Relevant bibliographies by topics / Parallel Multigrid Smoothers

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  2. Dissertations / Theses
  3. Book chapters
  4. Conference papers
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Academic literature on the topic 'Parallel Multigrid Smoothers'

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Published: 6 September 2023

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Journal articles on the topic "Parallel Multigrid Smoothers"

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CIEGIS, R., F. GASPAR, and C. RODRIGO. "On The Parallel Multiblock Geometric Multigrid Algorithm." Computational Methods in Applied Mathematics 8, no. 3 (2008): 223–36. http://dx.doi.org/10.2478/cmam-2008-0016.

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Abstract The application of a parallel multiblock geometric multigrid is consid-ered. It is applied to solve a two-dimensional poroelastic model. This system of PDEs is approximated by a special stabilized monotone finite-difference scheme. The obtained system of linear algebraic equations is solved by a multigrid method, when a domain is partitioned into structured blocks. A new strategy for the solution of the discrete problem on the coarsest grid is proposed and the efficiency of the obtained algorithm is investigated. The geometrical structure of the sequential multigrid method is used to develop a parallel version of the multigrid algorithm. The convergence properties of several smoothers are investigated and some computational results are presented.
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Xie, Dexuan. "Analysis of a Class of Parallel Multigrid Smoothers." BIT Numerical Mathematics 44, no. 4 (December 2004): 813–28. http://dx.doi.org/10.1007/s10543-004-3830-y.

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Yang, Xiang, and Rajat Mittal. "Efficient relaxed-Jacobi smoothers for multigrid on parallel computers." Journal of Computational Physics 332 (March 2017): 135–42. http://dx.doi.org/10.1016/j.jcp.2016.12.010.

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Axelsson, O. "Analysis of incomplete matrix factorizations as multigrid smoothers for vector and parallel computers." Applied Mathematics and Computation 19, no. 1-4 (July 1986): 3–22. http://dx.doi.org/10.1016/0096-3003(86)90094-9.

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John, Volker, and Lutz Tobiska. "Numerical performance of smoothers in coupled multigrid methods for the parallel solution of the incompressible Navier-Stokes equations." International Journal for Numerical Methods in Fluids 33, no. 4 (2000): 453–73. http://dx.doi.org/10.1002/1097-0363(20000630)33:4<453::aid-fld15>3.0.co;2-0.

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Gaidamour, Jérémie, Jonathan Hu, Chris Siefert, and Ray Tuminaro. "Design Considerations for a Flexible Multigrid Preconditioning Library." Scientific Programming 20, no. 3 (2012): 223–39. http://dx.doi.org/10.1155/2012/310508.

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MueLu is a library within the Trilinos software project [An overview of Trilinos, Technical Report SAND2003-2927, Sandia National Laboratories, 2003] and provides a framework for parallel multigrid preconditioning methods for large sparse linear systems. While providing efficient implementations of modern multigrid methods based on smoothed aggregation and energy minimization concepts, MueLu is designed to be customized and extended. This article gives an overview of design considerations for the MueLu package: user interfaces, internal design, data management, usage of modern software constructs, leveraging Trilinos capabilities, linear algebra operations and advanced application.
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Kashi, Aditya, Syam Vangara, Siva Nadarajah, and Patrice Castonguay. "Asynchronous fine-grain parallel implicit smoother in multigrid solvers for compressible flow." Computers & Fluids 198 (February 2020): 104255. http://dx.doi.org/10.1016/j.compfluid.2019.104255.

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Dünnebacke, Jonas, Stefan Turek, Christoph Lohmann, Andriy Sokolov, and Peter Zajac. "Increased space-parallelism via time-simultaneous Newton-multigrid methods for nonstationary nonlinear PDE problems." International Journal of High Performance Computing Applications 35, no. 3 (April 1, 2021): 211–25. http://dx.doi.org/10.1177/10943420211001940.

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We discuss how “parallel-in-space & simultaneous-in-time” Newton-multigrid approaches can be designed which improve the scaling behavior of the spatial parallelism by reducing the latency costs. The idea is to solve many time steps at once and therefore solving fewer but larger systems. These large systems are reordered and interpreted as a space-only problem leading to multigrid algorithm with semi-coarsening in space and line smoothing in time direction. The smoother is further improved by embedding it as a preconditioner in a Krylov subspace method. As a prototypical application, we concentrate on scalar partial differential equations (PDEs) with up to many thousands of time steps which are discretized in time, resp., space by finite difference, resp., finite element methods. For linear PDEs, the resulting method is closely related to multigrid waveform relaxation and its theoretical framework. In our parabolic test problems the numerical behavior of this multigrid approach is robust w.r.t. the spatial and temporal grid size and the number of simultaneously treated time steps. Moreover, we illustrate how corresponding time-simultaneous fixed-point and Newton-type solvers can be derived for nonlinear nonstationary problems that require the described solution of linearized problems in each outer nonlinear step. As the main result, we are able to generate much larger problem sizes to be treated by a large number of cores so that the combination of the robustly scaling multigrid solvers together with a larger degree of parallelism allows a faster solution procedure for nonstationary problems.
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Guo, Rongwen, Yongfei Wang, Gary D. Egbert, Jianxin Liu, Rong Liu, Kejia Pan, Jian Li, and Hang Chen. "An efficient multigrid solver based on a four-color cell-block Gauss-Seidel smoother for 3D magnetotelluric forward modeling." GEOPHYSICS 87, no. 3 (March 2, 2022): E121—E133. http://dx.doi.org/10.1190/geo2021-0275.1.

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Practical application of 3D magnetotelluric inversion requires efficient forward modeling of electromagnetic (EM) fields in the earth. To resolve realistic 3D structures, large computational domains and extremely large linear systems of equations are required. The iterative solvers, which are almost exclusively used to solve these systems, can be inefficient due to the abundant null space of the curl-curl operator. Multigrid (MG) solvers are considered a potentially efficient technique for solving such problems. However, due to the abundant null solution space and existence of the air layer, MG solvers can still converge slowly or even diverge. We have developed an efficient MG solver for finite-difference frequency-domain EM solution. In this algorithm, the excellent smoothing property of an efficient four-color cell-block Gauss-Seidel (GS) is exploited to remove the short-range errors effectively, and the interpolation and prolongation operators are used to handle the long-range errors. They work as a whole to speed the convergence of our algorithm remarkably. Because all of the nodes for the four-color cell-block GS are grouped into four colors and the edge components attached to different nodes in each color are completely decoupled, this can be used to develop a highly vectorized or parallelized algorithm. Another important property is that our algorithm is locally current divergence free, effectively eliminating spurious solutions in the null space of the curl-curl operator. The accuracy and efficiency of the algorithm are verified by comparing the numerical solutions obtained with our MG solver to those from the biconjugate gradient stabilized solver with different preconditioners based on synthetic models and a model from 3D inversion. Comparisons, in terms of iteration number and computational time, indicate that our algorithm is extremely stable and efficient relative to the other solvers. Our MG algorithm will be suitable for massively parallel computing as well.
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Munch, Peter, Timo Heister, Laura Prieto Saavedra, and Martin Kronbichler. "Efficient distributed matrix-free multigrid methods on locally refined meshes for FEM computations." ACM Transactions on Parallel Computing, January 24, 2023. http://dx.doi.org/10.1145/3580314.

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This work studies three multigrid variants for matrix-free finite-element computations on locally refined meshes: geometric local smoothing, geometric global coarsening (both h -multigrid), and polynomial global coarsening (a variant of p -multigrid). We have integrated the algorithms into the same framework—the open-source finite-element library deal.II —, which allows us to make fair comparisons regarding their implementation complexity, computational efficiency, and parallel scalability as well as to compare the measurements with theoretically derived performance metrics. Serial simulations and parallel weak and strong scaling on up to 147,456 CPU cores on 3,072 compute nodes are presented. The results obtained indicate that global-coarsening algorithms show a better parallel behavior for comparable smoothers due to the better load balance, particularly on the expensive fine levels. In the serial case, the costs of applying hanging-node constraints might be significant, leading to advantages of local smoothing, even though the number of solver iterations needed is slightly higher. When using p - and h -multigrid in sequence ( hp -multigrid), the results indicate that it makes sense to decrease the degree of the elements first from a performance point of view due to the cheaper transfer.
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Dissertations / Theses on the topic "Parallel Multigrid Smoothers"

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Iyer, Neha Mohan. "Parallel Smoothers for Multigrid Method in Heterogeneous CPU-GPU Environment." Thesis, 2020. https://etd.iisc.ac.in/handle/2005/4423.

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Real-world applications require the solution of large a sparse system of algebraic equations that arise from the discretization of partial di erential equations with the help of supercomputers. Modern supercomputers are heterogeneous - each node composed of multi-core CPUs and many-core GPUs. Porting existing sequential applications speci cally to the GPU architecture has lead to poor utilization of CPU computing power. In this respect, we develop hybrid parallel smoothers for the geometric multigrid method which is one of the most e cient solvers for a system of equations. We study the performance of multigrid method in terms of total execution time by employing di erent hybrid parallel approaches, viz. accelerating the smoothing operation using only GPU across all multigrid levels, alternately switching between GPU and CPU based on the multigrid level and our proposed novel approach of using combination of GPU and CPU across all multigrid levels. The performance of the hybrid parallel approaches, implemented using MPI, CUDA and OpenMP, is compared against the MPI only approach. In the rst part of the work, we have implemented the hybrid parallel approaches for the Jacobi and Gauss-Seidel smoothers and tested it to solve the system arising from the discretization of the Poisson equation. A coloring strategy is developed to color the degrees of freedom (DOFs) such that the independent set of DOFs are assigned the same color and are updated in parallel on GPU. We adopted two of the commonly used techniques, CSR Scalar and CSR Vector, that perform sparse matrix-vector multiplication on GPU, to implement the smoothing iterations. Further, the strong scaling behavior of the hybrid parallel smoothers is studied across di erent problem sizes, nite element types, standard and multilevel multigrid. In the second part of the work, we have implemented the hybrid parallel approaches for Vanka-type smoother which are typically used to solve the saddle-point problem arising from the Navier-Stokes equation. We studied the time taken by the two operations viz. assembling and solving of the local systems in cell and nodal Vanka. We have accelerated the operation of assembling the local system using the hybrid parallel approaches. A similar coloring strategy is developed to assign the same color to independent cells or pressure DOFs. The task of determining the neighbors of each cell or DOF is o oaded to GPU as it is an O(N2) operation. The operation of solving the assembled local systems is parallelized using OpenMP on CPU. Similar to the rst part, experiments are performed to study the strong scaling results across di erent problem sizes, number of OpenMP threads, standard and multilevel multigrid. The experimental results for the di erent smoothers show that the scaling performance of the hybrid parallel approaches is bounded by the degree of achievable thread parallelism which in turn is dependent on the parallel workload per process and the algorithm itself. To improve the scaling behavior, we propose a combination approach that uses a workload heuristic to decide the best approach to be applied at each level of multigrid. The combination approach improves the scaling behavior in addition to resulting in a signi cant speedup under appropriate workload and number of MPI processes compared to the MPI only approach.
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Book chapters on the topic "Parallel Multigrid Smoothers"

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Bolten, Matthias, and Oliver Letterer. "Increasing Arithmetic Intensity in Multigrid Methods on GPUs Using Block Smoothers." In Parallel Processing and Applied Mathematics, 515–25. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-32149-3_48.

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Douglas, Craig C., Sachit Malhotra, and Martin H. Schultz. "“Transpose Free” Alternating Direction Smoothers for Serial and Parallel Multigrid Methods." In Advances in Computational Mathematics, 39–52. Boca Raton: CRC Press, 2023. http://dx.doi.org/10.1201/9781003419839-3.

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Krzyżanowski, Piotr. "A Class of Block Smoothers for Multigrid Solution of Saddle Point Problems with Application to Fluid Flow." In Parallel Processing and Applied Mathematics, 1006–13. Berlin, Heidelberg: Springer Berlin Heidelberg, 2004. http://dx.doi.org/10.1007/978-3-540-24669-5_130.

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Reiter, Sebastian, Andreas Vogel, Arne Nägel, and Gabriel Wittum. "A Massively Parallel Multigrid Method with Level Dependent Smoothers for Problems with High Anisotropies." In High Performance Computing in Science and Engineering ´16, 667–75. Cham: Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-47066-5_45.

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Heuveline, Vincent, Dimitar Lukarski, Nico Trost, and Jan-Philipp Weiss. "Parallel Smoothers for Matrix-Based Geometric Multigrid Methods on Locally Refined Meshes Using Multicore CPUs and GPUs." In Facing the Multicore - Challenge II, 158–71. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012. http://dx.doi.org/10.1007/978-3-642-30397-5_14.

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Kawai, Masatoshi, Takeshi Iwashita, Hiroshi Nakashima, and Osni Marques. "Parallel Smoother Based on Block Red-Black Ordering for Multigrid Poisson Solver." In Lecture Notes in Computer Science, 292–99. Berlin, Heidelberg: Springer Berlin Heidelberg, 2013. http://dx.doi.org/10.1007/978-3-642-38718-0_29.

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Iyer, Neha, and Sashikumaar Ganesan. "Parallel Smoothers in Multigrid Method for Heterogeneous CPU-GPU Environment." In Parallel Computing: Technology Trends. IOS Press, 2020. http://dx.doi.org/10.3233/apc200031.

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Modern-day supercomputers are equipped with sophisticated graphics processing units (GPUs) along with high-performance CPUs. Adapting existing algorithms specifically to GPU has resulted in under-utilization of CPU computing power. In this respect, we parallelize Jacobi and successive-over relaxation (SOR), which are used as smoother in multigrid method to maximize the combined utilization of both CPUs and GPUs. We study the performance of multigrid method in terms of total execution time by employing different hybrid parallel approaches, viz. accelerating the smoothing operation using only GPU across all multigrid levels, alternately switching between GPU and CPU based on the multigrid level and our proposed novel approach of using combination of GPU and CPU across all multigrid levels. Our experiments demonstrate a significant speedup using the hybrid parallel approaches, across different problem sizes and finite element types, as compared to the MPI only approach. However, the scalability challenge persists for the hybrid parallel multigrid smoothers.
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MLLORENTE, I., B. DISKIN, and N. DUANEMELSON. "Parallel multigrid solvers with block-wise smoothers for multiblock grids." In Parallel Computational Fluid Dynamics 1999, 297–304. Elsevier, 2000. http://dx.doi.org/10.1016/b978-044482851-4/50037-2.

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Llorente, Ignacio M., Boris Diskin, and N. Duane Melson. "Parallel multigrid solvers with block-wise smoothers for multiblock grids * *This research was supported by the National Aeronautics and Space Administration under NASA Contract No. NAS1-19480 while the first authors were in residence at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23681-2199." In Parallel Computational Fluid Dynamics 1999, 297–304. Elsevier, 2000. http://dx.doi.org/10.1016/b978-044482851-4.50037-2.

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Conference papers on the topic "Parallel Multigrid Smoothers"

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Douglas, Craig C., and Gundolf Haase. "Parallel ADI Smoothers for Multigrid." In 2013 12th International Symposium on Distributed Computing and Applications to Business, Engineering & Science (DCABES). IEEE, 2013. http://dx.doi.org/10.1109/dcabes.2013.25.

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Tuminaro, R. S., and C. Tong. "Parallel Smoothed Aggregation Multigrid : Aggregation Strategies on Massively Parallel Machines." In ACM/IEEE SC 2000 Conference. IEEE, 2000. http://dx.doi.org/10.1109/sc.2000.10008.

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Mehraban, Arash, Jed Brown, Henry Tufo, Jeremy Thompson, Rezgar Shakeri, and Richard Regueiro. "Efficient Parallel Scalable Matrix-Free 3D High-Order Finite Element Simulation of Neo-Hookean Compressible Hyperelasticity at Finite Strain." In ASME 2021 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2021. http://dx.doi.org/10.1115/imece2021-70768.

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Abstract The paper investigates matrix-free high-order implementation of finite element discretization with p-multigrid preconditioning for the compressible Neo-Hookean hyperelasticity problem at finite strain on unstructured 3D meshes in parallel. We consider two formulations for the matrix-free action of the Jacobian in Neo-Hookean hyperelasticity: (i) working in the reference configuration to define the second Piola-Kirchhoff tensor as a function of the Green-Lagrange strain S(E) (or equivalently, the right Cauchy-Green tensor C = I+2E), and (ii) working in the current configuration to define the Kirchhoff stress in terms of the left Cauchy-Green tensor τ(b). The proposed efficient algorithm utilizes the Portable, Extensible Toolkit for Scientific Computation (PETSc), along with the libCEED library for efficient compiler optimized tensor-product-basis computation to demonstrate an efficient nonlinear solution algorithm. We utilize p-multigrid preconditioning on the high-order problem with algebraic multigrid (AMG) on the assembled linear Q1 coarse grid operator. In contrast to classical geometric multigrid, also known as h-multigrid, each level in p-multigrid is related to a different approximation polynomial order p, instead of the element size h. A Chebyshev polynomial smoother is used on each multigrid level. AMG is then applied to the assembled Q1 (trilinear hexahedral elements), which allows low storage that can be efficiently used to accelerate convergence to a solution. For the compressible Neo-Hookean hyperelastic constitutive model we exploit the stored energy density function to compute the stored elastic energy density of the Neo-Hookean material as it relates to the deformation gradient. Based on our formulation, we consider four different algorithms each with different storage strategies. Algorithms 1 and 3 are implemented in the reference and current configurations respectively and store ∇Xξ, det(∇ξX), and ∇Xu. Algorithm 2 in the reference configuration stores, ∇Xξ, det(∇ξX), ∇Xu, C−1, and λ log (J). Algorithm 4, in the current configuration, stores det(∇ξX), ∇xξ, τ, and μ – λ log(J). x refers to the current coordinates, X to the reference coordinates, and ξ to the natural coordinates. We perform 3D bending simulations of a tube composed of aluminum (modulus of elasticity E = 69 GPa, Poisson’s ratio v = 0.3) using unstructured meshes and polynomials of order p = 1 through p = 4 under mesh refinement. We explore accuracy-time-cost tradeoffs for the prediction of strain energy across the range of polynomial degrees and Jacobian representations. In all cases, Algorithm 4 using the current configuration formulation outperforms the other three algorithms and requires less storage. Similar simulations for large deformation compressible Neo-Hookean hyperelasticity as applied to the same aluminum material are conducted with ABAQUS, a commercial finite element software package which is a state-of-the-art engineering software package for finite element simulations involving large deformation. The best results from the proposed implementations and ABAQUS are compared in the case of p = 2 on an Intel system with @2.4 GHz and 128 GB RAM. Algorithm 4 outperforms ABAQUS for polynomial degree p = 2.
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Mehraban, Arash, Jed Brown, Valeria Barra, Henry Tufo, Jeremy Thompson, and Richard Regueiro. "Efficient Residual and Matrix-Free Jacobian Evaluation for Three-Dimensional Tri-Quadratic Hexahedral Finite Elements With Nearly-Incompressible Neo-Hookean Hyperelasticity Applied to Soft Materials on Unstructured Meshes in Parallel, With PETSc and libCEED." In ASME 2020 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2020. http://dx.doi.org/10.1115/imece2020-24522.

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Abstract Soft materials such as rubber, elastomers, and soft biological tissues mechanically deform at large strain isochorically for all time, or during their initial transient (when a pore fluid, typically incompressible such as water, does not have time to flow out of the deforming polymer or soft tissue porous skeleton). Simulating these large isochoric deformations computationally, such as with the Finite Element Method (FEM), requires higher order (typically quadratic) interpolation functions and/or enhancements through hybrid/mixed methods to maintain stability. Lower order (linear) finite elements with hybrid/mixed formulation may not perform stably for all mechanical loading scenarios involving large isochoric deformations, whereas quadratic finite elements with or without hybrid/mixed formulation typically perform stably, especially when large bending or folding deformations are being simulated. For topology-optimization design of soft robotics, for instance, the FEM solid mechanics solver must run efficiently and stably. Stability is ensured by the higher order finite element formulation (with possible enhancement), but efficiency for higher order FEM remains a challenge. Thus, this paper addresses efficiency from the perspective of computer science algorithms and programming. The proposed efficient algorithm utilizes the Portable, Extensible Toolkit for Scientific Computation (PETSc), along with the libCEED library for efficient compiler optimized tensor-product-basis computation to demonstrate an efficient nonlinear solution algorithm. For preconditioning, a scalable p-multigrid method is presented whereby a hierarchy of levels is constructed. In contrast to classical geometric multigrid, also known as h-multigrid, each level in p-multigrid is related to a different approximation polynomial order, p, instead of the element size, h. A Chebyshev polynomial smoother is used on each multigrid level. Algebraic MultiGrid (AMG) is then applied to the assembled Q1 (linear) coarse mesh on the nodes of the quadratic Q2 (quadratic) mesh. This allows low storage that can be efficiently used to accelerate the convergence to solution. For a Neo-Hookean hyperelastic problem, we examine a residual and matrix-free Jacobian formulation of a tri-quadratic hexahedral finite element with enhancement. Efficiency estimates on AVX-2 architecture based on CPU time are provided as a comparison to similar simulation (and mesh) of isochoric large deformation hyperelasticity as applied to soft materials conducted with the commercially-available FEM software program ABAQUS. The particular problem in consideration is the simulation of an assistive device in the form of finger-bending in 3D.
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Nilsen, H. M., E. Ahmed, A. F. Rasmussen, K. Bao, and T. Skille. "Constrained Pressure Residual Preconditioner Including Wells for Reservoir Simulation." In SPE Reservoir Simulation Conference. SPE, 2023. http://dx.doi.org/10.2118/212172-ms.

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Abstract We present a new practical constrained pressure residual (CPR) preconditioner including well degrees of freedom (DOFs). The action of the new CPR preconditioner applies only to the reservoir DOFs of the linear system, and includes well-reservoir coupling by solving an extended linear system for the pressure. This extended pressure system is similar to the one proposed in Zhou et al. (Comp. geosci 17(2), 2013). The preconditioner is suitable for a linear system which only includes reservoir DOFs as unknowns and where the effect of the wells is included by Schur complement in the linear operator, without explicit fill-in in the matrix. The main feature is that the pressure system is extended to include well DOFs. The full preconditioner then combines block ILU0 on the reservoir matrix as fine smoother with the new extended pressure CPR system, using standard AMG cycles on the latter. The new preconditioner has been implemented in the open-source reservoir simulator OPM Flow. The approach is compared with several different CPR approaches on conceptual and real-field cases including open benchmark cases, looking at accuracy, tolerances, performance and parallel scalability. Compared to applying CPR to the reservoir matrix without well fill-in, the new method yields lower linear iteration counts, similar to those that result from applying CPR to the reservoir matrix with well fill-in (explicit Schur complement). However, each iteration is less costly since one avoids the fill-in, which adds k2 extra nonzero matrix elements for a well with k perforations. An advantage of the approach is that the structural complexity introduced with the well system is included only in the scalar CPR pressure system in a way suitable for algebraic multigrid (AMG) preconditioning. All other complexity of the wells is handled in the linear operators used in the Krylov solvers. The new method can be implemented in reservoir simulators by building on existing preconditioner components, and can improve simulation times for complex cases, in particular those with many wells and perforations.
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Reports on the topic "Parallel Multigrid Smoothers"

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Baker, A. H., R. D. Falgout, T. V. Kolev, and U. M. Yang. Multigrid Smoothers for Ultra-Parallel Computing: Additional Theory and Discussion. Office of Scientific and Technical Information (OSTI), June 2011. http://dx.doi.org/10.2172/1122232.

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