2D Fast Poisson Solver for High-Performance Computing

Abstract

Among elliptic boundary value problems, the class of problems with separable variables can be solved fast and directly. Elliptic problems with separable variables usually assume that the computational domains are simple e.g., rectangle or circle, and constant coefficients [1]. This kind of problems can serve to generate preconditioners in iterative procedures for more complex methods. They can also be used in low-accuracy models similar to the ones used in Numerical Weather Simulations. A straightforward implementation of solvers for such problems makes them considered as too simple to pay much attention to them. For instance, NETLIB* contains the codes of various solvers for problems with separable variables. We are not aware if such solvers are provided as separate solvers in other software packages except NETLIB Fishpack* and Intel$\textcircled{\tiny{\,R}}$ Math Kernel Library (Intel$\textcircled{\tiny{\,R}}$ MKL) together with Intel$\textcircled{\tiny{\,R}}$ Cluster Poisson Solver Library (Intel$\textcircled{\tiny{\,R}}$ CPSL). The problems with the separable variables can be suboptimal in the sense that they require slightly more arithmetic operations (up to logarithmic factor) than the number of unknowns to compute the solution to the problem. This statement is true if, for example, the sizes of the discretized problems are powers of 2. Computational Mathematics suggests that we take into consideration not only arithmetic operations, but also the cost of memory operations as well. Modern computers can perform computations at a very high speed, while lacking the ability to deliver data to the computational units. Keeping in mind that a memory operation can easily be dozen to hundred times slower than an arithmetic one, a computationally optimal algorithm could compute the solution slower than memory optimal algorithm. The recent developments in processor industry resulted in multicore processors become standard processors not only for powerful clusters, but also for home computers and laptops. Therefore, the algorithm can also be non-optimal from the parallelization point of view. Optimality here can be understood in terms of the number of synchronization points and/or in terms of the amount of data that needs to be transferred between different cores/processors. In summary, the modern computational algorithm should focus on 3 key aspects, namely, parallelization, memory operations, and arithmetic costs. The purpose of this paper is to demonstrate on a simple 2D problem with separable variables that taking into account modern model the solution can be computed efficiently and fast. This would also help developers to compute solution to the problem with separable variables really negligible with respect to other computations. To complete our goal, we will evaluate software provided by Intel Corporation. In particular, we focus on the comparison (where possible) of NETLIB Fishpack* and Intel$\textcircled{\tiny{\,R}}$ CPSL provided at [2] as well as Intel$\textcircled{\tiny{\,R}}$ MKL provided at [3]. This paper is organized as follows. First we are solving a 2D Helmholtz problem using single precision on a shared memory machine with two 4-Core Intel$\textcircled{\tiny{\,R}}$ Xeon$\textcircled{\tiny{\,R}}$ Processors E5462. Next we do the same but use double precision arithmetic. Lastly we use double precision on a distributed memory machine using MPI.