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

A robust adaptive algebraic multigrid linear solver for structural mechanics

The numerical simulation of structural mechanics applications via finite elements usually requires the solution of large-size and ill-conditioned linear systems, especially when accurate results are sought for derived variables interpolated with lower order functions, like stress or deformation fields. Such task represents the most time-consuming kernel in commercial simulators; thus, it is of significant interest the development of robust and efficient linear solvers for such applications. In this context, direct solvers, which are based on LU factorization techniques, are often used due to their robustness and easy setup; however, they can reach only superlinear complexity, in the best case, thus, have limited applicability depending on the problem size. On the other hand, iterative solvers based on algebraic multigrid (AMG) preconditioners can reach up to linear complexity for sufficiently regular problems but do not always converge and require more knowledge from the user for an efficient setup. In this work, we present an adaptive AMG method specifically designed to improve its usability and efficiency in the solution of structural problems. We show numerical results for several practical applications with millions of unknowns and compare our method with two state-of-the-art linear solvers proving its efficiency and robustness.Comment: 50 pages, 16 figures, submitted to CMAM

Preconditioners for Soil-Structure Interaction Problems with Significant Material Stiffness Contrast

Ph.DDOCTOR OF PHILOSOPH

Preconditioners for iterative solutions of large-scale linear systems arising from Biot's consolidation equations

Ph.DDOCTOR OF PHILOSOPH

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

Numerical models for the large-scale simulation of fault and fracture mechanics

The possible activation of pre-existing faults and the generation of new fractures in the subsurface may play a critical role in several fields of great social interest, such as the management and the exploitation of groundwater resources, especially in arid areas, the hydrocarbon recovery and storage, and the monitoring of the seismic activity in the Earth’s crust. The sliding and/or opening of a fault can create preferential leakage paths for the pore fluid escape, causing a matter of great concern in the process of storing fluids and hydrocarbons underground. The most challenging effect connected to a fault activation is the possible earthquake triggering. Many earthquakes associated with the production and injection of fluids have been recently reported. Similar issues arise also in the development of unconventional hydrocarbon reservoirs, that has recently experienced a dramatic increase thanks to the deployment of the “fracking” technology, which is based on the massive generation of fractures through the injection of fluids at high pressures. The use of this technique in densely populated areas has raised a large scientific debate on the possible connected environmental risks. The over-exploitation of fresh aquifers in arid regions has caused the generation of significant ground fissures. In this thesis, a novel formulation based on the use of Lagrange multipliers has been developed for the stable and robust numerical modeling of fault mechanics. A fault or fracture is simulated as a pair of inner surfaces included in a 3D geological formation where Lagrange multipliers are used to prescribe the contact constraints. The standard variational formulation of the contact problem with Lagrange multipliers is modified to take into account the energy dissipated by the frictional work along the activated fault portion. This term is computed by making use of the principle of maximum plastic dissipation, whose application defines the direction of the limiting shear stress vector. The novel approach has been verified against analytical solutions and applied in a number of real-world problems. In particular, we test the novel approach in four cases: (i) mechanics of two adjacent blocks, to investigate the numerical properties of the algorithm; (ii-iii) ground fractures due to groundwater withdrawal, with different geometries; (iv) fault reactivation in an underground reservoir subject to primary production and Underground Gas Storage cycles. The results are analyzed and commented. In the fourth case, the possible magnitude of the seismic events triggered by fault reactivation is computed, in order to evaluate whether underground human activities may generate seismicity. The application of the fault model to large-scale problems gives rise to a set of sparse discrete systems of linearized equations with a generalized non-symmetric saddle point structure. The second part of this thesis is devoted to the development of efficient algorithms for the iterative solution of this kind of system. We focus on a preconditioning technique, denoted as “constraint preconditioning”, which exploits the native block structure of the Jacobian. The quality and performance of the preconditioner relies on two steps: (i) the preconditioning of the leading block and (ii) the Schur complement computation. In this work, novel preconditioning techniques for the leading block based on a multilevel framework are developed and tested. The main idea behind the multilevel preconditioner is to improve the quality of the factorized approximate inverses borrowing the scheme of incomplete factorizations, thus introducing some sequentially in perfectly parallelizable algorithms. The proposed approach is robust, from a theoretical point of view, and very efficient in parallel environment. As to the latter point, i.e. the Schur complement computation, it can be done with the aid of different approximations. The main difference is whether the Jacobian is symmetrized or not. The computation can be founded on the FSAI approximation of the leading block inverse or on a physically-based block diagonal block algorithm. The Schur complement must be inverted, thus other possibilities come in. The approximate Schur complement can be inverted through FSAI, if symmetric, or an incomplete factorization, if non-symmetric, but it can also be solved exactly, thanks to a direct solver. The performances of the proposed algorithms are finally investigated and discussed in a set of real-world numerical examples