Computational meshes, matrices, conjugate gradients, and mesh partitioning Some slides are from David Culler, Jim Demmel, Bob Lucas, Horst Simon, Kathy Yelick, et al., UCB CS267 CS267 Lecture 2

Parallelism in Regular meshes Computing a Stencil on a regular mesh need to communicate mesh points near boundary to neighboring processors. Often done with ghost regions Surface-to-volume ratio keeps communication down, but Still may be problematic in practice Implemented using “ghost” regions. Adds memory overhead

Irregular mesh: NASA Airfoil in 2D

Composite Mesh from a Mechanical Structure

Adaptive Mesh Refinement (AMR) Adaptive mesh around an explosion Refinement done by calculating errors Parallelism Mostly between “patches,” dealt to processors for load balance May exploit some within a patch (SMP)

Adaptive Mesh fluid density Shock waves in a gas dynamics using AMR (Adaptive Mesh Refinement) See: http://www.llnl.gov/CASC/SAMRAI/

Irregular mesh: Tapered Tube (Multigrid)

Challenges of Irregular Meshes for PDE’s How to generate them in the first place E.g. Triangle, a 2D mesh generator by Jonathan Shewchuk 3D harder! E.g. QMD by Stephen Vavasis How to partition them ParMetis, a parallel graph partitioner How to design iterative solvers PETSc, a Portable Extensible Toolkit for Scientific Computing Prometheus, a multigrid solver for finite element problems on irregular meshes How to design direct solvers SuperLU, parallel sparse Gaussian elimination These are challenges to do sequentially, more so in parallel

Converting the Mesh to a Matrix

Sparse matrix data structure (stored by rows) 31 53 59 41 26 31 53 59 41 26 1 3 2 Full: 2-dimensional array of real or complex numbers (nrows*ncols) memory Sparse: compressed row storage about (2*nzs + nrows) memory

Distributed row sparse matrix data structure Each processor stores: # of local nonzeros range of local rows nonzeros in CSR form P2 Pn

Conjugate gradient iteration to solve A*x=b x0 = 0, r0 = b, d0 = r0 for k = 1, 2, 3, . . . αk = (rTk-1rk-1) / (dTk-1Adk-1) step length xk = xk-1 + αk dk-1 approx solution rk = rk-1 – αk Adk-1 residual βk = (rTk rk) / (rTk-1rk-1) improvement dk = rk + βk dk-1 search direction One matrix-vector multiplication per iteration Two vector dot products per iteration Four n-vectors of working storage

Parallel Dense Matrix-Vector Product (Review) y = A*x, where A is a dense matrix Layout: 1D by rows Algorithm: Foreach processor j Broadcast X(j) Compute A(p)*x(j) A(i) is the n by n/p block row that processor Pi owns Algorithm uses the formula Y(i) = A(i)*X = Sj A(i)*X(j) P0 P1 P2 P3 x P0 P1 P2 P3 y

Parallel sparse matrix-vector product Lay out matrix and vectors by rows y(i) = sum(A(i,j)*x(j)) Only compute terms with A(i,j) ≠ 0 Algorithm Each processor i: Broadcast x(i) Compute y(i) = A(i,:)*x Optimizations Only send each proc the parts of x it needs, to reduce comm Reorder matrix for better locality by graph partitioning Worry about balancing number of nonzeros / processor, if rows have very different nonzero counts x y P0 P1 P2 P3 P0 P1 P2 P3

Other memory layouts for matrix-vector product Column layout of the matrix eliminates the broadcast But adds a reduction to update the destination – same total comm Blocked layout uses a broadcast and reduction, both on only sqrt(p) processors – less total comm Blocked layout has advantages in multicore / Cilk++ too P0 P1 P2 P3 P0 P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 P13 P14 P15

Irregular mesh: NASA Airfoil in 2D

Graphs and Sparse Matrices Sparse matrix is a representation of a (sparse) graph 1 2 3 4 5 6 1 1 1 2 1 1 1 3 1 1 1 4 1 1 5 1 1 6 1 1 3 2 4 1 5 6 Matrix entries are edge weights Number of nonzeros per row is the vertex degree Edges represent data dependencies in matrix-vector multiplication

Graph partitioning Assigns subgraphs to processors Determines parallelism and locality. Tries to make subgraphs all same size (load balance) Tries to minimize edge crossings (communication). Exact minimization is NP-complete. edge crossings = 6 edge crossings = 10 Nodes and edges in graph may be weighted Optimal graph partitioning is NP-complete (define for non-CS audience) Good graph partitioning techniques exist More on this in a later lecture CS267 Lecture 2

Link analysis of the web 1 5 2 3 4 6 7 1 2 3 4 7 6 5 Web page = vertex Link = directed edge Link matrix: Aij = 1 if page i links to page j

Web graph: PageRank (Google) [Brin, Page] An important page is one that many important pages point to. Markov process: follow a random link most of the time; otherwise, go to any page at random. Importance = stationary distribution of Markov process. Transition matrix is p*A + (1-p)*ones(size(A)), scaled so each column sums to 1. Importance of page i is the i-th entry in the principal eigenvector of the transition matrix. But the matrix is 1,000,000,000,000 by 1,000,000,000,000.

A Page Rank Matrix Importance ranking of web pages Stationary distribution of a Markov chain Power method: matvec and vector arithmetic Matlab*P page ranking demo (from SC’03) on a web crawl of mit.edu (170,000 pages)

Social Network Analysis in Matlab: 1993 Co-author graph from 1993 Householder symposium

Social Network Analysis in Matlab: 1993 Sparse Adjacency Matrix Which author has the most collaborators? >>[count,author] = max(sum(A)) count = 32 author = 1 >>name(author,:) ans = Golub

Social Network Analysis in Matlab: 1993 Have Gene Golub and Cleve Moler ever been coauthors? >> A(Golub,Moler) ans = 0 No. But how many coauthors do they have in common? >> AA = A^2; >> AA(Golub,Moler) ans = 2 And who are those common coauthors? >> name( find ( A(:,Golub) .* A(:,Moler) ), :) ans = Wilkinson VanLoan