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S.: Efficient finite element geometric multigrid solvers for unstructured grids on GPUs
 Second Int. Conf. on Parallel, Distributed, Grid and Cloud Computing for Engineering, p. 22 (2011). DOI 10.4203/ccp.95.22
"... Abstract. Fast, robust and efficient multigrid solvers are a key numerical tool in the solution of partial differential equations discretised with finite elements. The vast majority of practical simulation scenarios requires that the underlying grid is unstructured, and that highorder discretisatio ..."
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Abstract. Fast, robust and efficient multigrid solvers are a key numerical tool in the solution of partial differential equations discretised with finite elements. The vast majority of practical simulation scenarios requires that the underlying grid is unstructured, and that highorder discretisations are used. On the other hand, hardware is quickly evolving towards parallelism and heterogeneity, even within a single workstation. Commodity CPUs have multiple cores, and GPUs are the most prominent example of current finegrained parallel architectures. We are convinced that geometric multigrid methods are superior to algebraic multigrid methods, if their components are designed with respect to the underlying finite element discretisation. Such an approach, which we call finite element geometric multigrid (FEGMG), allows the design and development of numerically optimal solvers. While many multigrid components can be parallelised in a straight forward manner, two components pose severe challenges: Robust and strong smoothers are inherently
S.: Towards a complete FEMbased simulation toolkit on GPUs: Geometric multigrid solvers
 In: 23rd Int. Conf. on Parallel Computational Fluid Dynamics (ParCFD’11) (2011
"... Abstract: We describe a GPU and multicoreoriented implementation technique for a key component of finite element based simulation toolkits for partial differential equations on unstructured grids: Geometric Multigrid solvers. We use efficient sparse matrixvector multiplications throughout the sol ..."
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Cited by 4 (1 self)
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Abstract: We describe a GPU and multicoreoriented implementation technique for a key component of finite element based simulation toolkits for partial differential equations on unstructured grids: Geometric Multigrid solvers. We use efficient sparse matrixvector multiplications throughout the solver pipeline: within the coarsegrid solver, smoothers and even grid transfers. Our implementation can handle several low and highorder finite element spaces in 2D and 3D, and for representative benchmark problems, we achieve close to an order of magnitude speedup on a single GPU over a multithreaded CPU code. In addition we present preliminary results for experiments with strong smoothers for unstructured problems on the GPU, aiming at augmenting numerical and computational efficiency simultaneously.
Parallel Smoothers for Matrixbased Multigrid Methods on Unstructured Meshes Using Multicore CPUs and GPUs
, 2011
"... Multigrid methods are efficient and fast solvers for problems typically modeled by partial differential equations of elliptic type. For problems with complex geometries and local singularities stenciltype discrete operators on equidistant Cartesian grids need to be replaced by more flexible concept ..."
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Cited by 3 (0 self)
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Multigrid methods are efficient and fast solvers for problems typically modeled by partial differential equations of elliptic type. For problems with complex geometries and local singularities stenciltype discrete operators on equidistant Cartesian grids need to be replaced by more flexible concepts for unstructured meshes in order to properly resolve all probleminherent specifics and for maintaining a moderate number of unknowns. However, flexibility in the meshes goes along with severe drawbacks with respect to parallel execution – especially with respect to the definition of adequate smoothers. This point becomes in particular pronounced in the framework of finegrained parallelism on GPUs with hundreds of execution units. We use the approach of matrixbased multigrid that has high flexibility and adapts well to the exigences of modern computing platforms. In this work we investigate multicolored GaußSeidel type smoothers, the power(q)pattern enhanced multicolored ILU(p) smoothers with fillins,
PARALLEL UNSMOOTHED AGGREGATION ALGEBRAIC MULTIGRID ALGORITHMS ON GPUS
"... Abstract. We design and implement a parallel algebraic multigrid method for isotropic graph Laplacian problems on multicore Graphical Processing Units (GPUs). The proposed AMG method is based on the aggregation framework. The setup phase of the algorithm uses a parallel maximal independent set algor ..."
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Abstract. We design and implement a parallel algebraic multigrid method for isotropic graph Laplacian problems on multicore Graphical Processing Units (GPUs). The proposed AMG method is based on the aggregation framework. The setup phase of the algorithm uses a parallel maximal independent set algorithm in forming aggregates and the resulting coarse level hierarchy is then used in a Kcycle iteration solve phase with a `1Jacobi smoother. Numerical tests of a parallel implementation of the method for graphics processors are presented to demonstrate its effectiveness. 1.
Petascale elliptic solvers for anisotropic pdes on gpu clusters, CoRR abs/1402.3545
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VelocityPressure Coupling on GPU
, 2013
"... Abstract. We explore the possibilities to accelerate simulations in computational fluid dynamics (CFD) by additional graphics processing units (GPUs). By examining some examples of stationary incompressible flows from the industrial practice we demonstrate that the potential speedup obtained by depl ..."
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Abstract. We explore the possibilities to accelerate simulations in computational fluid dynamics (CFD) by additional graphics processing units (GPUs). By examining some examples of stationary incompressible flows from the industrial practice we demonstrate that the potential speedup obtained by deploying GPU accelerated linear solvers alone is limited if standard segregated algorithms are used. However, recently presented velocitypressure coupling algorithms are an attractive alternative to these segregated algorithms. We present an efficient AMG solver for the coupled linear system of mixed elliptichyperbolic character and show that the GPUaccelerated version of this linear solver accelerates the velocitypressure coupling scheme by almost 50 % compared to a competitive CPU implementation. Compared to standard segregated methods, the computing time of the whole simulation is reduced by up to 75%. 1.
Towards a complete FEMbased simulation toolkit on GPUs: Unstructured Grid Finite Element Geometric Multigrid solvers with strong smoothers based on Sparse Approximate Inverses
"... Abstract: We describe our FEgMG solver, a finite element geometric multigrid approach for problems relying on unstructured grids. We augment our GPU and multicoreoriented implementation technique based on cascades of sparse matrixvector multiplication by applying strong smoothers. In particula ..."
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Abstract: We describe our FEgMG solver, a finite element geometric multigrid approach for problems relying on unstructured grids. We augment our GPU and multicoreoriented implementation technique based on cascades of sparse matrixvector multiplication by applying strong smoothers. In particular, we employ Sparse Approximate Inverse (SPAI) and Stabilised Approximate Inverse (SAINV) techniques. We focus on presenting the numerical efficiency of our smoothers in combination with low and highorder finite element spaces as well as the hardware efficiency of the FEgMG. For a representative problem and computational grids in 2D and 3D, we achieve a speedup of an average of 5 on a single GPU over a multithreaded CPU code in our benchmarks. In addition, our strong smoothers can deliver a speedup of 3.5 depending on the element space, compared to simple Jacobi smoothing. This can even be enhanced to a factor of 7 when combining the usage of Approximate Inversebased smoothers with clever sorting of the degrees of freedom. In total the FEgMG solver can outperform a simple, (multicore)CPUbased multigrid by a total factor of over 40.
ACCELERATING PRECONDITIONED ITERATIVE LINEAR SOLVERS ON GPU
"... Abstract. Linear systems are required to solve in many scientific applications and the solution of these systems often dominates the total running time. In this paper, we introduce our work on developing parallel linear solvers and preconditioners for solving large sparse linear systems using NVIDIA ..."
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Abstract. Linear systems are required to solve in many scientific applications and the solution of these systems often dominates the total running time. In this paper, we introduce our work on developing parallel linear solvers and preconditioners for solving large sparse linear systems using NVIDIA GPUs. We develop a new sparse matrixvector multiplication kernel and a sparse BLAS library for GPUs. Based on the BLAS library, several Krylov subspace linear solvers, and algebraic multigrid (AMG) solvers and commonly used preconditioners are developed, including GMRES, CG, BICGSTAB, ORTHOMIN, classical AMG solver, polynomial preconditioner, ILU(k) and ILUT preconditioner, and domain decomposition preconditioner. Numerical experiments show that these linear solvers and preconditioners are efficient for solving the large linear systems. Key words. Krylov subspace solver, algebraic multigrid solver, parallel preconditioner, GPU computing, sparse matrixvector multiplication, HEC
Karl{Franzens Universitat Graz Technische
"... DouglasRachford splitting methods for saddlepoint problems with applications in image denoising and deblurring ..."
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DouglasRachford splitting methods for saddlepoint problems with applications in image denoising and deblurring
Unstructured Forests of Octrees
"... Abstract—We present a parallel multigrid method for solving variablecoefficient elliptic partial differential equations on arbitrary geometries using highly adapted meshes. Our method is designed for meshes that are built from an unstructured hexahedral macro mesh, in which each macro element is ..."
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Abstract—We present a parallel multigrid method for solving variablecoefficient elliptic partial differential equations on arbitrary geometries using highly adapted meshes. Our method is designed for meshes that are built from an unstructured hexahedral macro mesh, in which each macro element is adaptively refined as an octree. This forestofoctrees approach enables us to generate meshes for complex geometries with arbitrary levels of local refinement. We use geometric multigrid (GMG) for each of the octrees and algebraic multigrid (AMG) as the coarse grid solver. We designed our GMG sweeps to entirely avoid collectives, thus minimizing communication cost. We present weak and strong scaling results for the 3D variablecoefficient Poisson problem that demonstrate high parallel scalability. As a highlight, the largest problem we solve is on a nonuniform mesh with 100 billion unknowns on 262,144 cores of NCCS’s Cray XK6 “Jaguar”; in this solve we sustain 272 TFlops/s. I.