CS 584

Review n Systems of equations and finite element methods are related.

Gauss-Seidel Iteration n Gauss-Seidel parallelism can be increased by using graph coloring.

Partitioning P0P0 P1P1 P2P2 P3P3

Communication P0P0 P1P1 P2P2 P3P3

Finite Element Method n Used for deriving approximate numerical solutions to partial differential equations over a discretized domain. n Two sources of error –numerical (due to approximation) –discretization

Finite Element Method n Domain is bounded n Some grid is imposed on the domain n Equations are generated to indicate the relationships between grid points n Jacobi or Gauss-Seidel iteration is used to relax the system and converge to the solution

The Heat Equation n The steady-state temperature u at any point (x,y) on a metal sheet is governed by: u       yx u n Discretization allows us to transform this into a system of linear equations which will approximate the partial derivatives.

The Finite Element Method n For most applications, the finite element graph is not a regular structure. n The corresponding system of equations is, most often, very large and very sparse. n Efficient parallelization of the system depends heavily on the way the domain is partitioned among processors.

Partitioning Techniques n Regular grids –One dimensional striping –Two dimensional blocking n Generalized Graphs –Levelization –Scattered Decomposition –Recursive Bisection

Levelization n Begin with a boundary –Number these nodes 1 n All nodes connected to a level 1 node are labeled 2, etc. n Partitioning is performed –determine the number of nodes per processor –count off the nodes of a level until exhausted –proceed to the next level

Levelization

n Want to insure nearest neighbor comm. n If p is # processors and n is # nodes. n Let r i be the sum of the number of nodes in contiguous levels i and i + 1 n Let r = max{r 1, r 2, …, r n } n Nearest neighbor communication is assured if n/p > r

Scattered Decomposition n Used for highly irregular grids n Partition load into a large number r of rectangular clusters such that r >> p n Each processor is given a disjoint set of r/p clusters. n Communication overhead can be a problem for highly irregular problems.

Recursive Bisection n Recursively divide the domain in two pieces at each step. n 3 Methods –Recursive Coordinate Bisection –Recursive Graph Bisection –Recursive Spectral Bisection

Recursive Coordinate Bisection n Divide the domain based on the physical coordinates of the nodes. n Pick a dimension and divide in half. n RCB uses no connectivity information –lots of edges crossing boundaries –partitions may be disconnected n Some new research based on graph separators overcomes some problems.

Recursive Graph Bisection n Based on graph distance rather than coordinate distance. n Determine the two furthest separated nodes n Organize and partition nodes according to their distance from extremities. n Computationally expensive –Can use approximation methods.

Recursive Spectral Bisection n Uses the discrete Laplacian n Let A be the adjacency matrix n Let D be the diagonal matrix where –D[i,i] is the degree of node I n L G = A - D

Recursive Spectral Bisection n LG is negative semidefinite n Its largest eigenvalue is zero and the corresponding eigenvector is all ones. n The magnitude of the second largest eigenvalue gives a measure of the connectivity of the graph. n Its corresponding eigenvector gives a measure of distances between nodes.

Recursive Spectral Bisection n The eigenvector corresponding to the second largest eigenvalue is the Fiedler vector. n Calculation of the Fiedler vector is computationally intensive. n RSB yields connected partitions that are very well balanced.

RCB 529 edges cutRGB 618 edges cut RSB 299 edges cut