Many problems encountered in BoxLib applications require solutions to linear system, e.g., elliptic partial differential equations such as the Poisson equation for self-gravity, and the diffusion equation. BoxLib therefore includes geometric multigrid solvers for solving problems which use both cell-centered and nodal data. For this project, we have focused on the cell-centered solver due to its relative simplicity compared to the nodal solver.
Geometric multigrid is an iterative method for solving linear problems which contains roughly 4 steps:
- coarse-grid linear solve (either approximate or exact)
Although here we will not discuss the details of the geometric multigrid method, we summarize each of these steps below as they pertain to computational algorithms. Although these steps are algorithmically unique, we note that all of them feature low arithmetic intensity and are thus sensitive to cache and memory bandwidth.
A relaxation consists of one or more iterations of an approximate solution to the system of linear equations. In geometric multigrid, common algorithms used here include Jacobi and Gauss-Seidel. By default, the BoxLib solver uses a variation on Gauss-Seidel called Gauss-Seidel red-black ("GSRB"). GSRB deviates from the original Gauss-Seidel method by exploiting a symmetry in the data dependence among matrix elements, such that an update sweep of all matrix elements follows a stride-2 pattern rather than stride-1. (This property manifests in the innermost loop of the kernel shown below).
do k = lo(3), hi(3) do j = lo(2), hi(2) ioff = MOD(lo(1) + j + k + redblack,2) do i = lo(1) + ioff,hi(1),2 gamma = alpha*a(i,j,k) & + dhx*(bX(i,j,k)+bX(i+1,j,k)) & + dhy*(bY(i,j,k)+bY(i,j+1,k)) & + dhz*(bZ(i,j,k)+bZ(i,j,k+1)) g_m_d = gamma & - (dhx*(bX(i,j,k)*cf0 + bX(i+1,j,k)*cf3) & + dhy*(bY(i,j,k)*cf1 + bY(i,j+1,k)*cf4) & + dhz*(bZ(i,j,k)*cf2 + bZ(i,j,k+1)*cf5)) & rho = dhx*( bX(i ,j,k)*phi(i-1,j,k) & + bX(i+1,j,k)*phi(i+1,j,k) ) & + dhy*( bY(i,j ,k)*phi(i,j-1,k) & + bY(i,j+1,k)*phi(i,j+1,k) ) & + dhz*( bZ(i,j,k )*phi(i,j,k-1) & + bZ(i,j,k+1)*phi(i,j,k+1) ) & res = rhs(i,j,k) - (gamma*phi(i,j,k) - rho) phi(i,j,k) = phi(i,j,k) + omega/g_m_d * res end do end do end do
The algorithm above uses a 7-point cell-centered discretization of the 3-D variable-coefficient Helmholtz operator. The diffusion operator is one type of Helmholtz operator; the Laplace operator, which appears in the Poisson equation for self-gravity, is a simplified version, with constant coefficients.
The GSRB method for a 7-point discretization of the Helmholtz operator exhibits a low arithmetic intensity, requiring several non-contiguous loads from memory to evaluate the operator.
The relaxation step and the coarse grid solve (discussed below) often feature similar computational and data access patterns, because both are effectively doing the same thing - solving a linear system. The primary difference between them is that the relaxation method applies the iterative kernel only a handful of times, whereas the coarse grid solve often iterates all the way to convergence.
During a restriction, the value of a field on a fine grid is approximated on a coarser grid. This is typically done by averaging values of the field on fine grid points onto the corresponding grid points on the coarse grid. In BoxLib, the algorithm is the following:
do k = lo(3), hi(3) k2 = 2*k k2p1 = k2 + 1 do j = lo(2), hi(2) j2 = 2*j j2p1 = j2 + 1 do i = lo(1), hi(1) i2 = 2*i i2p1 = i2 + 1 c(i,j,k) = ( $ + f(i2p1,j2p1,k2 ) + f(i2,j2p1,k2 ) $ + f(i2p1,j2 ,k2 ) + f(i2,j2 ,k2 ) $ + f(i2p1,j2p1,k2p1) + f(i2,j2p1,k2p1) $ + f(i2p1,j2 ,k2p1) + f(i2,j2 ,k2p1) $ )*eighth end do end do end do
f is the field on the fine grid and
c is the field on the coarse
grid. (This multigrid solver always coarsens grids by factors of two in each
dimension.) For each evaluation of a coarse grid point, the algorithm must load
8 values from the fine grid. However, there is significant memory locality in
this algorithm, as many of the fine grid points for coarse grid point
c(i,j,k) also contribute to the point
Prolongation (also called interpolation) is the opposite of restriction: one approximates the value of a field on a coarse grid on a finer grid. The prolongation kernel in the BoxLib solver is as follows:
do k = lo(3), hi(3) k2 = 2*k k2p1 = k2 + 1 do j = lo(2), hi(2) j2 = 2*j j2p1 = j2 + 1 do i = lo(1), hi(1) i2 = 2*i i2p1 = i2 + 1 f(i2p1,j2p1,k2 ) = c(i,j,k) + f(i2p1,j2p1,k2 ) f(i2 ,j2p1,k2 ) = c(i,j,k) + f(i2 ,j2p1,k2 ) f(i2p1,j2 ,k2 ) = c(i,j,k) + f(i2p1,j2 ,k2 ) f(i2 ,j2 ,k2 ) = c(i,j,k) + f(i2 ,j2 ,k2 ) f(i2p1,j2p1,k2p1) = c(i,j,k) + f(i2p1,j2p1,k2p1) f(i2 ,j2p1,k2p1) = c(i,j,k) + f(i2 ,j2p1,k2p1) f(i2p1,j2 ,k2p1) = c(i,j,k) + f(i2p1,j2 ,k2p1) f(i2 ,j2 ,k2p1) = c(i,j,k) + f(i2 ,j2 ,k2p1) end do end do end do
In 3-D, the same value on the coarse grid contributes equally to eight neighboring points in the fine grid. (The symmetry arises from the constraint in the solver that the cells must be cubic.)
Exact linear solve¶
The multigrid solver in BoxLib recursively coarsens grids until the grid reaches a sufficiently small size, often \(2^3\) if the problem domain is cubic. On the coarsest grid, the solver then solves the linear system exactly, before propagating the solution back up to finer grids. The solution algorithm chosen for this step is rarely influential on the overall performance of the multigrid algorithm, because the problem size at the coarsest grid is so small. In BoxLib, the default coarse grid solver algorithm is BiCGSTAB, a variation on the conjugate-gradient iterative method.