33 research outputs found
Augmented Block-Arnoldi Recycling CFD Solvers
One of the limitations of recycled GCRO methods is the large amount of
computation required to orthogonalize the basis vectors of the newly generated
Krylov subspace for the approximate solution when combined with those of the
recycle subspace. Recent advancements in low synchronization Gram-Schmidt and
generalized minimal residual algorithms, Swirydowicz et
al.~\cite{2020-swirydowicz-nlawa}, Carson et al. \cite{Carson2022}, and Lund
\cite{Lund2022}, can be incorporated, thereby mitigating the loss of
orthogonality of the basis vectors. An augmented Arnoldi formulation of
recycling leads to a matrix decomposition and the associated algorithm can also
be viewed as a {\it block} Krylov method. Generalizations of both classical and
modified block Gram-Schmidt algorithms have been proposed, Carson et
al.~\cite{Carson2022}. Here, an inverse compact modified Gram-Schmidt
algorithm is applied for the inter-block orthogonalization scheme with a block
lower triangular correction matrix at iteration . When combined with a
weighted (oblique inner product) projection step, the inverse compact
scheme leads to significant (over 10 in certain cases) reductions in
the number of solver iterations per linear system. The weight is also
interpreted in terms of the angle between restart residuals in LGMRES, as
defined by Baker et al.\cite{Baker2005}. In many cases, the recycle subspace
eigen-spectrum can substitute for a preconditioner
Block Iterative Methods and Recycling for Improved Scalability of Linear Solvers
International audienceContemporary large-scale Partial Differential Equation (PDE) simulations usually require the solution of large and sparse linear systems. Moreover, it is often needed to solve these linear systems with different or multiple Right-Hand Sides (RHSs). In this paper, various strategies will be presented to extend the scalability of existing linear solvers using appropriate recycling strategies or block methods—i.e., by treating multiple right-hand sides simultaneously. The scalability of this work is assessed by performing simulations on up to 8,192 cores for solving linear systems arising from various physical phenomena modeled by Poisson's equation, the system of linear elasticity, or Maxwell's equation. This work is shipped as part of on open-source software, readily available and usable in any C, C++, or Python code. In particular, some simulations are performed on top of a well-established library, PETSc, and it is shown how our approaches can be used to decrease time to solution down by 30%
Block Iterative Methods and Recycling for Improved Scalability of Linear Solvers
International audienceContemporary large-scale Partial Differential Equation (PDE) simulations usually require the solution of large and sparse linear systems. Moreover, it is often needed to solve these linear systems with different or multiple Right-Hand Sides (RHSs). In this paper, various strategies will be presented to extend the scalability of existing linear solvers using appropriate recycling strategies or block methods—i.e., by treating multiple right-hand sides simultaneously. The scalability of this work is assessed by performing simulations on up to 8,192 cores for solving linear systems arising from various physical phenomena modeled by Poisson's equation, the system of linear elasticity, or Maxwell's equation. This work is shipped as part of on open-source software, readily available and usable in any C, C++, or Python code. In particular, some simulations are performed on top of a well-established library, PETSc, and it is shown how our approaches can be used to decrease time to solution down by 30%
A General Algorithm for Reusing Krylov Subspace Information. I. Unsteady Navier-Stokes
A general algorithm is developed that reuses available information to accelerate the iterative convergence of linear systems with multiple right-hand sides A x = b (sup i), which are commonly encountered in steady or unsteady simulations of nonlinear equations. The algorithm is based on the classical GMRES algorithm with eigenvector enrichment but also includes a Galerkin projection preprocessing step and several novel Krylov subspace reuse strategies. The new approach is applied to a set of test problems, including an unsteady turbulent airfoil, and is shown in some cases to provide significant improvement in computational efficiency relative to baseline approaches
Improving Pseudo-Time Stepping Convergence for CFD Simulations With Neural Networks
Computational fluid dynamics (CFD) simulations of viscous fluids described by
the Navier-Stokes equations are considered. Depending on the Reynolds number of
the flow, the Navier-Stokes equations may exhibit a highly nonlinear behavior.
The system of nonlinear equations resulting from the discretization of the
Navier-Stokes equations can be solved using nonlinear iteration methods, such
as Newton's method. However, fast quadratic convergence is typically only
obtained in a local neighborhood of the solution, and for many configurations,
the classical Newton iteration does not converge at all. In such cases,
so-called globalization techniques may help to improve convergence.
In this paper, pseudo-transient continuation is employed in order to improve
nonlinear convergence. The classical algorithm is enhanced by a neural network
model that is trained to predict a local pseudo-time step. Generalization of
the novel approach is facilitated by predicting the local pseudo-time step
separately on each element using only local information on a patch of adjacent
elements as input. Numerical results for standard benchmark problems, including
flow through a backward facing step geometry and Couette flow, show the
performance of the machine learning-enhanced globalization approach; as the
software for the simulations, the CFD module of COMSOL Multiphysics is
employed
Contribution to the study of efficient iterative methods for the numerical solution of partial differential equations
Multigrid and domain decomposition methods provide efficient algorithms for the numerical solution of partial differential equations arising in the modelling of many applications in Computational Science and Engineering. This manuscript covers certain aspects of modern iterative solution methods for the solution of large-scale problems issued from the discretization of partial differential equations. More specifically, we focus on geometric multigrid methods, non-overlapping substructuring methods and flexible Krylov subspace methods with a particular emphasis on their combination. Firstly, the combination of multigrid and Krylov subspace methods is investigated on a linear partial differential equation modelling wave propagation in heterogeneous media. Secondly, we focus on non-overlapping domain decomposition methods for a specific finite element discretization known as the hp finite element, where unrefinement/refinement is allowed both by decreasing/increasing the step size h or by decreasing/increasing the polynomial degree p of the approximation on each element. Results on condition number bounds for the domain decomposition preconditioned operators are given and illustrated by numerical results on academic problems in two and three dimensions. Thirdly, we review recent advances related to a class of Krylov subspace methods allowing variable preconditioning. We examine in detail flexible Krylov subspace methods including augmentation and/or spectral deflation, where deflation aims at capturing approximate invariant subspace information. We also present flexible Krylov subspace methods for the solution of linear systems with multiple right-hand sides given simultaneously. The efficiency of the numerical methods is demonstrated on challenging applications in seismics requiring the solution of huge linear systems of equations with multiple right-hand sides on parallel distributed memory computers. Finally, we expose current and future prospectives towards the design of efficient algorithms on extreme scale machines for the solution of problems coming from the discretization of partial differential equations
Kernel solver design of FPGA-based real-time simulator for active distribution networks
The field-programmable gate array (FPGA)-based real-time simulator takes advantage of many merits of FPGA, such as small time-step, high simulation precision, rich I/O interface resources, and low cost. The sparse linear equations formed by the node conductance matrix need to be solved repeatedly within each time-step, which introduces great challenges to the performance of the real-time simulator. In this paper, a fine-grained solver of the FPGA-based real-time simulator for active distribution networks is designed to meet the computational demand. The framework of the solver, offline process design on PC and online process design on FPGA are proposed in detail. The modified IEEE 33-node system with photovoltaics is simulated on a 4-FPGA-based real-time simulator. Simulation results are compared with PSCAD/EMTDC under the same conditions to validate the solver design