Optimization of a parallel Monte Carlo method for linear algebra problems

Abstract

Many problems in science and engineering can be represented by Systems of Linear Algebraic Equations (SLAEs). Numerical methods such as direct or iterative ones are used to solve these kind of systems. Depending on the size and other factors that characterize these systems they can be sometimes very difficult to solve even for iterative methods, requiring long time and large amounts of computational resources. In these cases a preconditioning approach should be applied. Preconditioning is a technique used to transform a SLAE into a equivalent but simpler system which requires less time and effort to be solved. The matrix which performs such transformation is called the preconditioner [7]. There are preconditioners for both direct and iterative methods but they are more commonly used among the later ones. In the general case a preconditioned system will require less effort to be solved than the original one. For example, when an iterative method is being used, less iterations will be required or each iteration will require less time, depending on the quality and the efficiency of the preconditioner. There are different classes of preconditioners but we will focused only on those that are based on the SParse Approximate Inverse (SPAI) approach. These algorithms are based on the fact that the approximate inverse of a given SLAE matrix can be used to approximate its result or to reduce its complexity. Monte Carlo methods are probabilistic methods, that use random numbers to either simulate a stochastic behaviour or to estimate the solution of a problem. They are good candidates for parallelization due to the fact that many independent samples are used to estimate the solution. These samples can be calculated in parallel, thereby speeding up the solution finding process [27]. In the past there has been a lot of research around the use of Monte Carlo methods to calculate SPAI preconditioners [1] [27] [10]. In this work we present the implementation of a SPAI preconditioner that is based on a Monte Carlo method. This algorithm calculates the matrix inverse by sampling a random variable which approximates the Neumann Series expansion. Using the Neumman series it is possible to calculate the matrix inverse of a system A by performing consecutive additions of the powers of a matrix expressed by the series expansion of (I − A) −1 . Given the stochastic approach of the Monte Carlo algorithm, the computational effort required to find an element of the inverse matrix is independent from the size of the matrix. This allows to target systems that, due to their size, can be prohibitive for common deterministic approaches [27]. Great part of this work is focused on the enhancement of this algorithm. First, the current errors of the implementation were fixed, making the algorithm able to target larger systems. Then multiple optimizations were applied at different stages of the implementation making a better use of the resources and improving the performance of the algorithm. Four optimizations, with consistently improvements have been performed: 1. An inefficient implementation of the realloc function within the MPI library was provoking the application to rapidly run out of memory. This function was replaced by the malloc function and some slight modifications to estimate the size of matrix A. 2. A coordinate format (COO) was introduced within the algorithm’s core to make a more efficient use of the memory, avoiding several unnecessary memory accesses. 3. A method to produce an intermediate matrix P was shown to produce similar results to the default one and with matrix P being reduced to a single vector, thus requiring less data. Given that this was a broadcast data a diminishing on it, translated into a reduction of the broadcast time. 4. Four individual procedures which accessed the whole initial matrix memory, were merged into two processes, reducing this way the number of memory accesses. For each optimization applied, a comparison was performed to show the particular improvements achieved. A set of different matrices, representing different SLAEs, was used to show the consistency of these improvements. In order to provide with insights about the scalability issues of the algorithm, other approaches are presented to show the particularities of the algorithm’s scalability: 1. Given that the original version of this algorithm was designed for a cluster of single-core machines, an hybrid approach of MPI + openMP was proposed to target the nowadays multi-core architectures. Surprisingly this new approach did not show any improvement but it was useful to show a scalability problem related to the random pattern used to access the memory. 2. Having that common MPI implementations of the broadcast operation do not take into account the different latencies between inter-node and intra-node communications [25]. Therefore, we decided to implement the broadcast in two steps. First by reaching a single process in each of the compute nodes and then using those processes to perform a local broadcast within their compute nodes. Results on this approach showed that this method could lead to improvements when very big systems are used. Finally a comparison is carried out between the optimized version of the Monte Carlo algorithm and the state of the art Modified SPAI (MSPAI). Four metrics are used to compare these approaches: 1. The amount of time needed for the preconditioner construction. 2. The time needed by the solver to calculate the solution of the preconditioned system. 3. The addition of the previous metrics, which gives a overview of the quality and efficiency of the preconditioner. 4. The number of cores used in the preconditioner construction. This gives an idea of the energy efficiency of the algorithm. Results from previous comparison showed that Monte Carlo algorithm can deal with both symmetric and nonsymmetric matrices while MSPAI only performs well with the nonsymetric ones. Furthermore the time for Monte Carlo’s algorithm is always faster for the preconditioner construction and most of the times also for the solver calculation. This means that Monte Carlo produces preconditioners of better or same quality than MSPAI. Finally, the number of cores used in the Monte Carlo approach is always equal or smaller than in the case of MSPAI.