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Flexible Solvers and Emergent Answers
6
What is a parametrically defined design?
"A design where meaningful properties
are controlled through constants."
Fitness Function must provide
a single number for every state.
Stochastics allow us to break free of the grid.
Heuristics allow us to do so in a clever way.
- Limited run due to cooling schedule
- Good at finding many local optima
- Good at navigating rough landscapes
- More prone to getting stuck in local optima
- Good at refining a solution
- Allows user interaction
{0.0000, 0.0001, 0.0002, ... , 99.9998, 99.9999, 100.0000}
1 million possible values per parameter
Observable universe contains roughly 10 cubic millimeters
Flexible Solvers & Emergent Answers
How stochastics and meta-heuristics can help us find solutions to parametrically defined problems
David Rutten (Robert McNeel & Associates)
Design parameters
Phase-Space
What do we mean by "best"?
Fragment
unit
Property
Lighting
Sunlight & daylight
~50,000 lux
Cooling
Sunlight & Insulation
~1000 watts × m²
Price
Purchase & Transport
€
(Size)
Weight & Size
~500 kg
Installation
(Position)
Construction
~1 m
Location & Size
Size & Accessibility
~2 m²
Cleaning
0~100 %
Retail
Quality of view
Discontinuous
Topologies
Multiple local optima
Fractal
Needle-in-a-haystack
~10 million rods & cones
~20 frames/second
~0.5 second
————————— ×
~100 million samples
Stable
Filament
Incomplete → Fragmentary
Unstable
Optimum
Brute-Force Runtimes
(100 samples per axis)
Duration
Dimension
(assuming 0.1 seconds per sample)
The curse of
dimensionality
1
10 seconds
16.7 minutes
2
3
28 hours
4
116 days
5
33 years
6
3251 years
91
50
3.2 × 10 years
Simulated Annealing
Simulated Evolution
Generation X
Generation 1
Generation 2
Generation 3
Warm
Hot
Cold
Frozen
Mutations allow 'fresh' genes
to enter the population.
Coalescence determines how genomes
are combined to produce offspring.
Selection determines which animals
are allowed to survive & mate.
Gene-complexes & Speciation
Solution
Random
Progression of a single Simulated Annealing run
Construction of an evolutionary generation
Phase-Space "Pressure" due to existing runs
Second annealing run
First annealing run
What then?
Offspring
Rejected
Quality
Fitness
Parent
Offspring
Phase-Space
User/Solver Interaction
High-dimensional spaces
Basins of attraction
Competing populations
Nearest neighbour
Nearest 3 neighbours
Nearest 2 neighbours
A
C
B
D
2
{A + B + C + ... + Z }
E
y
Exact solution no
longer possible
Radius
x
Fitness Landscape of a Circle Fitter
{
}
50 parameters
Each with a range from 0 to 100
Each with an accuracy of 4 decimal places
Position
300
10 possible distinct states in Phase-space
90
Radius