Solution of Sparse Linear Systems A Case Study in Computational Science & Engineering Solution of Sparse Linear Systems Direct Methods Systematic transformation of system of equations into equivalent systems, until the unknown variables are easily solved for. Iterative methods Starting with an initial “guess” for the unknown vector, successively “improve” the guess, until it is “sufficiently” close to the solution.

Direct Solution of Linear Systems Gaussian Elimination div by 2 *(-1) *(-3) Unknowns solved by back-substitution after Gaussian Elimination

LU Decomposition More efficient than Gaussian Eimination when solving many systems with the same coefficient matrix. First A is decomposed into product: A = LU To solve linear system Ax=b, we need to solve (LU)x=b Let z=Ux; we have L(Ux)=b, or Lz=b. This can be solved for z by forward-substitution. Since Ux=z, and z is now known, we can solve for x by back-substitution. =

Cholesky Factorization If A is symmetric and positive definite , it can be factored in the form Cholesky factorization requires only around half as many arithmetic operations as LU decomposition. The forward and back-substitution process is the same as with LU decomposition. =

Sparse Linear Systems A significant fraction of matrix elements are known to be zero, e.g. matrix arising from a finite-difference discretization of a PDE: At most 5 non-zero elements in any row of the matrix, irrespective of the size of the matrix (number of grid points). Sparse matrix is represented in some compact form that keeps information about the non-zero elements. 1 2 3 4 5 6 1 4 -1 0 -1 0 0 2 -1 4 -1 0 -1 0 3 0 -1 4 0 -1 -1 4 -1 0 0 4 -1 0 5 0 -1 0 -1 4 -1 6 0 0 -1 0 -1 4 1 2 3 4 5 6

Sparse Linear Systems For a 100 by 100 grid, with a finite difference discretization using a 5-point stencil, less than .05% of the matrix elements are non-zero. n 2 1 1 n n 2 Physical nxn Grid n 2 Resulting n 2 x sparse matrix

Compressed Sparse Row Format A commonly used representation for sparse matrices: 0 1 2 3 4 5 0 4 -1 0 -1 0 0 1 -1 4 -1 0 -1 0 2 0 -1 4 0 0 -1 3 -1 0 0 4 -1 0 4 0 -1 0 -1 4 -1 5 0 0 -1 0 -1 4 rb 0 3 7 10 13 17 20 a 4 -1 -1 -1 4 -1 -1 -1 4 -1 -1 4 -1 -1 -1 4 -1 -1 -1 4 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 0 1 3 0 1 2 4 1 2 5 0 3 4 1 3 4 5 2 4 5 col for (i = 0; i<n; i++) for(j=rb[i];j<rb[i+1];j++) y[i] += a[j]*x[col[j]]; Sparse MV Multiply for (i = 0; i<n; i++) for(j=0;j<n;j++) y[i] += a[i][j]*x[j]; Dense MV Multiply

Fill-in Non-Zeros During solution of sparse linear system (by GE or LU or Cholesky), row-updates often result in creation of non-zero entries that were originally zero. Row updates using row-1 result in fill-in non-zeros (F).

Effect of reordering on fill-in Re-ordering the equations (rows) or unknowns (columns) can result in significant change in the number of fill-in non-zeros, and hence time for matrix factorization. Fill-in with GE Reorder rows/cols No fill-in with GE

Associated graph of matrix A graph-based view of matrix’s sparsity structure is extremely useful in generating low-fill re-orderings. The associated graph of a symmetric sparse matrix has a vertex corresponding to each row/col. of matrix, and an edge corresponding to each non-zero matrix entry. 3 2 6 5 4 1

Fill-in and graph transformation Row-i updates row-j, j>i iff Aji is non-zero; in the associated graph a matrix non-zero corresponds to an edge. Row-update(i->j) could cause fill-in non-zero Ajk corresponding to all non-zeros Aik. After all updates from row-i, all neighbors of vertex i in the associated graph form a clique. l l i j i j k k

Fill-in and graph transformation Each row’s effect on fill-in generation is captured by the “clique” transformation on the associated graph. The graph view is valuable in suggesting matrix re-ordering approaches. 3 2 6 5 4 1 3 2 6 5 4 1 3 2 6 5 4 1 4 1 2 3 6 5

Matrix re-ordering: Minimum Degree Graph-based algorithm for generating low-fill re-ordering. Matrix permutation is viewed as node-numbering problem in associated graph. Low-degree nodes are numbered early - so that they are removed without adding many fill-in edges. For example, minimum-degree finds a no-fill ordering. d=1 d=3 d=1 1 d=2 d=3 2 d=1 1 d=2 d=3 2 3 1 d=2 2 3 1 4 d=1

Re-ordered matrix 3 2 6 5 4 1 1 4 5 6 3 2

Matrix re-ordering: Nested Dissection Find a minimal vertex-separator to bisect associated graph; number those nodes last; recursively apply to both halves. Property: Given a numbering of nodes, fill-in Aij exists, j>i, iff there is a path from i to j in graph using only lower numbered vertices. No fill-in edges between one half and other half of partition. 43 49 1-21 22-42 19 21 40 42

Comparison of Ordering Schemes Number of non-zeros after fill-in Sparse matrix factorization time