Non Linear Model Reduction

1
NON-LINEAR MODEL REDUCTION STRATEGIES FOR
GENERATING DATA DRIVEN STOCHASTIC INPUT MODELS
Nicholas Zabaras Materials Process Design and
Control Laboratory Sibley School of Mechanical
and Aerospace Engineering 188 Rhodes Hall Cornell
University Ithaca, NY 14853-3801 Email
zabaras_at_cornell.edu URL http//mpdc.mae.cornell.e
du/
2
ANALYSIS OF HETEROGENEOUS MEDIA
- Thermal and fluid transport in heterogeneous
media are ubiquitous - Range from large scale
systems (geothermal systems) to the small scale -
Most critical devices/applications utilize
heterogeneous/polycrystalline/functionally graded
materials
Thermal transport through polycrystalline and
functionally graded materials

• Properties depend on the distribution of
  material/microstructure
• But only possess limited information about the
  microstructure/property distribution (e.g. 2D
  images)

• Incorporate limited information into stochastic
  analysis
• worst case scenarios
• variations on physical properties

Hydrodynamic transport through heterogeneous
permeable media
3
FRAMEWORK FOR ANALYSIS OF HETEROGENEOUS MEDIA
1. Property extraction
2. Microstructure/property reconstruction
Extract properties P1, P2, .. Pn, that the
structure satisfies. These properties are usually
statistical Volume fraction, 2 Point
correlation, auto correlation
Reconstruct realizations of the structure
satisfying the correlations.
Construct a reduced stochastic model of property
variations from the data. This model must be able
to approximate the class of structures.
Solve the heterogeneous property problem in the
reduced stochastic space for computing property
variations.
4. Stochastic analysis
3. Reduced model
B. Ganapathysubramanian and N. Zabaras,
"Modelling diffusion in random heterogeneous
media Data-driven models, stochastic collocation
and the variational multi-scale method", Journal
of Computational Physics, in press
4
INPUT STOCHASTIC MODELS LINEAR APPROACH

• Methodology for creating linear models from data
• Given some limited information (either in terms
  of statistical correlation functions or sample
  microstructure/property variations)
• Utilize some reconstruction methodology to
  create a finite set of realizations of the
  property/microstructure.
• Utilize this data set to construct a model of
  this variability

Use Proper Orthogonal Decomposition (POD),
Principal Component Analysis (PCA) to construct a
reduced order model of the data.

an
a1
a2
..

Convert variability of property/microstructure to
variability of coefficients. Not all
combinations allowed. Developed subspace reducing
methodology1 to find the space of allowable
coefficients that reconstruct plausible
microstructures
B. Ganapathysubramanian and N. Zabaras,
"Modelling diffusion in random heterogeneous
media Data-driven models, stochastic collocation
and the variational multi-scale method", Journal
of Computational Physics, in press
5
LINEAR APPROACH TO MODEL GENERATION
Successfully applied to investigate effect of
heterogeneous media in diffusion phenomena in
two-phase microstructures. PCA based methods are
easy to implement and are well understood.
But PCA based methods are linear projection
methods Only guaranteed to discover the true
structure of data lying on a linear subspace of
the high dimensional input space
PCA works very well when the input space is linear
What about when the input space is
curved/non-linear? PCA based techniques tend to
overestimate the dimensionality of the model
of eigen vectors

• Further related issues
• How to generalize it to other properties/structure
  s? Can PCA be applied to other classes of
  microstructures, say, polycrystals?
• How does convergence change as the amount of
  information increases? Computationally?

of samples
6
TOWARDS A NON-LINEAR APPROACH TO MODEL GENERATION

• - Find structure of data lying on a possibly
  non-linear subspace of the high-dimensional input
  data
• PCA finds a low-dimensional embedding of the
  data points that best preserves their variance as
  measured in the high-dimensional input space.
  Variance is measured based on Euclidian distance.
  This results in a linear subspace approximation
  of the data

PCA
Pt A
3D data
Pt B

• But what if the data lie on nonlinear curve in
  high-dimensional space?
• Have to unfold/unravel the curve
• Need non-linear approaches

7
NONLINEAR REDUCTION THE KEY IDEA
Set of images. Each image 64x64 4096
pixels Each image is a point in 4096 dimensional
space. But each and every image is related (they
are pictures of the same object). Same object but
different poses. That is, all these images lie on
a unique curve (manifold) in ?4096. Can we get a
parametric representation of this curve? Problem
Can the parameters that define this manifold be
extracted, ONLY given these images (points in
?4096) Solution Each image can be uniquely
represented as a point in 2D space (UD,
LR). Strategy based on the manifold learning
problem
Different images of the same object changes in
up-down (UD) and left-right (LR) poses
8
NONLINEAR REDUCTION THE KEY IDEA (Contd.)
Based on principles from psychology and cognitive
sciences. How does the brain interpret/store/recal
l high-resolution images at different poses,
locations? A low-dimensional parameterization of
this very high-dimensional space Effectively
constructs a surrogate space of the actual
space. Performs all operations on this surrogate
low-dimensional parametric space and maps back to
the input space These ideas are being
increasingly used in problems in vision
enhancement, speech and motor control, data
compression
Different images of the same object changes in
up-down (UD) and left-right (LR) poses
9
NONLINEAR REDUCTION EXTENSION TO INPUT MODELS
Given some experimental correlation that the
microstructure/property variation
satisfies. Construct several plausible images
of the microstructure/property. Each of these
images consists of, say, n pixels. Each image
is a point in n-dimensional space. But each and
every image is related. That is, all these
images lie on a unique curve (manifold) in
?n. Can a low-dimensional parameterization of
this curve be computed? Strategy based on a
variant of the manifold learning problem.
Different microstructure realizations satisfying
some experimental correlations
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A FORMAL DEFINITION OF THE PROBLEM
State the problem as a parameterization problem
(also called the manifold learning problem)
Given a set of N unordered points belonging to a
manifold ? embedded in a high-dimensional space
?n, find a low-dimensional region ? ? ?d that
parameterizes ?, where d ltlt n
Classical methods in manifold learning have been
methods like the Principle Component Analysis
(PCA) and multidimensional scaling (MDS). These
methods have been shown to extract optimal
mappings when the manifold is embedded linearly
or almost linearly in the input space. In most
cases of interest, the manifold is nonlinearly
embedded in the input space, making the classical
methods of dimension reduction highly
approximate. Two approaches developed that can
extract non-linear structures while maintaining
the computational advantage offered by PCA1,2.

• J. B. Tenenbaum, V. De Silva, J. C. Langford, A
  global geometric framework for nonlinear
  dimension reduction Science 290 (2000),
  2319-2323.
• S Roweis, L. Saul., Nonlinear Dimensionality
  Reduction by Locally Linear Embedding, Science
  290 (2000) 2323--2326.

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AN INTUITIVE PICTURE OF THE STRATEGY

• Attempt to reduce dimensionality while
  preserving the geometry at all scales.
• Ensure that nearby points on the manifold map to
  nearby points in the low-dimensional space and
  faraway points map to faraway points in the
  low-dimensional space.

3D data
PCA
Non-linear approach unraveling the curve
Linear approach
12
KEY CONCEPT

• Geometry can be preserved if the distances
  between the points are preserved Isometric
  mapping.
• The geometry of the manifold is reflected in the
  geodesic distance between points
• First step towards reduced representation is to
  construct the geodesic distances between all the
  sample points

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MATHEMATICAL DETAILS AN OVERVIEW
Start from N samples lying on a curve that is
embedded in a high- dimensional space. What are
the properties of this curve ? ?
Generate an isometric transformation
(parameterization) from this curve to a
low-dimensional region. Proof of existence of
this mapping
Next step is to extract some knowledge of the
geometry of this curve. Utilize the concept of
geodesic distance to encode this knowledge of the
geometry. How is this geodesic distance defined
and computed?
How is this information used in numerically
constructing the required mapping?
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PROPERTIES OF THE CURVE ?
Given a set of N sample points lying on ?
Must have a notion of distance between these
sample points. Define an appropriate function
that determines the difference between any two
points
Ensure that it satisfies the properties of
positive-definiteness, symmetry and the triangle
inequality.
Lemma 1 (?, ?) is a metric space
Lemma 2a (?, ?) is a bounded metric space
Lemma 2b (?, ?) is dense
Lemma 2c (?, ?) is complete
Theorem (?, ?) is compact1.
A compact manifold embedded in a high-dimensional
space can be isometrically mapped to a region in
a low-dimensional space2.

• B. Ganapathysubramanian and N. Zabaras, "A
  non-linear dimension reduction methodology for
  generating data-driven stochastic input models",
  Journal of Computational Physics, submitted.
• V. de Silva, J. B. Tenenbaum, Unsupervised
  Learning of Curved Manifolds, In Proceedings of
  the MSRI workshop on nonlinear estimation and
  classification. Springer Verlag, 2002.

15
THE ISOMETRIC MAPPING COMPUTING THE GEODESIC
A compact manifold embedded in a high-dimensional
space can be isometrically mapped to a region in
a low-dimensional space. Isometry encoded into
the geodesic distances
Have no notion of the geometry of the manifold to
start with. Hence cannot construct true geodesic
distances!
Approximate the geodesic distance using the
concept of graph distance ?G(i,j) the distance
of points far away is computed as a sequence of
small hops. This approximation, ?G,
asymptotically matches the actual geodesic
distance ??. In the limit of large number of
samples1,2. (Theorem 4.5 in Ref. 1)

• B. Ganapathysubramanian and N. Zabaras, " A
  non-linear dimension reduction methodology for
  generating data-driven stochastic input models ",
  submitted to Journal of Computational Physics.
• M.Bernstein, V. deSilva, J.C.Langford,
  J.B.Tenenbaum, Graph approximations to geodesics
  on embedded manifolds, Dec 2000

16
THE ISOMETRIC MAPPING COMPUTING THE GEODESIC
Proof of theorem is based on ideas from geometry,
differential algebra and simple probability
arguments
Using this approximation can compute the pair
wise geodesic distance matrix, M, between all the
N sample points
17
FROM THE MATRIX, M, TO A LOW-DIMENSIONAL MAP
Have N objects, the preceding developments
provided a means of computing the pair wise
distances between them. Convert this set of pair
wise distances into points in a low-dimensional
space
This is a straightforward problem in multivariate
statistical analysis1. Can easily solve the
problem using the concept of Multi Dimensional
Scaling2 (MDS)

• J. T. Kent, J. M. Bibby, K. V. Mardia,
  Multivariate Analysis (Probability and
  Mathematical Statistics), Elsevier (2006).
• T. F. Cox, M. A. A. Cox, Multidimensional
  scaling, 1994, Chapman and Hall

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MULTI DIMENSIONAL SCALING
Given the N x N matrix of the geodesic distances,
M, with elements dij.
Compute the symmetric N x N matrix, A, with
elements aij -1/2 dij2.
Suppose that the N low-dimensional points that we
are interested in finding (the parameters of the
N objects in high-dimensional space) are
Denote the N x N matrix of the scalar product of
these N low- dimensional points as B,
Can show that A, and B, are related as
Where H is the centering matrix,
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MULTI DIMENSIONAL SCALING (contd.)
B is a positive definite matrix and can be
written in terms of its eigen values and eigen
vectors as
Where ? is the diagonal matrix of the eigenvalues
and G is the corresponding matrix of
eigenvectors. B will have a decaying
eigen-spectrum. From and we get
The above equation is an estimate of Y in terms
of the largest d eigenvalues of the eigenvalue
decomposition of the squared geodesic distance
matrix
From N points in a high-dimensional space to N
points in a low-dimensional space
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THE NONLINEAR MODEL REDUCTION ALGORITHM
Given a set of N sample points lying on ?
The complete algorithm consists of three simple
steps
Step 1 - Construct the neighborhood graph
Determine which points are neighbors, Construct
the weighted graph with edges given weights
corresponding to the distances, ?, between the
points
Step 2 - Estimate the shortest intrinsic path-
the geodesic lengths Compute the shortest path
length, M, between the points in the weighted
graph, using, say, Floyds algorithm
Step 3 - Construct the d-dimensional embedding
Perform classical Multi Dimensional Scaling on
the geodesic distance matrix, M, to extract a set
of N points in a low-dimensional space
21
CHOOSING THE OPTIMAL DIMENSIONALITY, d
The MDS procedure in the dimension reduction
methodology constructs the eigen-value
decomposition of a matrix. The low-dimensional
representation correspond to the largest d
eigenvalues of this matrix. How is the value of d
chosen?
Use ideas from recent development on estimating
the intrinsic dimensionality of manifolds (Costa
and Hero1) Elegant ideas from differential
geometry and graph theory. Connect dimensionality
to the rate of convergence of a graph theoretic
quantity

• J.A.Costa, A.O.Hero, Geodesic Entropic Graphs for
  Dimension and Entropy Estimation in Manifold
  Learning, IEEE Trans. on Signal Processing, 52
  (2004) 2210--2221.

22
CHOOSING THE OPTIMAL DIMENSIONALITY, d
Based on very elegant ideas linking graph theory
with differential geometry. Based on the theorem
of Beardwood-Halton-Hammersley1,2. Based on
concepts in geometric probability, the theorem
relates the structure/geometry of embedded
manifolds to the graph structure of a finite set
of points on these embedded manifolds
The theorem states that the length functional of
the minimal spanning graph is related to the
entropy of density of distribution of a finite
set of points in a embedded manifold (with some
specific properties)
Costa and Hero1 extended/applied this theorem to
general embedded manifolds Ganapathysubramanian
and Zabaras3 applied it to the model reduction
problem

• J.A.Costa, A.O.Hero, Manifold Learning with
  Geodesic Minimal Spanning Trees,
  arXivcs/0307038v1
• J. Beardwood, J. H. Halton, and J. M. Hammersley,
  The shortest path through many points, Proc.
  Cambridge Philosophical Society, 55 (1959)
  299327.
• B. Ganapathysubramanian and N. Zabaras, " A
  non-linear dimension reduction methodology for
  generating data-driven stochastic input models ",
  submitted to Journal of Computational Physics.

23
CHOOSING THE OPTIMAL DIMENSIONALITY, d (Contd.)
The length functional of the minimal spanning
tree of the geodesic matrix, M, is related to the
intrinsic dimensionality of the low-dimensional
representation of the manifold 1,2.

• B. Ganapathysubramanian and N. Zabaras, "A
  non-linear dimension reduction methodology for
  generating data-driven stochastic input models",
  submitted to Journal of Computational Physics.
• J.A.Costa, A.O.Hero, Geodesic Entropic Graphs for
  Dimension and Entropy Estimation in Manifold
  Learning, IEEE Trans. on Signal Processing, 52
  (2004) 2210--2221.

24
CHOOSING THE OPTIMAL DIMENSIONALITY, d (Contd.)
From the above theorem, can easily extract the
dimensionality. Taking the logarithm of the main
result of the algorithm,
with
Where L is the length functional of the minimal
spanning tree of the geodesic matrix, N is the
number of samples and d is the optimal
dimensionality.
Use readily available algorithms (Kruskals
algorithm or Prims algorithm) to compute L for
different sample sizes
Perform a least squares fit for the value of a
25
THE NONLINEAR MODEL REDUCTION FRAMEWORK
? ? ?d
?? ?n.
Given N unordered samples
N points in a low dimensional space
The procedure results in N points in a
low-dimensional space. The geodesic distance
MDS step (Isomap algorithm1) results in a
low-dimensional convex, connected space2, ? ? ?d.
Using the N samples, the reduced space is given
as
? serves as the surrogate space for ?. Access
variability in ? by sampling over ?. BUT have
only come up with ? ?? map . Need ??? map too

• J. B. Tenenbaum, V. De Silva, J. C. Langford, A
  global geometric framework for nonlinear
  dimension reduction ,Science 290 (2000),
  2319-2323.
• B. Ganapathysubramanian and N. Zabaras, "A
  non-linear dimension reduction methodology for
  generating data-driven stochastic input models",
  submitted to Journal of Computational Physics.

26
THE REDUCED ORDER STOCHASTIC MODEL
Only have N pairs to construct ??? map. Various
possibilities based on specific problem at hand.
But have to be conscious about computational
effort and efficiency. Illustrate 3 such
possibilities below. Error bounds can be
computed1.
? ? ?d
?? ?n
? ? ?d
?? ?n
2. Local linear interpolation
1. Nearest neighbor map
? ? ?d
?? ?n
3. Local linear interpolation with projection

• B. Ganapathysubramanian and N. Zabaras, "A
  non-linear dimension reduction methodology for
  generating data-driven stochastic input models",
  submitted to Journal of Computational Physics.

27
THE LOW DIMENSIONAL STOCHASTIC MODEL

• Algorithm consists of two parts.
• Compute the low-dimensional representation of a
  set of N unordered sample points belonging to a
  high-dimensional space

N points in a low dimensional space
Given N unordered samples
Compute pairwise geodesic distance
Perform MDS on this distance matrix
For using this model in a stochastic collocation
framework, must sample points in ??? 2) For an
arbitrary point ? ? must find the corresponding
point x ?. Compute the mapping from ???
? ? ?d
?? ?n.
28
NUMERICAL EXAMPLE
Given an experimental image of a two-phase
metal-metal composite (Silver-Tungsten
composite). Find the variability in temperature
arising due to the uncertainty in the knowledge
of the exact 3D material distribution of the
specified microstructure.
Problem strategy Extract pertinent statistical
information form the experimental
image Reconstruct dataset of plausible 3D
microstructures Construct a low-dimensional
parametrization of this space of
microstructures Solve the SPDE for temperature
evolution using this input model in a stochastic
collocation framework
T -0.5
T 0.5
1. S. Umekawa, R. Kotfila, O. D. Sherby, Elastic
properties of a tungsten-silver composite above
and below the melting point of silver J. Mech.
Phys. Solids 13 (1965) 229-230
29
TWO PHASE MATERIAL
Experimental image
Experimental statistics
Realizations of 3D microstructure
GRF statistics
30
NON LINEAR DIMENSION REDUCTION
The developments detailed before are applied to
find a low-dimensional representation of these
1000 microstructure samples. The optimal
representation of these points was a
9-dimensional region
Able to theoretically show that these points in
9D space form a convex region in ?9. This convex
region now represents the low- dimensional
stochastic input space Use sparse grid
collocation strategies to sample this space.
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COMPUTATIONAL DETAILS
The construction of the stochastic solution
through sparse grid collocation level 5
interpolation scheme used Number of deterministic
problems solved 26017
Computational domain of each deterministic
problem 65x65x65 pixels
Total number of DOF 653x26017 7x109
Computational platform 50 nodes on local Linux
cluster (x2 3.2 GHz) Total time 210 minutes
32
Coupling data driven model generation with a
Multiscale stochastic modeling framework
Seamlessly couple stochastic analysis with
multiscale analysis. Multiscale framework (large
deformation/thermal evolution) Adaptive
stochastic collocation framework Provides roadmap
to efficiently link any validated multiscale
framework Coupled with a data-driven input model
strategy to analyze realistic stochastic
multiscale problems.
Mean statistics
Stochastic multiscale framework
T -0.5
T 0.5
Higher-order statistics
Statistics extraction model generation
Limited data
33
OTHER APPLICATIONS OF THIS FRAMEWORK
? ? ?d
From unordered samples in high-dimensional space
to a convex space representing the
parameterization
?? ?n

• This methodology has significant applications to
  problems where working in high-dimensional spaces
  is computationally intractable.
• Can pose the problem in a low-dimensional space

Property-structure space
Property-process space
Process-structure space
visualizing property evolution, process-property
maps, searching and contouring, representing
input uncertainty, data mining
A1000
A80
a
Process paths
A100
34
Stochastic design/optimization framework
Most critical components in many
applications/devices are usually fabricated from
polycrystalline/ functionally graded/
heterogeneous materials
The properties at the device-scale (thermal and
electrical conductivity, elastic moduli and
failure mechanisms) depend on the microstructure
and material distribution at the meso-scale
Have to design the operating conditions/parameters
taking into account this limited information
about the microstructure.
Robust design in the presence of topological
uncertainty
Given limited topological information about the
microstructure, design the optimal (stochastic)
heat flux to be applied on one end of the device
such that a required (stochastic) temperature is
maintained at the other end.

• Design criterion
• Maintain a specified thermal profile on the
  right wall
• This thermal profile is given in terms of a pdf,
  usually specified in terms of moments

• Additional uncertainties
• Do not know the exact microstructure that the
  device is made up of.
• Only know certain statistical correlations that
  the microstructure satisfies.
• Consider the microstructure to be a random field
  ?.

• Design variables
• Must design the pdf of the optimal heat flux

35
Stochastic design/optimization framework Some
details
Denote the stochastic space in which the flux,
q(x) exists as ?q. Convert the problem of
constructing the stochastic function q ?q into
designing the values of the stochastic function
at a finite number of points in the stochastic
space (collocation based design)
Convert the problem into an optimization problem
Solution exists to the problem in the sense of
Tikhonov.
Use gradient based minimization methods. Need to
compute gradient. Notion of the directional
derivative
Need to compute the directional derivative of the
temperature
36
Stochastic design/optimization framework (contd.)
Leads to the definition of the continuum
stochastic sensitivity equations. Linear in
design variable perturbation part of the
dependant variables
The stochastic direct problem is converted into a
set of n-decoupled deterministic equation (the
collocation based approach to solve SPDEs) The
stochastic temperature sensitivity equations are
simply directional derivatives of the n
deterministic that form the above stochastic
direct problem. The stochastic sensitivity is
then represented in terms of these decoupled
stochastic equations as
Novel, highly-efficient way to compute stochastic
sensitivities based on parallel sparse grid
collocation schemes. Major advance in
non-intrusive stochastic optimization any
stochastic optimization problem can now be posed
as a deterministic optimization problem in a
large-dimensional space. Classical gradient and
gradient-free optimization methods are directly
applicable.
37
Stochastic design/optimization framework
Zabaras et al. JCP, 2007b
Standard deviation of temperature
Mean temperature (right wall)
Physical domain 20 µm x 20µm x
20µm Computational domain 65x65x65 Microstructure
stochastic space 1177 collocation points The
design stochastic heat flux represented as a
Bezier surface number of parameters 25
Designed mean heat flux (left wall)
Run optimization problem on 36 nodes of our Linux
cluster Each iteration 3 hours Solve 10593
deterministic problems in each iteration Each
deterministic problem has 65x65x65 275625
DOF. Total number of design variables 225
Bounds on the flux variation (left wall)
38
Impact 3D Multi-scale Design of Deformation
Processes

• A two-length scale continuum sensitivity
  framework has been developed that allows the
  efficient and accurate computation of the effects
  of perturbations of macroscopic design variables
  (e.g. dies and preforms) on microscopic variables
  (slip system resistances, crystallographic
  texture, etc.)
• Using this sensitivity framework, the first 3D
  multiscale deformation process design simulator
  has been developed for the control of
  texture-dependent material properties and applied
  to complex processes.

Macro-sensitivity problem driven by perturbation
to macro-design variable (b)
Micro-sensitivity problem driven by sensitivity
of deformation gradient

• Desired product shape
• Desired microstructure-sensitive properties

Bn1
Bo
X
Bn1
Micro-scale representation (Orientation
distribution function)
39
Multi-scale design Design for uniform yield
strength during 3D forging process
Initial guess Large variation in yield strength,
incomplete fill
Strong z-axis lt110gt fibers (compression texture)
Uniform strength, Desired shape obtained
ODF evolution at the center of the workpiece
Distribution of (z- axis tension) yield strength
obtained at various design iterations
40
Impact Application to designing critical
components with extremal properties

• - Significant developments towards the design of
  processing paths and materials for manufacturing
  critical components that have extremal
  properties.
• Dimension reduction strategy has significant
  applications to problems where working in high
  dimensional spaces is computationally
  intractable visualizing property evolution,
  process-property maps, searching and contouring.
• Stochastic multiscale framework provides an
  approach to incorporate in the multiscale design
  framework shown earlier operational variabilities
  across multiple scales. Provide rigorous failure
  criteria and the associated probabilities.

41
Future Plans

• Address a number of critical issues on non-linear
  model reduction techniques basis representation,
  projection, solution of PDEs, SPDEs, etc.
• Utilize model reduction strategy to compute
  realistic input stochastic models of
  polycrystalline materials based on limited
  information.
• Develop a multiscale framework for modeling the
  effect of microscale (topological) uncertainty on
  the performance of macro-components. Investigate
  uncertainty propagation and interaction of
  uncertainties across multiple scales.
• Utilize model reduction strategy as a technique
  for reducing problems in high- dimensional spaces
  (visualizing property evolution, process-property
  maps, searching and contouring) into more
  tractable low-dimensional surrogate spaces.
• Address many open issues on stochastic
  optimization using sparse grid collocation. Many
  technologically important applications.
• Multi-element collocation wavelets as an
  alternative to Lagrange polynomials (extension to
  Smolyak quadrature selection rule).