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Computational issues in nanotechnology and stochastic computing. Ashok Srinivasan Department of Computer Science Florida State University. Motivation. Research areas Parallel algorithms Scientific computing Discrete algorithms Applications. Motivation ... 2.
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Computational issues in nanotechnology and stochastic computing Ashok Srinivasan Department of Computer Science Florida State University
Motivation • Research areas • Parallel algorithms • Scientific computing • Discrete algorithms • Applications
Motivation ... 2 • Applications needing large amounts of computational power • Nanotechnology • Pharmaceuticals • Finance • Defense
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
Motivation ... 4 • Algorithms • Scalable • Latency tolerant • Enabling technologies • Fault tolerant • Usable software tools
Outline • Applications • Nanotechnology • Background • Sequential computation • Parallelization • Research issues • Algorithms • Stochastic techniques • Scalable parallelization • Linear algebra • Applications
Applications • Nanotechnology • Background • Sequential computation • Parallelization • Research issues
Background • Uses of Carbon nanotubes • Materials • NEMS • Transistors • Displays • Etc • www.ipt.arc.nasa.gov
Sequential computation • Molecular dynamics, using Brenner’s 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
Force computations • Pair interactions • Bond angles • Dihedral • Multibody
Profile of execution time • 1: Force • 2: Neighbor list • 3: Predictor/corrector • 4: Thermostating • 5: Miscellaneous
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
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)
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
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
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
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
Load balancing • Atom based decomposition • For each atom, compute forces due to each bond, angle, and dihedral • Load not balanced
Load balancing ... 2 • Bond based decomposition • For each bond, compute forces due to that bond, angles, and dihedrals • Finer grained • Load still not balanced!
Load balancing ... 3 • Load imbalance was not caused by granularity • Symmetry is used to reduce calculations through • If i > j, don’t compute for bond (i,j) • So threads get unequal load • Change condition to • If i+j is even, don’t compute bond (i,j) if i > j • If i+j is odd, don’t compute bond (i,j) if i < j • Does not work, due to regular structure of nanotube • Use a different condition to balance load
Load balancing ... 4 • Load is much better balanced now • ... at least for this simple configuration
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
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
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
Algorithms • Stochastic techniques • Scalable parallelization • Linear algebra • Applications
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
Linear algebra • Linear solvers • Matrix-vector multiplication • Smallest eigenvalue and eigenvector • Largest eigenvalue and eigenvector
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 state a by pab = |aab|/Si |aai| • Define random variables 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).
MC inverse iterations • Obtain the eigenvector for the smallest eigenvalue as • (A-1)i h, as i approaches infinity for some h • Repeatedly solve: Axk+1 = 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.
Applications • Graph partitioning • Seriation
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.
Seriation • Applications in • DNA sequencing • Matrix envelope reduction • Archaeological dating Given a similarity function f, find a permutation psuch that p(i) < p(j) < p(k) implies f(i,j) > f(i,k) Heuristic: Compute the Fiedler vector of L. Order vertices by the values of the corresponding components of the Fiedler vector.
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
Edge cut and time using deflated matrix, relative to exact Fiedler vector, for inverse iterations. Solid line – test.graph, dash-dotted line – hammond.graph.
Comparison of current stationary process (solid line), with Jacobi (dash-dotted) and Gauss-Seidel (dashed), for test.graph.
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