A GPU-Accelerated Linear Solver for Massively Parallel Underground Simulations

Modern engineering applications require the solution of linear systems of millions or even billions of equations. The solution of the linear system takes most of the simulation for large scale simulations, and represent the bottleneck in developing scientific and technical software. Usually, preconditioned iterative solvers are preferred because of their low memory requirements and they can have a high level of parallelism. Approximate inverses have been proven to be robust and effective preconditioners in several contexts. In this communication, we present an adaptive Factorized Sparse Approximate Inverse (FSAI) preconditioner with a very high level of parallelism in both set-up and application. Its inherent parallelism makes FSAI an ideal candidate for a GPU-accelerated implementation, even if taking advantage of this hardware is not a trivial task, especially in the set-up stage. An extensive numerical experimentation has been performed on industrial underground applications. It is shown that the proposed approach outperforms more traditional preconditioners in challenging underground simulation, greatly reducing time-to-solution.

Grid independent convergence using multilevel circulant preconditioning: Poisson’s equation

We consider the iterative solution of the discrete Poisson’s equation with Dirichlet boundary conditions. The discrete domain is embedded into an extended domain and the resulting system of linear equations is solved using a fixed point iteration combined with a multilevel circulant preconditioner. Our numerical results show that the rate of convergence is independent of the grid’s step sizes and of the number of spatial dimensions, despite the fact that the iteration operator is not bounded as the grid is refined. The embedding technique and the preconditioner is derived with inspiration from theory of boundary integral equations. The same theory is used to explain the behaviour of the preconditioned iterative method.

Dimensionally Consistent Preconditioning for Saddle-Point Problems

The preconditioned iterative solution of large-scale saddle-point systems is of great importance in numerous application areas, many of them involving partial differential equations. Robustness with respect to certain problem parameters is often a concern, and it can be addressed by identifying proper scalings of preconditioner building blocks. In this paper, we consider a new perspective to finding effective and robust preconditioners. Our approach is based on the consideration of the natural physical units underlying the respective saddle-point problem. This point of view, which we refer to as dimensional consistency, suggests a natural combination of the parameters intrinsic to the problem. It turns out that the scaling obtained in this way leads to robustness with respect to problem parameters in many relevant cases. As a consequence, we advertise dimensional consistency based preconditioning as a new and systematic way to designing parameter robust preconditoners for saddle-point systems arising from models for physical phenomena.

A GPU-accelerated adaptive FSAI preconditioner for massively parallel simulations

The solution of linear systems of equations is a central task in a number of scientific and engineering applications. In many cases the solution of linear systems may take most of the simulation time thus representing a major bottleneck in the further development of scientific and technical software. For large scale simulations, nowadays accounting for several millions or even billions of unknowns, it is quite common to resort to preconditioned iterative solvers for exploiting their low memory requirements and, at least potential, parallelism. Approximate inverses have been shown to be robust and effective preconditioners in various contexts. In this work, we show how adaptive Factored Sparse Approximate Inverse (aFSAI), characterized by a very high degree of parallelism, can be successfully implemented on a distributed memory computer equipped with GPU accelerators. Taking advantage of GPUs in adaptive FSAI set-up is not a trivial task, nevertheless we show through an extensive numerical experimentation how the proposed approach outperforms more traditional preconditioners and results in a close-to-ideal behavior in challenging linear algebra problems.

Iterative method for solving one-dimensional fractional mathematical physics model via quarter-sweep and PAOR

This paper will solve one of the fractional mathematical physics models, a one-dimensional time-fractional differential equation, by utilizing the second-order quarter-sweep finite-difference scheme and the preconditioned accelerated over-relaxation method. The proposed numerical method offers an efficient solution to the time-fractional differential equation by applying the computational complexity reduction approach by the quarter-sweep technique. The finite-difference approximation equation will be formulated based on the Caputo’s time-fractional derivative and quarter-sweep central difference in space. The developed approximation equation generates a linear system on a large scale and has sparse coefficients. With the quarter-sweep technique and the preconditioned iterative method, computing the time-fractional differential equation solutions can be more efficient in terms of the number of iterations and computation time. The quarter-sweep computes a quarter of the total mesh points using the preconditioned iterative method while maintaining the solutions’ accuracy. A numerical example will demonstrate the efficiency of the proposed quarter-sweep preconditioned accelerated over-relaxation method against the half-sweep preconditioned accelerated over-relaxation, and the full-sweep preconditioned accelerated over-relaxation methods. The numerical finding showed that the quarter-sweep finite difference scheme and preconditioned accelerated over-relaxation method can serve as an efficient numerical method to solve fractional differential equations.