Computational issues in nanotechnology and stochastic computing

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Computational issues in nanotechnology and
stochastic computing

• Ashok Srinivasan
• Department of Computer Science
• Florida State University

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Motivation

• Research areas
• Parallel algorithms
• Scientific computing
• Discrete algorithms
• Applications

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Motivation ... 2

• Applications needing large amounts of
  computational power
  
• Nanotechnology
• Pharmaceuticals
• Finance
• Defense

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Motivation ... 3

• New computational paradigms
• Grid computing
• Massive parallelism
• Need new algorithmic paradigms
• Develop algorithms and software tools for the
  computational environment 5 to 10 years from now
  

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Motivation ... 4

• Algorithms
• Scalable
• Latency tolerant
• Enabling technologies
• Fault tolerant
• Usable software tools

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Outline

• Applications
• Nanotechnology
• Background
• Sequential computation
• Parallelization
• Research issues
• Algorithms
• Stochastic techniques
• Scalable parallelization
• Linear algebra
• Applications

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Applications

• Nanotechnology
• Background
• Sequential computation
• Parallelization
• Research issues

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Background

• Uses of Carbon nanotubes
• Materials
• NEMS
• Transistors
• Displays
• Etc
• www.ipt.arc.nasa.gov

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Sequential computation

• Molecular dynamics, using Brenners potential
• Short-range interactions
• Neighbors can change dynamically during the
  course of the simulation
  
• Computational scheme
• Find force on each particle due to interactions
  with close neighbors
  
• Update position and velocity of each atom

Conventional particle methods, with pair-wise
interactions
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Force computations
Pair interactions

• Bond angles

Dihedral
Multibody
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Performance analysis
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Profile of execution time
1 Force 2 Neighbor list 3 Predictor/correc
tor
4 Thermostating 5 Miscellaneous
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Profile for force computations
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Neighbor search

• Neighbor lists
• Crude algorithm
• Compare each pair, and determine if they are
  close enough
  
• O(N2) for N atoms
• Cell based algorithm
• Divide space into cells
• Place atoms in their respective cells
• Compare atoms only in neighboring cells
• Problem
• Many empty cells
• Inefficient use of memory

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Computational geometry techniques

• Orthogonal search data structures
• K-d tree
• Tree construction time O(N log N)
• Worst case search overhead O(N2/3)
• Memory O(N)
• Range tree
• Tree construction time O(N log2N)
• Worst case search overhead O(log2N)
• Memory O(N log2N)

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Desired properties of search techniques

• Update should be efficient
• But the number of atoms does not change
• Position changes only slightly
• The queries are known too
• Use knowledge of the structure of the nanotube
• Account for periodic boundary conditions
• Parallelization

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Parallelization

• Shared memory
• Common memory
• Multiple threads divide the computation amongst
  themselves
  
• Distributed memory
• Distinct memory for each process
• Processes communicate to exchange data
• Distributed shared memory
• Memory physically distributed, but logically
  shared
  
• Data locality important

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Shared memory parallelization

• Do each of the following loops in parallel
• For each atom
• Update forces due to atom i
• If neighboring atoms are owned by other threads,
  update an auxiliary array
  
• For each thread
• Collect force terms for atoms it owns
• Srivastava, et al, SC-97 and CSE 2001
• Simulated 105 to 107 atoms
• Up to 32 processors
• Speedup around 16
• Include long-range forces too

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Message passing parallelization

• Decompose domain into cells
• Each cell contains its atoms
• Assign a set of adjacent cells to each processor
• Each processor computes values for its cells
• Communicates with neighbors when their data is
  needed
  
• Caglar and Griebel, World scientific, 1999
• Simulated 108 atoms on up to 512 processors
• Linear speedup for 160,000 atoms on 64 processors

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Load balancing

• Atom based decomposition
• For each atom, compute forces due to each bond,
  angle, and dihedral
  
• Load not balanced

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Load balancing ... 2

• Bond based decomposition
• For each bond, compute forces due to that bond,
  angles, and dihedrals
  
• Finer grained
• Load still not
• balanced!

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Load balancing ... 3

• Load imbalance was not caused by granularity
• Symmetry is used to reduce calculations through
• If i j, dont compute for bond (i,j)
• So threads get unequal load
• Change condition to
• If ij is even, dont compute bond (i,j) if i
  j
  
• If ij is odd, dont compute bond (i,j) if i
• Does not work, due to regular structure of
  nanotube
  
• Use a different condition to balance load

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Load balancing ... 4

• Load is much better balanced now
• ... at least for this simple configuration

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Locality

• Locality important to reduce cache misses

Current scheme based on lexical ordering
Alternate Decompose based on a breadth first
search traversal of the atom-interaction graph
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Locality ... 2
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Research issues

• Neighbor search
• More efficient data structures
• Update should be efficient
• But the number of atoms does not change
• Position changes only slightly
• The queries are known too
• May be able to use knowledge of the structure of
  the nanotube
  
• Account for periodic boundary conditions
• Parallelization

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Research issues ... 2

• Load balancing and locality
• Better graph based techniques
• Geometric partitioning
• Dynamic schemes
• Use structure of the tube
• Spectral partitioning
• Multi-scale
• Space
• Time

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Algorithms

• Stochastic techniques
• Scalable parallelization
• Linear algebra
• Applications

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Scalable parallelization

• Conventional Monte Carlo parallelization
• Perform identical computations on each processor,
  but with a different random number sequence
  
• Finally, combine the results
• Latency tolerant and fault tolerant

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Linear algebra

• Linear solvers
• Matrix-vector multiplication
• Smallest eigenvalue and eigenvector
• Largest eigenvalue and eigenvector

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Monte Carlo power method
Obtain the eigenvector for the largest
eigenvalue as Amh, as m approaches infinity for
some h Use a random walk of length m to estimate
Amh Initial probabilities given by Pa ha/Si
hi Transition probability from state b to sta
te a by pab aab/Si aai Define random varia
bles Wi as W0 hk0/Pk0, Wi
Wi-1 akiki-1 / pkiki-1, where ki i th state
of random walk Then E(Widaki) (Aih)a, where d
is the Kronecker delta function (dij 1 if i
j, and 0 otherwise).
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MC inverse iterations

• Obtain the eigenvector for the smallest
  eigenvalue as
  
• (A-1)i h, as i approaches infinity for some h
• Repeatedly solve Axk1 Axk, x0 h
• MC linear solve write A I C. Then
• yk Cyk-1 h S Ciy0, y0 h
• Estimate yk for large k, for example, using the
  matrix-vector product technique to estimate each
  Ciy0.

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Applications

• Graph partitioning
• Seriation

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Graph partitioning

• Applications in
• Parallel computing
• VLSI
• Databases
• Clustering
• Linear programming
• Matrix reordering

Partition the vertices into components of equal
size such the number of edges between vertices in
different components is minimized
Heuristic Compute the Fiedler vector of L.
Partition vertices such that all vertices with
Fiedler component smaller than the median are in
one component, and the rest in another.
Recursively apply this algorithm.
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Seriation

• Applications in
• DNA sequencing
• Matrix envelope reduction
• Archaeological dating

Given a similarity function f, find a permutation
p such that p(i)
f(i,k)
Heuristic Compute the Fiedler vector of L. Order
vertices by the values of the corresponding
components of the Fiedler vector.
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Acceleration techniques for Laplacian of a graph

• Deflation define H as
• hij -1 if j 1, hij 1 if j i 1, and 0
  otherwise
  
• HLH-1 yields a deflated matrix B.
• B is at least half as sparse as L, and can be
  computed in time proportional to the number of
  non-zero elements of B.
  
• The Fiedler vector is easily computed from the
  eigenvector of the smallest eigenvalue of B.
  
• Shift and use matrix-vector multiplication
• If D 2 Si di, compute largest eigenvalue of DI
  B
  

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Edge cut and time using deflated matrix, relative
to exact Fiedler vector, for inverse iterations.
Solid line test.graph, dash-dotted line
hammond.graph.
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Comparison of current stationary process (solid
line), with Jacobi (dash-dotted) and Gauss-Seidel
(dashed), for test.graph.
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Research issues

• We have developed non-Jacobi based techniques,
  with theoretically better properties
  
• Other stationary and non-stationary methods
• Use the structure of the application, for example
  the nanotube, to accelerate convergence