1/3
2. V2F
a
1-3a
a
1/3
a
1/3
4. F2V
(5. V2V)
c/k
c/k
1/k
1/k
c/k
1-c
1/k
1/k
1/k
c/k
c/k
Repeat 2. 4.(5.) another (n-1) times
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Irregular vertices Combined scheme for n2
Double pyramid k 3 (not C ) k 4
(barely C )
Tetrahedron k 3 (not C )
Double pyramid k 12 (C but too flat)
k 4 (already treated)
1
1
1
1
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V2V-modification Tetrahedron
before
after
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V2V-modification k3,4
after
before
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V2V-modification k12
before
after
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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!
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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 Griebels 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/t
eaching.html
(temporary download of todays USC talk)