How has Computational Geometry helped
Visualization and Analysis of Massive Scientific Data:
from the Design of Clean Fuels to Understanding Climate Change
Valerio Pascucci
Director, Center for Extreme Data Management Analysis and Visualization
Professor, SCI institute and School of Computing, University of Utah
Laboratory Fellow, Pacific Northwest National Laboratory
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Massive Scientific Models are Source of
Great Challenges and Opportunities
Subsurface
Modeling
BlueGene/L
Hydrodynamic
Instabilities
Combustion Simulations
Molecular Dynamics
Climate Modeling
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Material Sciences
Given Three Burning Hydrogen Flames,
Rank Them by Level of Turbulence
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Count the Number of Bubbles in a
Rayleigh–Taylor Instability
Rayleigh-Taylor
instabilities arise in
fusion, super-novae,
and other fundamental
phenomena:
– start: heavy fluid above,
light fluid below
– gravity drives the mixing
process
– the mixing region lies
between
the
upper
envelope surface (red)
and the lower envelope
surface (blue)
– 25 to 40 TB of data from
simulations
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Are these LES and DNS simulation codes
expressing equivalent fenomena?
LES
Large Eddy Simulation
DNS
Direct Numerical Simulation
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What is the Long Term Impact of Human
Activities on the Global Climate?
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What Conditions Cause Extinction and
Reignition in Premixed Hydrogen Flames
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We Develop General Purpose Tools for
Efficient and Reliable Data Understanding
Real functions are ubiquitous
in the representation of
scientific information.
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Traditional Data Analysis Tools are
Often Ineffective for Massive Models
• Massive models are challenging: Rayleigh–Taylor instability (Miranda)
– Sheer volume of information
– Complexity of the
information represented
– Complexity of presentation
• Tools do not scale with the
data sizes
• Difficult to capture multiple scales
• Numerical methods unstable and sensitive to noise
• Difficulty in providing error bounds associated
with the coarse scale analysis
• Need new abstractions and metaphors to covey
information reliably and efficiently
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We Develop New Techniques for Efficient
Data Management and Presentation
• Advanced data storage techniques:
– Data re-organization.
– Compression.
• Advanced algorithmic techniques:
– Streaming.
– Progressive multi-resolution.
– Out of core computations.
• Scalability across a wide range of running conditions:
– From laptop, to office desktop,
to cluster of PC, to BG/L.
– Memory, to disk, to remote
data access.
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We Demonstrated Performance and
Scalability in a Variety of Applications
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Our Framework is Based on Robust Topological
Computations for Quantitative Data Analysis
• Provably robust computation
• Provably complete feature
extraction and quantification
• Hierarchical topological
structures used to capture
multiple scales
• Error-bounded approximations
associated with each scale
• Formal mathematical definition
associated with each analysis
• Scalable performance in
association with streaming
techniques
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Hierarchical topology of a
2D Miranda vorticity field
Coarse scale:
blue = molecules
Medium scale:
red-blue = dipoles
Molecular dynamics simulation (left) with
abstract graph representation of its features
at two scales (right)
Motivation
• The presence of computational errors in streamline
computation using integration techniques can lead to
inconsistent results
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We Adopted a Combinatorial Approach to Morse
Theory for Provably Correct Computations
f
(
x
)
:
D

 F
(
x
)
:
S


Classical mathematical
definitions
domain
function
critical
point
D
f
S
smooth manifold
infinitely differentiable

f(
p
)

0
numerical
1D
Simulation of
differentiability
simplicial complex
F(x) PL-extension of f (xi)
d

1
Low
(
p
)

Be
combinatorial
2D
3D
Independent local computation yield globally consistent results
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We Use Robust Techniques for Critical
Point Detection and Classification
type
index
The Morse Lemma
Minimum
0
There are d+1 types
of critical points
Saddle
d-1
Maximum
d
Minimum

f(
p
)

0
d

k
d
2
2
i
j
i

1
j

d

k

1
Minimum
Maximum
Saddle
Maximum
f
(
x
)
|

f
(
p
)

x

x


p
Minimum
numerical
1-saddle
2-saddle
combinatorial
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Maximum
We Introduced the Morse–Smale
Complex for Complete Data Analysis
• The Morse–Smale complex
partitions the domain of f in
regions of uniform gradient
• Generalizes the notion of
monotonic interval
• Dimension of a region equal
index difference of source
and destination
• Remove inconsistency of
local gradient evaluations
1D
2D
3D
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Minimum
Saddle
Maximum
We Use Cancellations to Create a Multi-scale
Representation of the Trends in the Data
Cancellations:
Approximation:
Multi-scale:
critical points can be created or destroyed
in pairs that are connected 1-manifolds
error = persistence/2 (proven lower bound)
consistent gradient segmentation at all scales
persistence p
1D: cancellation=contraction
3D
2D: cancellation=contraction + edge removal
Time
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Contraction of extrema with saddles
allows to simplify the MS-complex.
0.5% persistence filter
1.0% persistence filter
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20% persistence filter
Contraction of extrema with saddles
allows to simplify the MS-complex.
0.5% persistence filter
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Contraction of extrema with saddles
allows to simplify the MS-complex.
1.0% persistence filter
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Contraction of extrema with saddles
allows to simplify the MS-complex.
20% persistence filter
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Topological Constructs Allow Building
Effective and Succinct Shape Descriptors
The Reeb graph is the graph
obtained by continuous
contraction
of
all
the
contours in a scalar field,
where each contour is
collapsed to a distinct point.
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We Use Streaming Techniques to Achieve
High Performance Analysis of Shapes
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We Exploit the Existing Locality of the
Input Mesh to Avoid Costly Reordering
• Computation of the Reeb graph for two layouts of
the same model
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We Develop Shape Signatures to Find
Defects in Large Scale Geometric Models
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1.E+05
1.E+04
Practical Tests Confirm Robustness and
Show Virtually Linear Scalability
1.E+03
100Ks
1.E+05
100MB
1.E+05
1.E+02
10MB
1.E+04
10Ks
1.E+04
1hour
1MB
1.E+01
1.E+03
1ks
1.E+03
100KB
1.E+02
1.E+00
100s
1.E+02
1min
10KB
1.E+01
1.E-01
1.E+00
1KB
10s
1.E+01
1.E-02
1.E-01
100B
1s
1.E+00
1.E-02
10B
1.E-03
0.1s
1.E-01
1.E-03
1B
1.E-04
1.E-04
1.E-02
1.E+031Kt
1.E+04
1.E+05
1.E+06
1.E+07
1.E+08
10Kt
100Kt
1Mt
100Mt
1Gt
10Mt
1.E+03
1.E+04
1.E+05
1.E+06
1.E+07
1.E+08
1.E+09
Min
Max
Time Lin.
Lin. Scale
Avg Run
Run Time
Max Run
Run Time
Time
Read T
MinMem
Mem
Max Mem
Mem
Time
Scale
Avg
Time
Max
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Demo C4H4
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Demo S3D Combustion Simulation
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Data Analysis Examples
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We Analyze High-Resolution
Rayleigh–Taylor Instability Simulations
• Large eddy simulation run on
Linux cluster: 1152 x 1152 x 1152
maximum
bubble
– ~ 40 G / dump
– 759 dumps, about 25 TB
• Direct numerical simulation run
on BlueGene/L: 3072 x 3072 x Z
– Z depends on width of mixing layer
– More than 40 TB
• Bubble-like structures are observed in laboratory
and simulations
• Bubble dynamics are considered an important
way to characterize the mixing process
– Mixing rate =
(# bubbles ) / t.
• There is no prevalent formal definition of bubbles
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We Compute the Morse–Smale
Complex of the Upper Envelope Surface
Maximum
F(x) = z
F(x) on the surface is
aligned against the
direction of gravity
which drives the flow
Morse complex
cells drawn in
distinct colors
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In each Morse
complex cell, all
steepest ascending
lines converge to
one maximum
A Hierarchal Model is Generated by
Simplification of Critical Points
• Persistence is varied to annihilate pairs of critical
points and produce coarser segmentations
• Critical points with higher persistence are
preserved at the coarser scales
Saddle
p1
p2
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The Segmentation Method is Robust
From Early Mixing to Late Turbulence
T=100
T=700
T=353
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We Evaluated Our Quantitative
Analysis at Multiple Scales
Under-segmented
Coarse scale
Medium scale
100000
Persistence
Threshold
00.93%
02.39%
04.84%
slope -0.65
slope -1.72
10000
Number of
bubbles
Over-segmented
slope -0.75
1000
100
10
53.0
Timestep
100
353
400
503
700
1000
Fine scale
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We Characterize Events that
Occur in the Mixing Process
death
birth
merge
split
Comp AD 07-DRC
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First Robust Bubble Tracking From
Beginning to Late Turbulent Stages
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First Time Scientists Can Quantify
Robustly Mixing Rates by Bubble Count
slope 1.536
slope 1.949
-0.5
-1
-1.5
-2
-2.5
-3 1
Bubble Count
Derivative
2
0.1
slope 2.575
Derivative of Bubble Count
0
3 4 5 6 7 8 9 10
t/ (log scale)
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Mode-Normalized Bubble
Count
(log scale)
1.0
(9524 bubbles)
0.5
0.01
20 30 40
We Provide the First Quantification of
Known Stages of the Mixing Process
mixing 1.0
transition
0.5
slope 1.536
slope 1.949
-0.5
-1
-1.5
-2
-2.5
weak
-3 1
turbulence
Bubble Count
Derivative
2
3
0.1
slope 2.575
Derivative of Bubble Count
0
linear
growth
4 5 6 7 8 9 10
t/ (log scale)
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Mode-Normalized Bubble
Count
(log scale)
(9524 bubbles)
0.01
20 30 40
strong
turbulence
We Provided the First Feature-Based
Validation of a LES with Respect to a DNS
LES (Large Eddy Simulation)
Mode-Normalized
Bubble Count
DNS (Direct Numerical Simulation)
Normalized Time
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Quantitative Analysis of the Impact of a
Micrometeoroid in a Porous Medium
• Many possible applications:
–
–
–
–
NASA’s Stardust Spacecraft
National Ignition Facility Targets
Light and Robust Materials
many more...
Micrometeoroid
impact
direction
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The Topological Reconstruction Method is
Validated with a Controlled Test Shape
Challenge: robust reconstruction of the structure of a porous medium
Preparation: we develop control test data to validate the approach
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We Report the Distribution of Topological
Features in the Full Resolution Data
saddle
p
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maximum
q
(f(p), f(q))
The Hierarchical Morse-Smale Complex
Has Very Good Reconstruction Properties
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We Compute the Complete Morse-Smale
Complex for the Porous Medium
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Need to Find Proper Threshold Values and
Characterize the Stability of the Solution
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Need to Find Proper Threshold Values and
Characterize the Stability of the Solution
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We Obtain a Robust Reconstruction of the
Filament Structures in the Material
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We Track the Evolution of the Filament
Structure of the Material Under Impact
Time comparison of the reconstructions
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Demo Porous Medium
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The Extracted Structures Allow to Quantify
the Change in Porosity of the Material
Density profiles
Decay in porosity of the material
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Understanding Turbulence for Low
Emission, High Efficiency Combustion
• Lean premixed H2 flames
• Low Swirl Combustion
(LSC) Burners
• Low pollution in energy
production
• High Efficiency in fuel
consumption
• Scalable from residential
to industrial use
• Each variable 3.9-4.5 TB
Experiment
Simulation
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We Take on the Challenge of Developing a
Quantitative Analysis Detecting Turbulence
Understanding combustion processes over
a broad range of burning conditions is an
important problem for designing engines
and power plants.
– Simulation with AMR mesh.
– Simulations of lean premixed hydrogen flames
with three degrees of turbulence.
– Can we identify precisely and track in time
burning regions?
– Can we discriminate the degree of turbulence
from a quantitative analysis?
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We Build a Reduced Topological Model of
H2 Consumption on an Isotermal Surfaces
domain
hierarchy
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We Build a Reduced Topological Model of
H2 Consumption on an Isotermal Surfaces
domain
hierarchy
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We Build a Reduced Topological Model of
H2 Consumption on an Isotermal Surfaces
domain
hierarchy
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We Build a Reduced Topological Model of
H2 Consumption on an Isotermal Surfaces
domain
hierarchy
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We Build a Reduced Topological Model of
H2 Consumption on an Isotermal Surfaces
domain
hierarchy
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We Build a Reduced Topological Model of
H2 Consumption on an Isotermal Surfaces
domain
hierarchy
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We Build a Reduced Topological Model of
H2 Consumption on an Isotermal Surfaces
domain
hierarchy
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We Build a Reduced Topological Model of
H2 Consumption on an Isotermal Surfaces
domain
hierarchy
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We track in time by interpolation in 4D and
contraction to a Reeb Graph
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We track in time by interpolation in 4D and
contraction to a Reeb Graph
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We track in time by interpolation in 4D and
contraction to a Reeb Graph
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We track in time by interpolation in 4D and
contraction to a Reeb Graph
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We track in time by interpolation in 4D and
contraction to a Reeb Graph
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We track in time by interpolation in 4D and
contraction to a Reeb Graph
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We track in time by interpolation in 4D and
contraction to a Reeb Graph
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Each Set of Parameters Results in a Robust
Segmentation and Tracking of Burning Cells
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We Allow Exploration of the Full Space
of Parameters Defining the Features
• interactive exploration
• comparison of
statistics
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Topological Segmentation Allows to Quantify
Turbulence as Slope of the Area Distributions
• Combustion & flame
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Exploration of High Dimensional
Functions for Sensitivity Analysis
Integrated presentation of statistics and topology
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Exploration of High Dimensional
Functions for Sensitivity Analysis
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Analysis of Combustion Simulations
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Pure fuel
The Framework Allows Detailed Visualization
and Analysis of High Dimensional Functions
10 dimensional data set describing the
heat release wrt. to various chemical
species in a combustion simulation
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Pure oxidizer
The Framework Allows Detailed Visualization
and Analysis of High Dimensional Functions
10 dimensional data set describing the
heat release wrt. to various chemical
species in a combustion simulation
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Local extinction
The Framework Allows Detailed Visualization
and Analysis of High Dimensional Functions
10 dimensional data set describing the
heat release wrt. to various chemical
species in a combustion simulation
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Analysis of Climate Data
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The Framework Reveals Relationship Between
Convection and Global Long Wave Flux
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Discrete Analysis of Vector Field Structures is
Enabling New Robust Analysis of Climate Data
How many eddies in the ocean transport
energy of a global climate model?
1
2
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Encoding Must Explicitly Capture Flow’s
Topological Features
H. Theisel, T. Weinkauf, H.-C. Hege, H.-P. Seidel. Gridindependent Detection of Closed Stream Lines in 2D Vector
Fields. VMV 2004.
J. L. Helman and L. Hesselink.
Representation and Display of
Vector Field Topology in Fluid Flow
Data Sets. IEEE Computer, 1989.
G. Scheuermann, X. Tricoche. Topological
Methods in Flow Visualization. Visualization
Handbook, 2004
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Detecting Features Robustly Is Critical
Data Credit: Mathew Maltrud from the
Climate, Ocean and Sea Ice Modelling
program at Los Alamos National Laboratory
(LANL) and the BER Office of Science UVCDAT team
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Classification of Closed Orbits
Attracting
Neutral
Repelling
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Quantized Features: Topological
Decompositions
Stable Manifolds
Topological Skeletons
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Comparison to Other Discretizations
Morse Sets
G. Chen, Q. Deng, A. Szymczak, R. S. Laramee, E. Zhang. Morse
set classification and hierarchical refinement using the Conley
index. IEEE TVCG, 2011.
PC Morse Sets
A. Szymczak. Stable Morse Decompositions
for piecewise constant vector fields on
surfaces. CGF 2011.
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Stable Manifolds of Ocean Data
time depth data:
3600x2400 mesh
~25k critical points
~39k cycles
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CG Enabled Fundamental Advances
in Scientific Data Analysis and Visualization
• Tight cycle of :
– basic research,
– software deployment
– user support
• Plenty of Open Problems:
– Combinatorial Methods for:
• Vector Fields
• Tensor Fields
• Dynamic models
First time scientists can quantify
robustly mixing rates by bubble count
slope 1.949
-0.5
-1.5
-2
-3 1
Comp AD 07-DRC
Bubble Count
Derivative
2
3 4 5 6 7 8 9 10
t/ (log scale)
0.1
slope 2.575
-1
Mode-Normalized Bubble
Count
(log scale)
slope 1.536
0
-2.5
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1.0
(9524 bubbles)
0.5
Derivative of Bubble Count
– Efficient approaches for high
dimensional data
– Simple data structure and
algorithms (fast in practice and
easier to validate)
0.01
20 30 40
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