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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, w...
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... In the context of preconditioning linear systems, much work has recently been devoted to low-rank matrix representations [35,23,36,22,58]. Let us recall the following result [36]. ...
Divergence constraints are present in the governing equations of many physical phenomena, and they usually lead to a Poisson equation whose solution typically is the main bottleneck of many simulation codes. Algebraic Multigrid (AMG) is arguably the most powerful preconditioner for Poisson's equation, and its effectiveness results from the complementary roles played by the smoother, responsible for damping high-frequency error components, and the coarse-grid correction, which in turn reduces low-frequency modes. This work presents several strategies to make AMG more compute-intensive by leveraging reflection, translational and rotational symmetries, often present in academic and industrial configurations. The best-performing method, AMGR, is based on a multigrid reduction framework that introduces an aggressive coarsening to the multigrid hierarchy, reducing the memory footprint, setup and application costs of the top-level smoother. While preserving AMG's excellent convergence, AMGR allows replacing the standard sparse matrix-vector product with the more compute-intensive sparse matrix-matrix product, yielding significant accelerations. Numerical experiments on industrial CFD applications demonstrated up to 70% speed-ups when solving Poisson's equation with AMGR instead of AMG. Additionally, strong and weak scalability analyses revealed no significant degradation.
... One of the motivations of this paper is to propose a method that is a move away from nested dissection. Similar low-rank correction ideas have also been exploited in [11,17,1,12]. A related class of methods is the class of rank structured matrix methods, which include the hierarchically off-diagonal low-rank matrix [3], the \scrH -matrix [4,6,7], the \scrH 2 -matrix [15], and hierarchically semiseparable (HSS) matrices [32]. ...
... Previous work on the computation of inverse factors has to a great extent focused on approximations used as preconditioners for iterative solution of linear systems (see Benzi & Tuma, 1999). Examples, besides the AINV algorithms already mentioned, include FSAI (Kolotilina & Yeremin, 1993) and more recent variants (Franceschini et al., 2018). Recursive partitioning of the matrix is used in direct methods for factorization such as multifrontal methods (see Davis et al., 2016). ...
We propose a localized divide and conquer algorithm for inverse factorization of Hermitian positive definite matrices S with localized structure, e.g. exponential decay with respect to some given distance function on the index set of S. The algorithm is a reformulation of recursive inverse factorization (Rubensson et al. (2008) Recursive inverse factorization. J. Chem. Phys., 128, 104105) but makes use of localized operations only. At each level of the recursion, the problem is cut into two subproblems and their solutions are combined using iterative refinement (Niklasson (2004) Iterative refinement method for the approximate factorization of a matrix inverse. Phys. Rev. B, 70, 193102) to give a solution to the original problem. The two subproblems can be solved in parallel without any communication and, using the localized formulation, the cost of combining their results is negligible compared to the overall cost for sufficiently large systems and appropriate partitions of the problem. We also present an alternative derivation of iterative refinement based on a sign matrix formulation, analyze the stability and propose a parameterless stopping criterion. We present bounds for the initial factorization error and the number of iterations in terms of the condition number of S when the starting guess is given by the solution of the two subproblems in the binary recursion. These bounds are used in theoretical results for the decay properties of the involved matrices. We demonstrate the localization properties of our algorithm for matrices corresponding to nearest neighbor overlap on one-, two- and three-dimensional lattices, as well as basis set overlap matrices generated using the Hartree–Fock and Kohn–Sham density functional theory electronic structure program Ergo (Rudberg et al. (2018) Ergo: an open-source program for linear-scaling electronic structure. SoftwareX, 7, 107). We evaluate the parallel performance of our implementation based on the chunks and tasks programming model, showing that the proposed localization of the algorithm results in a dramatic reduction of communication costs.
... These preconditioners approximate the Schur complement or its inverse by exploiting various low-rank corrections and because they are essentially approximate inverse methods they tend to perform rather well on indefinite linear systems. Similar ideas have also been exploited in [9]. A related class of methods is the class of rank structured matrix methods, which include the HOLDR-matrix [1], the H-matrix [2,4], the H 2 -matrix [11] and hierarchically semiseparable (HSS) matrices [5,18,24]. ...
... Proof. Since A is SPD, S and C 0 are also SPD and C 9) and this shows that λ(C −1 0 E s ) < 1. Now we prove the second part of this lemma. ...
An effective power based parallel preconditioner is proposed for general large sparse linear systems. The preconditioner combines a power series expansion method with some low-rank correction techniques, where the Sherman-Morrison-Woodbury formula is utilized. A matrix splitting of the Schur complement is proposed to expand the power series. The number of terms used in the power series expansion can control the approximation accuracy of the preconditioner to the inverse of the Schur complement. To construct the preconditioner, graph partitioning is invoked to reorder the original coefficient matrix, leading to a special block two-by-two matrix whose two off-diagonal submatrices are block diagonal. Variables corresponding to interface variables are obtained by solving a linear system with the coeffcient matrix being the Schur complement. For the variables related to the interior variables, one only needs to solve a block diagonal linear system. This can be performed efficiently in parallel. Various numerical examples are provided to illustrate that the efficiency of the proposed preconditioner.
... which shows that reaching the minimum of each [F AF T ] ii with \scrW \scrS \scrB becoming dense is equivalent to choosing \widetil F = W T ideal . Using an approach similar to the one presented in [26], we approximate the ideal prolongator by running Algorithm 3 with configuration parameters k p , \rho p , and \epsilon p , where in place of computing the row vectors g i , we calculate the column vectors w i of matrix W . In this way, at each step of the procedure we compute, for the current pattern, a minimizer of tr(P T AP ) and select the most promising entries to enlarge W . ...
... Usually, to limit the number of non-zeros in W , only strong connections are considered. However, the connectivity of the SoC matrix may vary significantly, especially in difficult problems, with the consequence that an a priori selected nonzero pattern may give rise to both overdetermined and underdetermined row systems (26). The first occurrence causes unnecessary large operator complexity, while the latter prevents the construction of an effective prolongation as the target range cannot be represented locally. ...
The numerical simulation of modern engineering problems can easily incorporate millions or even billions degrees of freedom. In several applications, these simulations require the solution to sparse linear systems of equations, and algebraic multigrid (AMG) methods are often standard choices as iterative solvers or preconditioners. This happens due to their high convergence speed guaranteed even in large size problems, which is a consequence of the AMG ability of reducing particular error components across their multilevel hierarchy. Despite carrying the name “algebraic”, most of these methods still rely on additional information other than the global assembled sparse matrix, as for instance the knowledge of the operator near kernel. This fact somewhat limits their applicability as black-box solvers. In this work, we introduce a novel AMG approach featuring the adaptive Factored Sparse Approximate Inverse (aFSAI) method as a flexible smoother as well as three new approaches to adaptively compute the prolongation operator. We assess the performance of the proposed AMG through the solution of a set of model problems along with real-world engineering test cases. Moreover, comparisons are made with the aFSAI and BoomerAMG preconditioners, showing that our new method proves to be superior to the first strategy and comparable to the second one, if not better as in the elasticity problems.
... Iterative solvers are efficient, but require a good preconditioner in order for the solution to converge [17,157,54,16,156,55,98]. The constrained preconditioner and solver presented in [55] was tested in this work. ...
... Iterative solvers are efficient, but require a good preconditioner in order for the solution to converge [17,157,54,16,156,55,98]. The constrained preconditioner and solver presented in [55] was tested in this work. This type of iterative procedure is able to treat the asymmetric matrix generated after discretization and preliminary results could be obtained. ...
Porous media are found in many engineering-relevant materials such as foam and cementitious composites, as well as in biological tissues. They present a complex nature, being composed of both solid and fluid phases, which interact with each other. This interaction can be caused by chemical reactions or due to drying, e.g. during hardening
of cement-based materials when the solid matrix is formed and deforms. The presence of inner heterogeneities, self or external restraints prevents the free deformation of the medium and might lead to cracking. Cracks induced by change of volume, or shrinkage, occur due to short- and long-term drying. In cementitious materials at early ages, these can compromise the durability of the construction. Aim of this thesis is to develop a numerical framework able to predict shrinkage-induced cracking in porous materials. In this work, porous media are modeled at the macroscopic scale, in other words pores
and heterogeneities are not modeled explicitly and average properties are taken into account instead. The poromechanical framework is based on the effective stress concept. A phase-field model of brittle fracture is utilized to model cracking and coupled to the poromechanical part through the effective stress and through an expression of the fracture energy depending on hydraulic variables. The developed mathematical framework is discretized with the finite element method using the Taylor-Hood element pair and implemented within the deal.II library. The first application of the framework deals
with the desiccation phenomenon in soils. A block of clay is subjected to drying at different configurations in a 2D setting. A sensitivity analysis of the problem with respect to the variation of input quantities is performed. The obtained behavior compares very well with the experimentally observed one. Numerical aspects and the extension to the 3D setting are also investigated. The second application deals with cracking induced by drying shrinkage in cementitious mortar. In order to obtain the most appropriate input data for the calibration of the framework and to pursue its independent validation, a set of original experiments is performed. The calibration tests encompass
mechanical as well as poromechanical tests, which aim at providing material properties and boundary conditions such as mass loss (flux) data for shrinkage-induced cracking simulations. For independent validation, ring tests are performed. A good agreement between computational and experimental results is found.
... Previous work on the computation of inverse factors has to large extent focused on approximations used as preconditioners for iterative solution of linear systems [5]. Examples, besides the AINV algorithms already mentioned, include FSAI [23] and more recent variants [11]. Methods making use of a recursive partitioning of the matrix is used in direct methods for factorization such as multifrontal methods [10]. ...
We propose a localized divide and conquer algorithm for inverse factorization of Hermitian positive definite matrices S with localized structure, e.g. exponential decay with respect to some given distance function on the index set of S. The algorithm is a reformulation of recursive inverse factorization [J. Chem. Phys., 128 (2008), 104105] but makes use of localized operations only. At each level of recursion, the problem is cut into two subproblems and their solutions are combined using iterative refinement [Phys. Rev. B, 70 (2004), 193102] to give a solution to the original problem. The two subproblems can be solved in parallel without any communication and, using the localized formulation, the cost of combining their results is proportional to the cut size, defined by the binary partition of the index set. This means that for cut sizes increasing as o(n) with system size n the cost of combining the two subproblems is negligible compared to the overall cost for sufficiently large systems. We also present an alternative derivation of iterative refinement based on a sign matrix formulation, analyze the stability, and propose a parameterless stopping criterion. We present bounds for the initial factorization error and the number of iterations in terms of the condition number of S when the starting guess is given by the solution of the two subproblems in the binary recursion. These bounds are used in theoretical results for the decay properties of the involved matrices. The localization properties of our algorithms are demonstrated for matrices corresponding to nearest neighbor overlap on one-, two-, and three-dimensional lattices as well as basis set overlap matrices generated using the Hartree-Fock and Kohn-Sham density functional theory electronic structure program Ergo [SoftwareX, 7 (2018), 107].
... Several effective off-the-shelf SPD algebraic preconditionersà −1 are already available, in the field of incomplete factorizations, approximate inverses, domain decomposition and multigrid methods, e.g. [33][34][35][36][37]. Unfortunately, designing robust and at the same time inexpensive preconditioners for the exact Schur complement -which is almost completely dense due to the term A −1 -is a challenging task and represents the cornerstone the global preconditioner behavior rests on. ...
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 saddle-point 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.
... Thus, the minimum of each [FAF T ] ii is equivalent to choosing F = W ideal . Similar to the idea presented in Franceschini et al. (2017), the ideal prolongator is approximated by running few iterations of the FSAI procedure (Janna and Ferronato, 2011), where in place of g i we compute w T i , the i-th row of W. We call this prolongation strategy ABF, due to its similarity with the Adaptive Block FSAI set-up (Janna et al., 2015), and the parameters used for its set-up are denoted as: ...
