Preconditioning for Sparse Linear Systems at the Dawn of the 21st Century: History, Current Developments, and Future Perspectives

Iterative methods are currently the solvers of choice for large sparse linear systems of equations. However, it is well known that the key factor for accelerating, or even allowing for, convergence is the preconditioner. The research on preconditioning techniques has characterized the last two decades. Nowadays, there are a number of different options to be considered when choosing the most appropriate preconditioner for the specific problem at hand. The present work provides an overview of the most popular algorithms available today, emphasizing the respective merits and limitations. The overview is restricted to algebraic preconditioners, that is, general-purpose algorithms requiring the knowledge of the system matrix only, independently of the specific problem it arises from. Along with the traditional distinction between incomplete factorizations and approximate inverses, the most recent developments are considered, including the scalable multigrid and parallel approaches which represent the current frontier of research. A separate section devoted to saddle-point problems, which arise in many different applications, closes the paper

Block preconditioning for fault/fracture mechanics saddle-point problems

The efficient simulation of fault and fracture mechanics is a key issue in several applications and is attracting a growing interest by the scientific community. Using a formulation based on Lagrange multipliers, the Jacobian matrix resulting from the Finite Element discretization of the governing equations has a non-symmetric generalized saddlepoint structure. In this work, we propose a family of block preconditioners to accelerate the convergence of Krylov methods for such problems. We critically review possible advantages and difficulties of using various Schur complement approximations, based on both physical and algebraic considerations. The proposed approaches are tested in a number of real-world applications, showing their robustness and efficiency also in large-size and ill-conditioned problems

A Novel Factorized Sparse Approximate Inverse Preconditioner with Supernodes

AbstractKrylov methods preconditioned by Factorized Sparse Approximate Inverses (FSAI) are an efficient approach for the solution of symmetric positive definite linear systems on massively parallel computers. However, FSAI often suffers from a high set-up cost, especially in ill-conditioned problems. In this communication we propose a novel algorithm for the FSAI computation that makes use of the concept of supernode borrowed from sparse LU factorizations and direct methods

preconditioning techniques for sparse linear systems

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A novel block non-symmetric preconditioner for mixed-hybrid finite-element-based flow simulations

In this work we propose a novel block preconditioner, labelled Explicit Decoupling Factor Approximation (EDFA), to accelerate the convergence of Krylov subspace solvers used to address the sequence of non-symmetric systems of linear equations originating from flow simulations in porous media. The flow model is discretized blending the Mixed Hybrid Finite Element (MHFE) method for Darcy's equation with the Finite Volume (FV) scheme for the mass conservation. The EDFA preconditioner is characterized by two features: the exploitation of the system matrix decoupling factors to recast the Schur complement and their inexact fully-parallel computation by means of restriction operators. We introduce two adaptive techniques aimed at building the restriction operators according to the properties of the system at hand. The proposed block preconditioner has been tested through an extensive experimentation on both synthetic and real-case applications, pointing out its robustness and computational efficiency

Efficient solvers for hybridized three-field mixed finite element coupled poromechanics

We consider a mixed hybrid finite element formulation for coupled poromechanics. A stabilization strategy based on a macro-element approach is advanced to eliminate the spurious pressure modes appearing in undrained/incompressible conditions. The efficient solution of the stabilized mixed hybrid block system is addressed by developing a class of block triangular preconditioners based on a Schur-complement approximation strategy. Robustness, computational efficiency and scalability of the proposed approach are theoretically discussed and tested using challenging benchmark problems on massively parallel architectures

Numerical investigation on a block preconditioning strategy to improve the computational efficiency of DFN models

[EN] The simulation of underground flow across intricate fracture networks can be addressed by means of discrete fracture network models. The combination of such models with an optimization formulation allows for the use of nonconforming and independent meshes for each fracture. The arising algebraic problem produces a symmetric saddle-point matrix with a rank-deficient leading block. In our work, we investigate the properties of the system to design a block preconditioning strategy to accelerate the iterative solution of the linearized algebraic problem. The matrix is first permuted and then projected in the symmetric positive-definite Schur-complement space. The proposed strategy is tested in applications of increasing size, in order to investigate its capabilities.Gazzola, L.; Ferronato, M.; Berrone, S.; Pieraccini, S.; Scialò, S. (2022). Numerical investigation on a block preconditioning strategy to improve the computational efficiency of DFN models. En Proceedings of the YIC 2021 - VI ECCOMAS Young Investigators Conference. Editorial Universitat Politècnica de València. 346-354. https://doi.org/10.4995/YIC2021.2021.12234OCS34635

A robust multilevel approximate inverse preconditioner for symmetric positive definite matrices

The use of factorized sparse approximate inverse (FSAI) preconditioners in a standard multilevel framework for symmetric positive definite (SPD) matrices may pose a number of issues as to the definiteness of the Schur complement at each level. The present work introduces a robust multilevel approach for SPD problems based on FSAI preconditioning, which eliminates the chance of algorithmic breakdowns independently of the preconditioner sparsity. The multilevel FSAI algorithm is further enhanced by introducing descending and ascending low-rank corrections, thus giving rise to the multilevel FSAI with low-rank corrections (MFLR) preconditioner. The proposed algorithm is investigated in a number of test problems. The numerical results show that the MFLR preconditioner is a robust approach that can significantly accelerate the solver convergence rate preserving a good degree of parallelism. The possibly large set-up cost, mainly due to the computation of the eigenpairs needed by low-rank corrections, makes its use attractive in applications where the preconditioner can be recycled along a number of linear solves

Finite element analysis of land subsidence above depleted reservoirs with pore pressure gradient and total stress formulations

SUMMARY The solution of the poroelastic equations for predicting land subsidence above productive gas/oil "elds may be addressed by the principle of virtual works using either the e!ective intergranular stress, with the pore pressure gradient regarded as a distributed body force, or the total stress incorporating the pore pressure. In the "nite element (FE) method both approaches prove equivalent at the global assembled level. However, at the element level apparently the equivalence does not hold, and the strength source related to the pore pressure seems to generate di!erent local forces on the element nodes. The two formulations are brie#y reviewed and discussed for triangular and tetrahedral "nite elements. They are shown to yield di!erent results at the global level as well in a three-dimensional axisymmetric porous medium if the FE integration is performed using the average element-wise radius. A modi"cation to both formulations is suggested which allows to correctly solve the problem of a "nite reservoir with an in"nite pressure gradient, i.e. with a pore pressure discontinuity on its boundary