A Roofline Performance Analysis of an Algebraic Multigrid PDE Solver

We present a performance analysis of a novel element-based algebraic multigrid (AMGe) method combined with a robust coarse-grid solution technique based on HSS lowrank sparse factorization. Our test datasets come from the SPE Comparative Solution Project for oil reservoir simulations. The current performance study focuses on one multicore node and on bound analysis using the roofline technique. We found that keeping a small value of spectral tolerance is most critical to achieve the best AMG solver performance. Our roofline bound estimate is within 23% accuracy compared to the actual runtime on a Cray XC30 12-cores processor. I. THE SA-ρAMGE ALGORITHM The Smoothed Aggregation Spectral Element Agglomeration AMG method was originally introduced by Brezina and Vassilevski [1] and implemented by Kalchev [2], [3]. It is a two-level algorithm that forms the prolongation operator P by first computing the eigenvectors of the low-end eigenvalues of the local stiffness matrices associated with the agglomerates and then multiplying by a matrix polynomial smoother (see [1] for the details of this step). The coarse-grid matrix is computed as Ac = PAP , where A is the finite-element stiffness matrix. The AMG solution cycle proceeds by alternating between the two levels. Specifically, cycle k transforms the current iterate xk into xk+1 using the following steps: 1) Pre-smoothing: yk ← xk +M(b−Axk) 2) Restriction: rc ← P (b−Ayk) 3) Coarse solution: xc ← A−1 c rc 4) Interpolation: zk ← yk + Pxc 5) Post-smoothing: xk+1 ← zk +M(b−Azk) In the above, the matrix M corresponds to a polynomial smoother such that M−1 = (I − pν(DA))A, where pν(t) is a polynomial and D is a diagonal matrix that can be efficiently computed from A. The polynomial is of degree 3ν+1, where ν is a user-specified parameter, and has the form pν(t) = ( 1− t τ1 )( 1− t τ2 ) · · · ( 1− t τ3ν+1 )