  Generally, discretization of partial differential equations (PDEs) creates a
sequence of linear systems $A_k x_k = b_k, k = 0, 1, 2, ..., N$ with well-known
and structured sparsity patterns. Preconditioners are often necessary to
achieve fast convergence When solving these linear systems using iterative
solvers. We can use preconditioner updates for closely related systems instead
of computing a preconditioner for each system from scratch. One such
preconditioner update is the sparse approximate map (SAM), which is based on
the sparse approximate inverse preconditioner using a least squares
approximation. A SAM then acts as a map from one matrix in the sequence to
another nearby one for which we have an effective preconditioner. To
efficiently compute an effective SAM update (i.e., one that facilitates fast
convergence of the iterative solver), we seek to compute an optimal sparsity
pattern. In this paper, we examine several sparsity patterns for computing the
SAM update to characterize optimal or near-optimal sparsity patterns for linear
systems arising from discretized PDEs.
