A generalized Block FSAI preconditioner for nonsymmetric linear systems

Ferronato, Massimiliano; Janna, Carlo; Pini, Giorgio

doi:10.1016/j.cam.2013.07.049

Cited by 31 publications

(26 citation statements)

References 35 publications

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“…Then the coarse grid operator A h =RAP can also be considered as an approximation to the Schur complement S. Besides the choices in (5.9), one can also choose to use Jacobi approach for restriction, that is W r = −A CF D −1 F F . Another option is to construct A −1 F F using incomplete factorizations (ILU) or sparse approximate inverse techniques, such as sparse approximate inverse (SPAI) [21], factored sparse approximate inverse (FSAI) [18], or minimal residual (MR) [11]. Although these methods could provide a better approximation to A −1 F F , and therefore better approximations for the restriction and interpolation operators, they tend to make these operators dense.…”

Section: Multigridmentioning

confidence: 99%

A Scalable Multigrid Reduction Framework for Multiphase Poromechanics of Heterogeneous Media

Bui¹,

Osei-Kuffuor²,

Castelletto³

et al. 2020

SIAM J. Sci. Comput.

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Simulation of multiphase poromechanics involves solving a multi-physics problem in which multiphase flow and transport are tightly coupled with the porous medium deformation. To capture this dynamic interplay, fully implicit methods, also known as monolithic approaches, are usually preferred. The main bottleneck of a monolithic approach is that it requires solution of large linear systems that result from the discretization and linearization of the governing balance equations. Because such systems are non-symmetric, indefinite, and highly ill-conditioned, preconditioning is critical for fast convergence. Recently, most efforts in designing efficient preconditioners for multiphase poromechanics have been dominated by physics-based strategies. Current state-ofthe-art "black-box" solvers such as algebraic multigrid (AMG) are ineffective because they cannot effectively capture the strong coupling between the mechanics and the flow sub-problems, as well as the coupling inherent in the multiphase flow and transport process. In this work, we develop an algebraic framework based on multigrid reduction (MGR) that is suited for tightly coupled systems of PDEs. Using this framework, the decoupling between the equations is done algebraically through defining appropriate interpolation and restriction operators. One can then employ existing solvers for each of the decoupled blocks or design a new solver based on knowledge of the physics. We demonstrate the applicability of our framework when used as a "black-box" solver for multiphase poromechanics. We show that the framework is flexible to accommodate a wide range of scenarios, as well as efficient and scalable for large problems.

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Section: Multigridmentioning

confidence: 99%

A Scalable Multigrid Reduction Framework for Multiphase Poromechanics of Heterogeneous Media

Bui¹,

Osei-Kuffuor²,

Castelletto³

et al. 2020

SIAM J. Sci. Comput.

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“…Besides the blocks associated with the PDEs, we also need to consider those in last row of the matrix in (19), which are derived from the discrete version of the complementarity constraint equation (9). When the gas phase does not exist, we have…”

Section: Linear Systemmentioning

confidence: 99%

Algebraic Multigrid Preconditioners for Multiphase Flow in Porous Media

Bui¹,

Elman²,

Moulton³

2017

SIAM J. Sci. Comput.

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Abstract. Multiphase flow is a critical process in a wide range of applications, including carbon sequestration, contaminant remediation, and groundwater management. Typically, this process is modeled by a nonlinear system of partial differential equations derived by considering the mass conservation of each phase (e.g., oil, water), along with constitutive laws for the relationship of phase velocity to phase pressure. In this study, we develop and study efficient solution algorithms for solving the algebraic systems of equations derived from a fully coupled and time-implicit treatment of models of multiphase flow. We explore the performance of several preconditioners based on algebraic multigrid (AMG) for solving the linearized problem, including "black-box" AMG applied directly to the system, a new version of constrained pressure residual multigrid (CPR-AMG) preconditioning, and a new preconditioner derived using an approximate Schur complement arising from the block factorization of the Jacobian. We show that the new methods are the most robust with respect to problem character as determined by varying effects of capillary pressures, and we show that the block factorization preconditioner is both efficient and scales optimally with problem size.1. Introduction. Multiphase flow is a feature of many physical systems and models of it are used in applications such as reservoir simulation, carbon sequestration, ground water management and contaminant transport. Modeling multiphase flow in highly heterogeneous media with complex geometries is difficult, especially when realistic processes such as capillary pressure are included. The system describing multiphase flow consists of nonlinear partial differential equations, constitutive laws and constraints. In this paper, we focus on the iterative solution of linear systems arising in a fully implicit cell-centered finite volume discretization of single component isothermal two-phase flow model with capillary pressure. This fully implicit time-stepping scheme is among the most robust for simulation of subsurface flow. Moreover, it can serve as a basis for modeling more complex processes in which the physical quantities are tightly coupled. This additional complexity could include adding more components, miscibility between components, thermal effects, and phase transitions.The fully implicit discretization gives rise to a nonlinear system of equations at each time step. We employ a variant of Newton's method with an exact Jacobian of the discretized equations to solve this system. For the linear system, we use a preconditioned generalized minimal residual (GMRES) method. There is a vast literature on different approaches to precondition the Jacobian system. A very popular approach is to use incomplete LU factorization (ILU) for constructing the preconditioner. Though popular for its general applicability, ILU-based preconditioners are neither effective nor scalable in many cases. Another approach is to consider decoupled preconditioners for the coupled system [6]. This...

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“…Popular preconditioners include the incomplete factorization preconditioners, the sparse approximate inverse (SPAI) preconditioners based on Frobenius norm minimization, the factorized sparse approximate inverse (FSAI) preconditioners, and the preconditioners that consist of an incomplete factorization, followed by an approximate inversion of the incomplete factors . However, the cost of constructing the preconditioners is generally very high for large‐scale problems.…”

Section: Introductionmentioning

confidence: 99%

An efficient sparse approximate inverse preconditioning algorithm on GPU

Yin

Gao

2019

Concurrency and Computation

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Summary The sparse approximate inverse (SPAI) preconditioner has proven to be effective in accelerating the convergence of iterative methods. Recently, accelerating it on the graphics processing unit (GPU) has attracted considerable attention due to the fact that the cost of constructing it is high. This motivates us to investigate how to accelerate the construction of SPAI preconditioners on GPU in this paper. We propose an efficient sparse approximate inverse algorithm on GPU, called SPAI‐Adaptive. For our proposed SPAI‐Adaptive, there are the following novelties: (1) an adaptive thread allocation strategy for SPAI‐Adaptive is proposed to assign the optimal thread number for each column of the preconditioner, and (2) Each component of the preconditioner, which includes finding indices I and J, constructing local submatrix, decomposing the local matrix into QR, and solving the upper triangular linear system, is computed in parallel inside a thread group of GPU. Experimental results show that the proposed SPAI‐Adaptive is effective, and has good performance and high parallelism.

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A generalized Block FSAI preconditioner for nonsymmetric linear systems

Cited by 31 publications

References 35 publications

A Scalable Multigrid Reduction Framework for Multiphase Poromechanics of Heterogeneous Media

A Scalable Multigrid Reduction Framework for Multiphase Poromechanics of Heterogeneous Media

Algebraic Multigrid Preconditioners for Multiphase Flow in Porous Media

An efficient sparse approximate inverse preconditioning algorithm on GPU

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