We present a scalable block preconditioning strategy for the trace system coming from the high-order hybridized discontinuous Galerkin (HDG) discretization of incompressible resistive magnetohydrodynamics (MHD). We construct the block preconditioner with a least squares commutator (BFBT) approximation for the inverse of the Schur complement that segregates out the pressure unknowns of the trace system. The remaining velocity, magnetic field, and Lagrange multiplier unknowns form a coupled nodal unknown block (the upper block), for which a system algebraic multigrid (AMG) is used for the approximate inverse. The complexity of the MHD equations together with the algebraic nature of the statically condensed HDG trace system makes the choice of smoother in the system AMG part critical for the convergence and performance of the block preconditioner. Our numerical experiments show GMRES preconditioned by ILU(0) of overlap zero as a smoother inside system AMG performs best in terms of robustness, time per nonlinear iteration and memory requirements. With several transient test cases in 2D and 3D including the island coalescence problem at high Lundquist number we demonstrate the robustness and parallel scalability of the block preconditioner. Additionally for the upper block a preliminary study of an alternate nodal block system solver based on a multilevel approximate nested dissection is presented. On a 2D island coalescence problem the multilevel approximate nested dissection preconditioner shows better scalability with respect to mesh refinement than the system AMG, but is relatively less robust with respect to Lundquist number scaling.
We propose a uniform block-diagonal preconditioner for condensed $H$(div)-conforming HDG schemes for parameter-dependent saddle point problems, including the generalized Stokes equations and the linear elasticity equations. An optimal preconditioner is obtained for the stiffness matrix on the global velocity/displacement space via the auxiliary space preconditioning (ASP) technique \cite{Xu96}. A spectrally equivalent approximation to the Schur complement on the element-wise constant pressure space is also constructed, and an explicit computable exact inverse is obtained via the Woodbury matri…
We present the lowest-order hybridizable discontinuous Galerkin schemes with numerical integration (quadrature), denoted as HDG-P0, for the reaction-diffusion equation and the generalized Stokes equations on conforming simplicial meshes in two- and three-dimensions. Here by lowest order, we mean that the (hybrid) finite element space for the global HDG facet degrees of freedom (DOFs) is the space of piecewise constants on the mesh skeleton. A discontinuous piecewise linear space is used for the approximation of the local primal unknowns. We give the optimal a priori error analysis of the propo…
In this study, we present an $hp$-multigrid preconditioner for a divergence-conforming HDG scheme for the generalized Stokes and the Navier-Stokes equations using an augmented Lagrangian formulation. Our method relies on conforming simplicial meshes in two- and three-dimensions. The $hp$-multigrid algorithm is a multiplicative auxiliary space preconditioner that employs the lowest-order space as the auxiliary space, and we developed a geometric multigrid method as the auxiliary space solver. For the generalized Stokes problem, the crucial ingredient of the geometric multigrid method is the equ…
We propose a multilevel approach for trace systems resulting from hybridized discontinuous Galerkin (HDG) methods. The key is to blend ideas from nested dissection, domain decomposition, and high-order characteristic of HDG discretizations. Specifically, we first create a coarse solver by eliminating and/or limiting the front growth in nested dissection. This is accomplished by projecting the trace data into a sequence of same or high-order polynomials on a set of increasingly $h-$coarser edges/faces. We then combine the coarse solver with a block-Jacobi fine scale solver to form a two-level s…
A balancing domain decomposition by constraints preconditioner for a hybridizable discontinuous Galerkin discretization of an elliptic optimal control problem
We present robust and highly parallel multilevel non-overlapping Schwarz preconditioners, to solve an interior penalty discontinuous Galerkin finite element discretization of a system of steady state, singularly perturbed reaction-diffusion equations with a singular reaction operator, using a GMRES solver. We provide proofs of convergence for the two-level setting and the multigrid V-cycle as well as numerical results for a wide range of regimes.
Domain decomposition (DD) methods are widely used as preconditioner techniques. Their effectiveness relies on the choice of a locally constructed coarse space. Thus far, this construction was mostly achieved using non-assembled matrices from discretized partial differential equations (PDEs). Therefore, DD methods were mainly successful when solving systems stemming from PDEs. In this paper, we present a fully algebraic multilevel DD method where the coarse space can be constructed locally and efficiently without any information besides the coefficient matrix. The condition number of the precon…
A hybrid recursive multilevel incomplete factorization preconditioner for solving general linear systems
Johann Bernoulli Institute for Mathematics and Computer Science - University of Groningen（格罗宁根大学）；School of Mathematical Sciences, University of Electronic Science and Technology of China（电子科技大学）
We propose a $p$-multilevel preconditioner for Hybrid High-Order discretizations (HHO) of the Stokes equation, numerically assess its performance on two variants of the method, and compare with a classical Discontinuous Galerkin scheme. We specifically investigate how the combination of $p$-coarsening and static condensation influences the performance of the $V$-cycle iteration for HHO. Two different static condensation procedures are considered, resulting in global linear systems with a different number of unknowns and non-zero elements. An efficient implementation is proposed where coarse le…
Multilevel Preconditioner with Stable Coarse Grid Corrections for the Helmholtz Equation
Xiamen University（厦门大学）；Nanjing University（南京大学）；Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所）
The multigroup neutron transport criticality calculations using modern supercomputers have been widely employed in a nuclear reactor analysis for studying whether or not a system is self-sustaining. However, the design and development of efficient parallel algorithms for the transport criticality calculations is challenging especially when the number of processor cores is large and an unstructured mesh is adopted. In particular, both the compute time and memory usage have to be carefully taken into consideration due to the dimensionality of the neutron transport equations. In this paper, we st…
We present the construction of additive multilevel preconditioners, also known as BPX preconditioners, for the solution of the linear system arising in isogeometric adaptive schemes with (truncated) hierarchical B-splines. We show that the locality of hierarchical spline functions, naturally defined on a multilevel structure, can be suitably exploited to design and analyze efficient multilevel decompositions. By obtaining smaller subspaces with respect to standard tensor-product B-splines, the computational effort on each level is reduced. We prove that, for suitably graded hierarchical meshes…
In this work, we provide a performance comparison between the Balancing Domain Decomposition by Constraints (BDDC) and the Algebraic Multigrid (AMG) preconditioners for cardiac mechanics on both structured and unstructured finite element meshes. The mechanical behavior of myocardium can be described by the equations of three-dimensional finite elasticity, which are discretized by finite elements in space and yield the solution of a large scale nonlinear algebraic system. This problem is solved by a Newton-Krylov method, where the solution of the Jacobian linear system is accelerated by BDDC/AM…
Parallel algorithms and simulators with good scalabilities are particularly important for large-scale reservoir simulations on modern supercomputers with a large number of processors. In this paper, we introduce and study a family of highly scalable multilevel restricted additive Schwarz (RAS) methods for the fully implicit solution of subsurface flows with Peng-Robinson equation of state in two and three dimensions. With the use of a second-order fully implicit scheme, the proposed simulator is unconditionally stable with the relaxation of the time step size by the stability condition. The in…
We propose an unconditionally robust and highly effective preconditioner for general symmetric positive definite (SPD) matrices based on structured incomplete factorization (SIF), called enhanced SIF (eSIF) preconditioner. The original SIF strategy proposed recently derives a structured preconditioner by applying block diagonal preprocessing to the matrix and then compressing appropriate scaled off-diagonal blocks. Here, we use an enhanced scaling-and-compression strategy to design the new eSIF preconditioner. Some subtle modifications are made, such as the use of two-sided block triangular pr…
We propose and analyze optimal additive multilevel solvers for isogeometric discretizations of scalar elliptic problems for locally refined T-meshes. Applying the refinement strategy in Morgenstern and Peterseim (2015, Comput. Aided Geom. Design, 34, 50-66) we can guarantee that the obtained T-meshes have a multilevel structure, and that the associated T-splines are analysis-suitable, for which we can define a dual basis and a stable projector. Taking advantage of the multilevel structure, we develop two BPX preconditioners: the first on the basis of local smoothing only for the functions affe…
The multigroup neutron transport equations has been widely used to study the interactions of neutrons with their background materials in nuclear reactors. High-resolution simulations of the multigroup neutron transport equations using modern supercomputers require the development of scalable parallel solving techniques. In this paper, we study a scalable transport method for solving the algebraic system arising from the discretization of the multigroup neutron transport equations. The proposed transport method consists of a fully coupled Newton solver for the generalized eigenvalue problems an…
We consider effective preconditioners for solving Laplacians of general weighted graphs. Theoretically, spectral sparsifiers (SSs) provide preconditioners of optimal computational complexity. However, they are not easy to use for real-world applications due to the implementation complications. Multigrid (MG) methods, on the contrary, are computationally efficient but lack of theoretical justifications. To bridge the gap between theory and practice, we adopt ideas of MG and SS methods and proposed preconditioners that can be used in practice with theoretical guarantees. We expand the original g…
Fast multilevel solvers for a class of discrete fourth order parabolic problems
Pacific Northwest National Laboratory（太平洋西北国家实验室）；Southwest Jiaotong University（西南交通大学）；Tufts University（塔夫茨大学）；University of California, Irvine（加州大学尔湾分校）；University of Maryland（马里兰大学）；Pennsylvania State University（宾夕法尼亚州立大学）
We present an implementation of a fully stage-parallel preconditioner for Radau IIA type fully implicit Runge--Kutta methods, which approximates the inverse of $A_Q$ from the Butcher tableau by the lower triangular matrix resulting from an LU decomposition and diagonalizes the system with as many blocks as stages. For the transformed system, we employ a block preconditioner where each block is distributed and solved by a subgroup of processes in parallel. For combination of partial results, we either use a communication pattern resembling Cannon's algorithm or shared memory. A performance mode…
Algorithms for studying transitions and instabilities in incompressible flows typically require the solution of linear systems with the full Jacobian matrix. Other popular approaches, like gradient-based design optimization and fully implicit time integration, also require very robust solvers for this type of linear system. We present a parallel fully coupled multilevel incomplete factorization preconditioner for the 3D stationary incompressible Navier-Stokes equations on a structured grid. The algorithm and software are based on the robust two-level method developed by Wubs and Thies. In this…
Multi-level preconditioners for Discontinuous Galerkin (DG) discretizations are widely used to solve elliptic equations, and a main ingredient of such solvers is the interpolation operator to transfer information from the coarse to the fine grid. Classical interpolation operators give continuous interpolated values, but since DG solutions are naturally discontinuous, one might wonder if one should not use discontinuous interpolation operators for DG discretizations. We consider a discontinuous interpolation operator with a parameter that controls the discontinuity, and determine the optimal ch…
Multilevel Preconditioning of Discontinuous-Galerkin Spectral Element Methods, Part I: Geometrically Conforming Meshes
RWTH Aachen（亚琛工业大学）；CNRS, UMR 7598, Laboratoire Jacques-Louis Lions（法国国家科学研究中心，联合研究单位7598，雅克-路易·利翁斯实验室）；UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions（巴黎第六大学，联合研究单位7598，雅克-路易·利翁斯实验室）；Politecnico di Torino（都灵理工大学）
Analysis of Block Jacobi/Gauss-Seidel and additive/multiplicative Schwarz preconditioning through the theory of GLT sequences, with applications to domain decomposition discretizations
University of Rome Tor Vergata（罗马托尔维加塔大学）；University Mohammed VI Polytechnic（穆罕默德六世polytechnic大学）；University of Insubria（因苏布里亚大学）；Uppsala University（乌普萨拉大学）
Runge-Kutta (RK) schemes, especially Gauss-Legendre and some other fully implicit RK (FIRK) schemes, are desirable for the time integration of parabolic partial differential equations due to their A-stability and high-order accuracy. However, it is significantly more challenging to construct optimal preconditioners for them compared to diagonally implicit RK (or DIRK) schemes. To address this challenge, we first introduce mathematically optimal preconditioners called block complex Schur decomposition (BCSD), block real Schur decomposition (BRSD), and block Jordan form (BJF), motivated by block…
We introduce a new class of preconditioners to enable flexible GMRES to find a least-squares solution, and potentially the pseudoinverse solution, of large-scale sparse, asymmetric, singular, and potentially inconsistent systems. We develop the preconditioners based on a new observation that generalized inverses (i.e., $\boldsymbol{A}^{g}\in\{\boldsymbol{G}\mid\boldsymbol{A}\boldsymbol{G}\boldsymbol{A}=\boldsymbol{A}\}$) enable the preconditioned Krylov subspaces to converge in a single step. We then compute an approximate generalized inverse (AGI) efficiently using a hybrid incomplete factori…
