Restricted Additive Schwarz Method Research Articles

Accurate implementation of two-level asynchronous domain decomposition solvers

Recently, asynchronous coarse-grid correction has been achieved within additive Schwarz-type and primal Schur domain decomposition frameworks. Both additive and multiplicative coarse-grid corrections were discussed, however, the implemented asynchronous Schwarz-type solver with additive correction relies on the specific design of the restricted additive Schwarz (RAS) method, and also requires an overlap between the subsets of unknowns. In this paper, we first highlight a gap between the theoretical analysis from the literature and the associated RAS implementation. It turns out that communications delays would actually need to be bounded in order to fit the theory. This has to be stressed since, despite the asynchronous nature of the solver, the coarse-grid correction requires non-blocking global synchronization, which is subject to communications bottleneck. Second, we propose an implementation approach which applies to a wider class of additive Schwarz-type methods while still coping with the bounded delays requirement.

Accelerating multivariate functional approximation computation with domain decomposition techniques

Modeling large datasets through Multivariate Functional Approximations (MFA) provide an elegant way to handle many visualization and scientific analysis workflows. The process necessitates scalable data partitioning methods to compute MFA representations efficiently without compromising the accuracy or continuity of the reconstructed solution. We propose a domain-decomposed method for computing the MFA with B-spline bases, which reduces the total work per task and uses a restricted Additive Schwarz (RAS) method to converge the control point data degrees-of-freedom along subdomain boundaries. We provide an in-depth analysis of the parallel approach with domain decomposition solvers, aiming to minimize local subdomain error residuals and recover high-order continuity at subdomain interfaces with appropriate choices of knot overlaps. The communication cost, determined by the overlap regions in the RAS implementation, is optimized to recover the numerical error profile of the single subdomain case. Our proposed method stands in contrast to previous methods, which typically only recover either C0 or at best C1 continuity for arbitrary B-spline degree expansions, or those that require post-processing to blend discontinuities in the reconstructed data. We demonstrate the effectiveness of our approach using analytical and real-world datasets in 1D, 2D, and 3D through both strong and weak scaling studies. The performance results indicate that the overall cost of computing the approximation is directly proportional to the underlying nearest-neighbor communication implementation, and is only weakly dependent on the overlap region size that determines the size of the messages. This finding underscores the efficiency and scalability of our proposed method, making it a promising solution for handling large datasets in scientific workflows.

Nonlinear parallel-in-time simulations of multiphase flow in porous media

As the number of processors on supercomputers has increased dramatically, there is a growing interest in developing scalable algorithms with a high degree of parallelism for extreme-scale reservoir simulation. However, traditional simulators and algorithms for such nonlinear problems are usually based on the family of time-marching methods, where parallelization is restricted to the spatial dimension only. In this paper, we propose a family of parallel-in-time (PinT) reservoir algorithms for solving multiphase flow problems in porous media to fully exploit the parallelism of supercomputers, in which the design of efficient nonlinear and linear space-time algorithms plays an essential role. More precisely, we first introduce a space-time mixed finite element method for the fully-implicit discretization. The nonlinear system arising at each parallel-in-time step is solved by a variant of bound-preserving Newton methods with a nonlinear elimination preconditioner, where the corresponding linear system is solved by the space-time restricted additive Schwarz (stRAS) method. Since the straightforward extension of one-level stRAS to multilevel does not work because of the pollution effects from the temporal direction, we accordingly present a nonstandard V-cycle multilevel stRAS method with a pollution-removing strategy to accelerate the convergence and improve the robustness. Numerical experiments are presented to demonstrate that the aforementioned parallel-in-time algorithm can not only achieve a high degree of parallelism in accurately resolving reservoir transport features in a heterogeneous medium, but also successfully circumvent the convergence and scalability issues associated with the constraint of large time steps and the boundedness requirement of the solution.

Asynchronous multisplitting-based primal Schur method

This paper presents a new scalable asynchronous domain decomposition method which significantly outperforms the recent two-level asynchronous restricted additive Schwarz (RAS) method. The novel approach consists of considering the classical primal Schur complement framework and solving the interface problem using a multisplitting relaxation based on a weighted additive Schwarz preconditioner. The computational model does not require any significant implementation effort compared to an asynchronous RAS solver. Experiments were conducted on a linear elasticity test case, and the results clearly show the practical superiority of the new asynchronous method, which is even competitive with its conjugate gradient counterpart.

Convergence of restricted additive Schwarz with impedance transmission conditions for discretised Helmholtz problems

The Restricted Additive Schwarz method with impedance transmission conditions, also known as the Optimised Restricted Additive Schwarz (ORAS) method, is a simple overlapping one-level parallel domain decomposition method, which has been successfully used as an iterative solver and as a preconditioner for discretised Helmholtz boundary-value problems. In this paper, we give, for the first time, a convergence analysis for ORAS as an iterative solver—and also as a preconditioner—for nodal finite element Helmholtz systems of any polynomial order. The analysis starts by showing (for general domain decompositions) that ORAS is an unconventional finite element approximation of a classical parallel iterative Schwarz method, formulated at the PDE (non-discrete) level. This non-discrete Schwarz method was recently analysed in [Gong, Gander, Graham, Lafontaine, and Spence, Convergence of parallel overlapping domain decomposition methods for the Helmholtz equation], and the present paper gives a corresponding discrete version of this analysis. In particular, for domain decompositions in strips in 2-d, we show that, when the mesh size is small enough, ORAS inherits the convergence properties of the Schwarz method, independent of polynomial order. The proof relies on characterising the ORAS iteration in terms of discrete ‘impedance-to-impedance maps’, which we prove (via a novel weighted finite-element error analysis) converge as h → 0 h\rightarrow 0 in the operator norm to their non-discrete counterparts.

Three-dimensional finite-difference finite-element frequency-domain wave simulation with multi-level optimized additive Schwarz domain-decomposition preconditioner: A tool for FWI of sparse node datasets

Efficient frequency-domain full-waveform inversion (FWI) of long-offset node data can be designed with a few discrete frequencies, which lead to modest data volumes to be managed during the inversion process. Moreover, attenuation effects can be straightforwardly implemented in the forward problem without the computational overhead. However, 3D frequency-domain seismic modeling is challenging because it requires solving a large and sparse linear indefinite system for each frequency with multiple right-hand sides (RHSs). This linear system can be solved by direct or iterative methods. The former allows efficient processing of multiple RHSs but may suffer from limited scalability for very large problems. Iterative methods equipped with a domain-decomposition preconditioner provide a suitable alternative to process large computational domains for sparse-node acquisition. We have investigated the domain-decomposition preconditioner based on the optimized restricted additive Schwarz (ORAS) method, in which a Robin or perfectly matched layer condition is implemented at the boundaries between the subdomains. The preconditioned system is solved by a Krylov subspace method, whereas a block low-rank lower-upper decomposition of the local matrices is performed at a preprocessing stage. Multiple sources are processed in groups with a pseudoblock method. The accuracy, the computational cost, and the scalability of the ORAS solver are assessed against several realistic benchmarks. In terms of discretization, we compare a compact wavelength-adaptive 27-point finite-difference stencil on a regular Cartesian grid with a P3 finite-element method on h-adaptive tetrahedral mesh. Although both schemes have comparable accuracy, the former is more computationally efficient, the latter being beneficial to comply with known boundaries such as bathymetry. The scalability of the method, the block processing of multiple RHSs, and the straightforward implementation of attenuation, which further improves the convergence of the iterative solver, make the method a versatile forward engine for large-scale 3D FWI applications from sparse node data sets.

Evaluating asynchronous Schwarz solvers on GPUs

With the commencement of the exascale computing era, we realize that the majority of the leadership supercomputers are heterogeneous and massively parallel. Even a single node can contain multiple co-processors such as GPUs and multiple CPU cores. For example, ORNL’s Summit accumulates six NVIDIA Tesla V100 GPUs and 42 IBM Power9 cores on each node. Synchronizing across compute resources of multiple nodes can be prohibitively expensive. Hence, it is necessary to develop and study asynchronous algorithms that circumvent this issue of bulk-synchronous computing. In this study, we examine the asynchronous version of the abstract Restricted Additive Schwarz method as a solver. We do not explicitly synchronize, but allow the communication between the sub-domains to be completely asynchronous, thereby removing the bulk synchronous nature of the algorithm. We accomplish this by using the one-sided Remote Memory Access (RMA) functions of the MPI standard. We study the benefits of using such an asynchronous solver over its synchronous counterpart. We also study the communication patterns governed by the partitioning and the overlap between the sub-domains on the global solver. Finally, we show that this concept can render attractive performance benefits over the synchronous counterparts even for a well-balanced problem.

Parallel multilevel restricted Schwarz preconditioners for implicit simulation of subsurface flows with Peng-Robinson equation of state

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 investigation then focuses on the development of several types of multilevel overlapping additive Schwarz methods for the preconditioning of the resultant linear system arising from the inexact Newton iteration, and some fast solver technologies are presented for the assurance of the multilevel approach efficiency and scalability. We numerically show that the proposed fully implicit framework is highly efficient for solving both standard benchmarks as well as realistic problems with several hundreds of millions of unknowns and scalable to 8192 processors on the Tianhe-2 supercomputer.

Additive and restricted additive Schwarz–Richardson methods for inequalities with nonlinear monotone operators

The main aim of this paper is to analyze in a comparative way the convergence of some additive and additive Schwarz–Richardson methods for inequalities with nonlinear monotone operators. We first consider inequalities perturbed by a Lipschitz operator in the framework of a finite dimensional Hilbert space and prove that they have a unique solution if a certain condition is satisfied. For these inequalities, we introduce additive and restricted additive Schwarz methods as subspace correction algorithms and prove their convergence, under a certain convergence condition, and estimate the error. The convergence of the restricted additive methods does not depend on the number of the used subspaces and we prove that the convergence rate of the additive methods depends only on a reduced number of subspaces which corresponds to the minimum number of colors required to color the subdomains such that the subdomains having the same color do not intersect with each other, but not on the actual number of subdomains. The convergence condition of the algorithms is more restrictive than the existence and uniqueness condition of the solution. We then introduce new additive and restricted additive Schwarz algorithms that have a better convergence and whose convergence condition is identical to the condition of existence and uniqueness of the solution. The additive and restricted additive Schwarz–Richardson algorithms for inequalities with nonlinear monotone operators are obtained by taking the Lipschitz operator of a particular form and the convergence results are deducted from the previous ones. In the finite element space, the introduced algorithms are additive and restricted additive Schwarz–Richardson methods in the usual sense. Numerical experiments carried out for three problems confirm the theoretical predictions.

A Krylov-Subspace-Based Exponential Time Integration Scheme for Discontinuous Galerkin Time-Domain Methods

Discontinuous Galerkin time-domain (DGTD) methods using explicit time integration schemes often suffer from unreasonably small time-step size when applied to simulations involving electrically small structures. In this paper, an exponential time integration (ETI) scheme based on Krylov subspace is proposed for DGTD. The proposed scheme is unconditionally stable and exhibits good accuracy for large time-step sizes. When compared with classic ETI schemes, in the proposed scheme, the dense matrix exponentials are absorbed in matrix–vector products, hence the prohibitive memory costs are avoided. Moreover, a preconditioner combining the shift-and-invert technique and restricted additive Schwarz method is also developed to accelerate the proposed scheme.

New approach for accelerating the nonlinear Schwarz iterations

The vector Epsilon algorithm is an effective extrapolation method used for accelerating the convergence of vector sequences. In this paper, this method is used to accelerate the convergence of Schwarz iterative methods for stationary linear and nonlinear partial differential equations (PDEs). The vector Epsilon algorithm is applied to the vector sequences produced by additive Schwarz (AS) or restricted additive Schwarz (RAS) methods after discretization. Some convergence analysis is presented, and several test-cases of analytical problems are performed in order to illustrate the interest of such algorithm. The obtained results show that the proposed algorithm yields much faster convergence than the classical Schwarz iterations.

Parallel domain decomposition methods for a quantum-corrected drift–diffusion model for MOSFET devices

In this paper, we describe parallel domain decomposition methods based on the restricted additive Schwarz (RAS) method for a quantum-corrected drift–diffusion (QCDD) model for MOSFET devices. We have developed hybrid Message Passing Interface (MPI)/OpenMP parallelization algorithms of the QCDD system. For internode parallelization, two extensions of the RAS method are newly developed for the QCDD model. For intranode parallelization, we combine the conjugate gradient (CG) and BiCGSTAB procedures with a splitting-up operator method to realize parallelization of the incomplete factorization. The parallel numerical results for a three-dimensional Si bulk n-MOSFET on a multi-core NEC SX-ACE parallel computer are demonstrated. The intranode parallel numerical results are further evaluated on a many-core Cray XC40 parallel computer.

The convergence theory for the restricted version of the overlapping Schur complement preconditioner

The restricted version of the overlapping Schur complement (SchurRAS) preconditioner was introduced by Li and Saad (2006) for the solution of linear system Ax=b, and numerical results have shown that the SchurRAS method outperforms the restricted additive Schwarz (RAS) method both in terms of iteration count and CPU time. In this paper, based on meticulous derivation, we give an algebraic representation of the SchurRAS preconditioner, and prove that the SchurRAS method is convergent under the condition that A is an M-matrix and it converges faster than the RAS method.

Efficient CPR-type preconditioner and its adaptive strategies for large-scale parallel reservoir simulations

With the rapid development of high performance parallel computers and the increasing demands of large-scale reservoir simulation, the development of effective and fast solvers and preconditioners for parallel reservoir simulation has become increasingly important. The incomplete LU factorization (ILU) preconditioners are the most commonly used preconditioners in reservoir simulations due to their low computational cost. They have been implemented in commercial simulators as the default preconditioners. However, for reservoirs with highly heterogeneous permeability, the CPR (constrained pressure residual)-type preconditioners are more effective than the ILU preconditioners, especially when parallel computing is employed. The CPR-type preconditioners consist of multi-stages of preconditioning processes, which lead to huge computational costs. Employing the algebraic multigrid (AMG) method to solve a pressure system and the ILU method to solve the whole system has two indispensable stages. For parallel computation, due to the serial nature of the ILU method, the ILU method is usually combined with the restricted additive Schwarz (RAS) method and used as a subdomain solver. Different settings of the AMG and RAS methods, such as a grid coarsening strategy, interpolation and smoothing in the AMG method, and a subdomain solver and an overlap level in the RAS method, as well as different CPR-type preconditioners, yield significantly different performance. This paper first gives an overview of the CPR-type preconditioners and the AMG and RAS methods. Then a detailed comparison between different CPR-type preconditioners with different settings of the AMG and RAS methods is presented. Based on the comparison, the efficient settings of the AMG and RAS methods and the most efficient CPR-type preconditioner are given. Furthermore, an adaptive preconditioning strategy is designed to save the computational time. The adaptive preconditioning strategy introduces an indicator to measure the difficulty level of solving a linear system. According to difficulties, a RAS–ILU preconditioner or a CPR-type preconditioner is automatically and dynamically selected to achieve both efficiency and effectiveness.

A multi-continuum multi-phase parallel simulator for large-scale conventional and unconventional reservoirs

Reservoir simulation is considered as the core of data analysis during the development of an oilfield. With higher requirements from reservoir engineers, high-resolution geological model problems are commonly used in reservoir simulation. To simulate this kind of large-scale reservoir problems, based on black-oil models and compositional model, a parallel simulator is developed in this paper. Cartesian and corner point grids are supported by our simulator to describe complex geology, including faults and pinch-outs. The multi-continuum models (including a dual-porosity model, a dual-porosity dual-permeability model, and a multiple interacting continua model) are used to simulate fractured reservoirs. Geomechanics effects are successfully involved through a flexible and convenient interface provided by our simulator. Krylov subspace linear solvers, restricted additive Schwarz method, incomplete LU factorization (ILU) preconditioner, and a family of CPR (constrained pressure residual)-type preconditioners are implemented to provide flexibility of the linear system solution processes. Moreover, a new adaptive preconditioning strategy, which can automatically select and change preconditioner between a CPR-type preconditioner and an ILU preconditioner, is designed to accelerate the solution processes and is considered as the most efficient preconditioning technique to our knowledge. MPI-based parallel implementation is employed for each module of the simulator. With the simulator we developed, various conventional and unconventional reservoir simulation problems with large-scale geological models can be achieved on parallel computers within practical computational time. The computational efficiency and parallel scalability are achieved by our high-quality parallel implementation and efficient nonlinear and linear solution techniques.

An Additive Schwarz Method Type Theory for Lions's Algorithm and a Symmetrized Optimized Restricted Additive Schwarz Method

Optimized Schwarz methods (OSMs) are very popular methods which were introduced by P. L. Lions in [On the Schwarz alternating method III: A variant for nonoverlapping subdomains, in 3rd International Symposium on Domain Decomposition Methods for Partial Differential Equations (Houston, TX, 1989), T. F. Chan, R. Glowinski, J. Periaux, and O. Widlund, eds., SIAM, Philadelphia, 1990, pp. 202--223] for elliptic problems and by B. Despres in [C. R. Acad. Sci. Paris Ser. I Math., 311 (1990), pp. 313--316] for propagative wave phenomena. We give here a theory for Lions's algorithm that is the genuine counterpart of the theory developed over the years for the Schwarz algorithm. The first step is to introduce a symmetric variant of the optimized restricted additive Schwarz (ORAS) algorithm [A. St-Cyr, M. J. Gander, and S. J. Thomas, SIAM J. Sci. Comput., 29 (2007), pp. 2402--2425] that is suitable for the analysis of a two-level method. Then we build a coarse space for which the convergence rate of the two-level m...

Overlapping restricted additive Schwarz method with damping factor for H-matrix linear complementarity problem

In this paper, we consider an overlapping restricted additive Schwarz method (RAS) with damping factor for solving H+-matrix linear complementarity problem. Moreover, we estimate the weighted max-norm bounds for iteration errors and show that the sequence generated by the overlapping restricted additive Schwarz method (RAS) with damping factor converges to the unique solution of the problem without any restriction on the initial point. Finally, we establish monotone convergence of the proposed method under appropriate conditions.

Block preconditioners for linear systems arising from multiscale collocation with compactly supported RBFs

SummarySymmetric collocation methods with RBFs allow approximation of the solution of a partial differential equation, even if the right‐hand side is only known at scattered data points, without needing to generate a grid. However, the benefit of a guaranteed symmetric positive definite block system comes at a high computational cost. This cost can be alleviated somewhat by considering compactly supported RBFs and a multiscale technique. But the condition number and sparsity will still deteriorate with the number of data points. Therefore, we study certain block diagonal and triangular preconditioners. We investigate ideal preconditioners and determine the spectra of the preconditioned matrices before proposing more practical preconditioners based on a restricted additive Schwarz method with coarse grid correction. Numerical results verify the effectiveness of the preconditioners. Copyright © 2015 John Wiley & Sons, Ltd.

Restricted Additive Schwarz Method for a Kind of Nonlinear Complementarity Problem

In this paper, a new Schwarz method called restricted additive Schwarz method (RAS) is presented and analyzed for a kind of nonlinear complementarity problem (NCP). The method is proved to be convergent by using weighted maximum norm. Besides, the effect of overlap on RAS is also considered. Some preliminary numerical results are reported to compare the performance of RAS and other known methods for NCP.

A scalable Helmholtz solver in GRAPES over large‐scale multicore cluster

SUMMARYThis paper discusses performance optimization on the dynamical core of global numerical weather prediction model in Global/Regional Assimilation and Prediction System (GRAPES). GRAPES is a new generation of numerical weather prediction system developed and currently used by Chinese Meteorology Administration. The computational performance of the dynamical core in GRAPES relies on the efficient solution of three‐dimensional Helmholtz equations, which lead to large‐scale and sparse linear systems formulated by the discretization in space and time. We choose generalized conjugate residual (GCR) algorithm to solve the corresponding linear systems and further propose algorithm optimizations for large‐scale parallelism in two aspects: (i) reduction of iteration number for solution and (ii) performance enhancement of each GCR iteration. The reduction of iteration number is achieved by advanced preconditioning techniques, combining block incomplete LU factorization‐k preconditioner over 7‐diagonals of the coefficient matrix with the restricted additive Schwarz method effectively . The improvement for GCR iteration is to reduce the global communication operations by refactoring the GCR algorithm, which decreases the communication overhead over large number of cores. Performance evaluation on the Tianhe‐1A system shows that the new preconditioning techniques reduce almost one‐third iterations for solving the linear systems, the proposed methods can obtain 25% performance improvement on average compared with the original version of Helmholtz solver in GRAPES, and the speedup with our algorithms can reach 10 using 2048 cores compared with 256 cores. Copyright © 2013 John Wiley & Sons, Ltd.