Schwarz Preconditioner Research Articles

A fully coupled two‐level Schwarz preconditioner based on smoothed aggregation for the transient multigroup neutron diffusion equations

SummaryThe multigroup neutron diffusion equations (an approximation of the neutron transport equation) are widely used for studying the motion of neutrons and their interactions with stationary background materials. Solving the multigroup neutron diffusion equations is challenging because the unknowns are tightly coupled through scattering and fission events, and solutions with high spatial resolution of full reactor cores in multiphysics environments are frequently required. In this paper, we focus on the development of a scalable, parallel preconditioner for solving the system of equations arising from the finite‐element discretization of the multigroup neutron diffusion equations in space and an implicit finite‐difference scheme in time. The parallel preconditioner (here referred to as the “fully coupled Schwarz preconditioner”) is constructed by monolithically applying the overlapping domain decomposition method together with a smoothed aggregation‐based coarse space to the coupled system. Our approach is different from the traditional block Gauss–Seidel sweep method that applies the preconditioner from the fast group to the thermal group sequentially, and we demonstrate that it provides significant improvements in terms of both the number of iterations required and the total compute time for a system of equations with millions of unknowns on a large supercomputer.

Isogeometric Schwarz preconditioners for the biharmonic problem

A scalable overlapping Schwarz preconditioner for the biharmonic Dirichlet problem discretized by isogeometric analysis is constructed, and its convergence rate is analyzed. The proposed preconditioner is based on solving local biharmonic problems on overlapping subdomains that form a partition of the CAD domain of the problem and on solving an additional coarse biharmonic problem associated with the subdomain coarse mesh. An h-analysis of the preconditioner shows an optimal convergence rate bound that is scalable in the number of subdomains and is cubic in the ratio between subdomain and overlap sizes. Numerical results in 2D and 3D confirm this analysis and also illustrate the good convergence properties of the preconditioner with respect to the isogeometric polynomial degree p and regularity k.

Additive average Schwarz with adaptive coarse spaces: scalable algorithms for multiscale problems

We present an analysis of the additive average Schwarz preconditioner with two newly proposed adaptively enriched coarse spaces, which were presented at the twenty-third international conference on domain decomposition methods in Korea, for solving second-order elliptic problems with highly varying and discontinuous coefficients. It is shown that the condition number of the preconditioned system is bounded independently of the variations and the jumps in the coefficient while depending only on a prescribed threshold for the eigenvalues of the coarse space, and it depends linearly on the mesh parameter ratio H/h that is the ratio between the subdomain size and the mesh size thereby retaining the same optimality and scalability of the original additive average Schwarz preconditioner.

A scalable fully implicit framework for reservoir simulation on parallel computers

The modeling of multiphase fluid flow in porous medium is of interest in the field of reservoir simulation. The promising numerical methods in the literature are mostly based on the explicit or semi-implicit approach, which both have certain stability restrictions on the time step size. In this work, we introduce and study a scalable fully implicit solver for the simulation of two-phase flow in a porous medium with capillarity, gravity and compressibility, which is free from the limitations of the conventional methods. In the fully implicit framework, a mixed finite element method is applied to discretize the model equations for the spatial terms, and the implicit Backward Euler scheme with adaptive time stepping is used for the temporal integration. The resultant nonlinear system arising at each time step is solved in a monolithic way by using a Newton–Krylov type method. The corresponding linear system from the Newton iteration is large sparse, nonsymmetric and ill-conditioned, consequently posing a significant challenge to the fully implicit solver. To address this issue, the family of additive Schwarz preconditioners is taken into account to accelerate the convergence of the linear system, and thereby improves the robustness of the outer Newton method. Several test cases in one, two and three dimensions are used to validate the correctness of the scheme and examine the performance of the newly developed algorithm on parallel computers.

Convergence of Krylov subspace solvers with Schwarz preconditioner for the exterior Maxwell problem

The consideration of an integral representation as an exact boundary condition for the finite element resolution of wave propagation problems in exterior domain induces algorithmic difficulties. In this paper, we are interested in the resolution of an exterior Maxwell problem in 3D. As a first step, we focus on the justification of an algorithm described in literature, using an interpretation as a Schwarz method. The study of the convergence indicates that it depends significantly on the thickness of the domain of computation. This analysis suggests the use of the finite element term of Schwarz method as a preconditioner for use of Krylov iterative solvers. An analytical study of the case of a spherical perfect conductor indicates the efficiency of such approach. The consideration of the preconditioner suggested by the Schwarz method leads to a superlinear convergence of the GMRES predicted by the analytical study and verified numerically.

Domain decomposition methods for space fractional partial differential equations

A two-level additive Schwarz preconditioner is proposed for solving the algebraic systems resulting from the finite element approximations of space fractional partial differential equations (SFPDEs). It is shown that the condition number of the preconditioned system is bounded by C(1+H/δ), where H is the maximum diameter of subdomains and δ is the overlap size among the subdomains. Numerical results are given to support our theoretical findings.

An arbitrary Lagrangian‐Eulerian framework with exact mass conservation for the numerical simulation of 2D rising bubble problem

SummaryAn arbitrary Lagrangian‐Eulerian framework, which combines the advantages of both Lagrangian and Eulerian methods, is presented to solve incompressible multiphase flow problems. The incompressible Navier‐Stokes equations are discretized using the side‐centered unstructured finite volume method, where the velocity vector components are defined at the midpoint of each cell face, while the pressure term is defined at element centroids. The pressure field is treated to be discontinuous across the interface with the discontinuous treatment of density and viscosity. The surface tension term at the interface is treated as a force tangent to the interface and computed with several different approaches including the use of Legendre polynomials. In addition, the several different discretizations of interface kinematic boundary conditions are investigated. For the application of the interface kinematic boundary condition, a special attention is given to satisfy both local and global discrete geometric conservation law to conserve the total mass of both species at machine precision. The mesh vertices are deformed by solving the linear elasticity equations due to the normal displacement of interface. The resulting algebraic equations are solved in a fully coupled manner, and a one‐level restricted additive Schwarz preconditioner with a block‐incomplete factorization within each partitioned subdomain is used for the resulting fully coupled system. The method is validated by simulating the classical benchmark problem of a single rising bubble in a viscous fluid due to buoyancy. The results of numerical simulations are found out to be in an excellent agreement with the earlier results in the literature. The mass of the bubble is conserved, and discontinuous pressure field is obtained to avoid errors due to the incompressibility condition in the vicinity of the interface, where the density and viscosity jumps occur.

A parallel monolithic approach for fluid-structure interaction in a cerebral aneurysm

Abstract A parallel fully-coupled numerical algorithm has been developed for the fluid-structure interaction problem in a cerebral artery with aneurysm. For the fluid part of the problem, an Arbitrary Lagrangian-Eulerian formulation based on the side-centered unstructured finite volume method is employed for the governing incompressible Navier–Stokes equations. The deformation of the solid domain is governed by the constitutive laws for the nonlinear Saint Venant-Kirchhoff material and the classical Galerkin finite element method is used to discretise the governing equations in a Lagrangian frame. The time integration method for the structure domain is based on the Newmark type generalized − α method while the second-order backward difference (BDF2) is used in the fluid domain. A special attention is given to construct an algorithm obeying the local/global discrete geometric conservation laws (DGCL) in order to conserve fluid volume at machine precision when the fluid domain is entirely enclosed by solid domain boundary. Therefore, a compatible kinematic boundary condition is applied at the interface between the solid and fluid domains. The parallel implementation of the present fully coupled unstructured fluid-structure solver is based on the PETSc library and a one-level restricted additive Schwarz preconditioner with a block-incomplete factorization within each partitioned sub-domains is utilized for the resulting fully coupled system. The proposed algorithm is initially validated for a pressure pulse propagating in a flexible tube and the mass conservation accuracy is tested for a thin elastic sphere filled with an incompressible fluid in a circular tube. Then the numerical method is applied to a complicated problem involving unsteady pulsatile blood flow in a cerebral artery with aneurysm as a realistic fluid-structure interaction problem encountered in biomechanics. Various hemodynamic quantities of interest like fluid velocities, blood pressure and wall shear stresses (WSS) are computed as well as the time dependent artery wall deformations.

Parallel Dynamics Simulation Using a Krylov-Schwarz Linear Solution Scheme

Fast dynamics simulation of large-scale power systems is a computational challenge because of the need to solve a large set of stiff, nonlinear differential-algebraic equations at every time step. The main bottleneck in dynamic simulations is the solution of a linear system during each nonlinear iteration of Newton’s method. In this paper, we present a parallel Krylov–Schwarz linear solution scheme that uses the Krylov subspace-based iterative linear solver GMRES with an overlapping restricted additive Schwarz preconditioner. Performance tests of the proposed Krylov–Schwarz scheme for several large test cases ranging from 2000 to 20000 buses, including a real utility network, show good scalability on different computing architectures.

Domain decomposition preconditioners for the discontinuous Petrov–Galerkin method

In this paper, we design some efficient domain decomposition preconditioners for the discontinuous Petrov–Galerkin (DPG) method. Due to the special properties of the DPG method, the boundary condition becomes crucial in both of its application and analysis. We mainly focus on one of the boundary conditions: the Robin boundary condition, which actually appears in some useful model problems like the Helmholtz equation. We first design a two-level additive Schwarz preconditioner for the Poisson equation with a Robin boundary condition and give a rigorous condition number estimate for the preconditioned algebraic system. Moreover we also construct an additive Schwarz preconditioner for solving the Helmholtz equation. Numerical results show that the condition number of the preconditioned system is independent of wavenumber ω and mesh size h .

A scalable nonlinear fluid–structure interaction solver based on a Schwarz preconditioner with isogeometric unstructured coarse spaces in 3D

Nonlinear fluid–structure interaction (FSI) problems on unstructured meshes in 3D appear in many applications in science and engineering, such as vibration analysis of aircrafts and patient-specific diagnosis of cardiovascular diseases. In this work, we develop a highly scalable, parallel algorithmic and software framework for FSI problems consisting of a nonlinear fluid system and a nonlinear solid system, that are coupled monolithically. The FSI system is discretized by a stabilized finite element method in space and a fully implicit backward difference scheme in time. To solve the large, sparse system of nonlinear algebraic equations at each time step, we propose an inexact Newton–Krylov method together with a multilevel, smoothed Schwarz preconditioner with isogeometric coarse meshes generated by a geometry preserving coarsening algorithm. Here “geometry” includes the boundary of the computational domain and the wet interface between the fluid and the solid. We show numerically that the proposed algorithm and implementation are highly scalable in terms of the number of linear and nonlinear iterations and the total compute time on a supercomputer with more than 10,000 processor cores for several problems with hundreds of millions of unknowns.

Isogeometric overlapping Schwarz preconditioners for the Bidomain reaction–diffusion system

We present and study an overlapping additive Schwarz preconditioner for the isogeometric discretization of a reaction–diffusion system modeling the heart bioelectrical activity, known as the Bidomain model, consisting of a degenerate system of parabolic and elliptic PDE. This model includes intramural fiber rotation, anisotropic conductivity coefficients and is coupled through the reaction term with a system of ODEs, which models the ionic currents of the cellular membrane. The overlapping Schwarz preconditioner is applied with a PCG accelerator to solve the linear system arising at each time step from the isogeometric discretization in space and a semi-implicit adaptive method in time. A theoretical convergence rate analysis shows that the resulting solver is scalable, optimal in the ratio of subdomain/element size and the convergence rate improves with increasing overlap size. Numerical tests in three-dimensional ellipsoidal domains confirm the theoretical estimates and additionally show the robustness with respect to jump discontinuities of the orthotropic conductivity coefficients.

Hybrid and Multiplicative Overlapping Schwarz Algorithms with Standard Coarse Spaces for Mixed Linear Elasticity and Stokes Problems

AbstractThe goal of this work is to construct and study hybrid and multiplicative two-level overlapping Schwarz algorithms with standard coarse spaces for the almost incompressible linear elasticity and Stokes systems, discretized by mixed finite and spectral element methods with discontinuous pressures. Two different approaches are considered to solve the resulting saddle point systems: a) a preconditioned conjugate gradient (PCG) method applied to the symmetric positive definite reformulation of the almost incompressible linear elasticity system obtained by eliminating the pressure unknowns; b) a GMRES method with indefinite overlapping Schwarz preconditioner applied directly to the saddle point formulation of both the elasticity and Stokes systems. Condition number estimates and convergence properties of the proposed hybrid and multiplicative overlapping Schwarz algorithms are proven for the positive definite reformulation of almost incompressible elasticity. These results are based on our previous study [8] where only additive Schwarz preconditioners were considered for almost incompressible elasticity. Extensive numerical experiments with both finite and spectral elements show that the proposed overlapping Schwarz preconditioners are scalable, quasi-optimal in the number of unknowns across individual subdomains and robust with respect to discontinuities of the material parameters across subdomains interfaces. The results indicate that the proposed preconditioners retain a good performance also when the quasi-monotonicity assumption, required by the available theory, does not hold.

Restricted Additive Schwarz Preconditioner for Elliptic Equations with Jump Coefficients

AbstractThis paper provides a proof of robustness of the restricted additive Schwarz preconditioner with harmonic overlap (RASHO) for the second order elliptic problems with jump coefficients. By analyzing the eigenvalue distribution of the RASHO preconditioner, we prove that the convergence rate of preconditioned conjugate gradient method with RASHO preconditioner is uniform with respect to the large jump and meshsize.

Two-Level Space–Time Domain Decomposition Methods for Flow Control Problems

For time-dependent control problems, the class of sub-optimal algorithms is popular and the parallelization is usually applied in the spatial dimension only. In the paper, we develop a class of fully-optimal methods based on space–time domain decomposition methods for some boundary and distributed control of fluid flow and heat transfer problems. In the fully-optimal approach, we focus on the use of an inexact Newton solver for the necessary optimality condition arising from the implicit discretization of the optimization problem and the use of one-level and two-level space–time overlapping Schwarz preconditioners for the Jacobian system. We show that the numerical solution from the fully-optimal approach is generally better than the solution from the sub-optimal approach in terms of meeting the objective of the optimization problem. To demonstrate the robustness and parallel scalability and efficiency of the proposed algorithm, we present some numerical results obtained on a parallel computer with a few thousand processors.

Joint influence of transmural heterogeneities and wall deformation on cardiac bioelectrical activity: A simulation study

The aim of this work is to investigate, by means of numerical simulations, the influence of myocardial deformation due to muscle contraction and relaxation on the cardiac repolarization process in presence of transmural intrinsic action potential duration (APD) heterogeneities. The three-dimensional electromechanical model considered consists of the following four coupled components: the quasi-static transversely isotropic finite elasticity equations for the deformation of the cardiac tissue; the active tension model for the intracellular calcium dynamics and cross-bridge binding; the anisotropic Bidomain model for the electrical current flow through the deforming cardiac tissue; the membrane model of ventricular myocytes, including stretch-activated channels. The numerical simulations are based on our finite element parallel solver, which employs Multilevel Additive Schwarz preconditioners for the solution of the discretized Bidomain equations and Newton-Krylov methods for the solution of the discretized non-linear finite elasticity equations. Our findings show that: (i) the presence of intrinsic transmural cellular APD heterogeneities is not fully masked by electrotonic current flow or by the presence of the mechanical deformation; (ii) despite the presence of transmural APD heterogeneities, the recovery process follows the activation sequence and there is no significant transmural repolarization gradient; (iii) with or without transmural APD heterogeneities, epicardial electrograms always display the same wave shape and discordance between the polarity of QRS complex and T-wave; (iv) the main effects of the mechanical deformation are an increase of the dispersion of repolarization time and APD, when computed over the total cardiac domain and over the endo- and epicardial surfaces, while there is a slight decrease along the transmural direction.

A Uniform Additive Schwarz Preconditioner for High-Order Discontinuous Galerkin Approximations of Elliptic Problems

In this paper we design and analyze a uniform preconditioner for a class of high-order Discontinuous Galerkin schemes. The preconditioner is based on a space splitting involving the high-order conforming subspace and results from the interpretation of the problem as a nearly-singular problem. We show that the proposed preconditioner exhibits spectral bounds that are uniform with respect to the discretization parameters, i.e., the mesh size, the polynomial degree and the penalization coefficient. The theoretical estimates obtained are supported by numerical tests.

Parallel subdomain solver strategies for the algebraic additive Schwarz preconditioner

Hybrid domain-decomposition method with two levels of domain partitioning.Solution of the local problem by ShyLU or other parallel approximation techniques.Proposal of an inexact QR factorization to be used within ShyLU.Comparisons of parallel performances between UMFPACK, ShyLU, and inexact QR. Domain-decomposition (DD) methods are used in most, if not all, modern parallel implementations of finite element modeling software. In the solver stage, the algebraic additive Schwarz (AAS) domain-decomposition preconditioner represents a fundamental component and its performance and scalability are key to the overall performance of the solution process. The established approach to construct the preconditioner in a parallel MPI setting is with a 1-to-1 correspondence between the number of MPI processes and the number of AAS subdomains.In this paper, we describe our attempts to extend this paradigm with the possibility to assign more than one MPI process per AAS subdomain, with the goal of improving the overall performance of the AAS preconditioner on supercomputers with multicore nodes.We discuss the implementation of the new AAS preconditioner framework, based on two levels of MPI parallelism, and the performance of different subdomain solver strategies. Finally, we examine the behavior of our novel approach for a series of benchmark problems, performed with the LifeV parallel finite element library.

Optimal additive Schwarz preconditioning for hypersingular integral equations on locally refined triangulations

For non-preconditioned Galerkin systems, the condition number grows with the number of elements as well as the quotient of the maximal and the minimal mesh-size. Therefore, reliable and effective numerical computations, in particular on adaptively refined meshes, require the development of appropriate preconditioners. We analyze and numerically compare multilevel additive Schwarz preconditioners for hypersingular integral equations, where 2D and 3D as well as closed boundaries and open screens are covered. The focus is on a new local multilevel preconditioner which is optimal in the sense that the condition number of the corresponding preconditioned system is independent of the number of elements, the local mesh-size, and the number of refinement levels.

Schwarz type preconditioners for the neutron diffusion equation

Domain decomposition is a mature methodology that has been used to accelerate the convergence of partial differential equations. Even if it was devised as a solver by itself, it is usually employed together with Krylov iterative methods improving its rate of convergence, and providing scalability with respect to the size of the problem.In this work, a high order finite element discretization of the neutron diffusion equation is considered. In this problem the preconditioning of large and sparse linear systems arising from a source driven formulation becomes necessary due to the complexity of the problem. On the other hand, preconditioners based on an incomplete factorization are very expensive from the point of view of memory requirements. The acceleration of the neutron diffusion equation is thus studied here by using alternative preconditioners based on domain decomposition techniques inside Schur complement methodology. The study considers substructuring preconditioners, which do not involve overlapping, and additive Schwarz preconditioners, where some overlapping between the subdomains is taken into account.The performance of the different approaches is studied numerically using two-dimensional and three-dimensional problems. It is shown that some of the proposed methodologies outperform incomplete LU factorization for preconditioning as long as the linear system to be solved is large enough, as it occurs for three-dimensional problems. They also outperform classical diagonal Jacobi preconditioners, as long as the number of systems to be solved is large enough in such a way that the overhead of building the preconditioner is less than the improvement in the convergence rate.