Computational cost of isogeometric multi-frontal solvers on parallel distributed memory machines

Woźniak, Maciej; Paszyński, Maciej; Pardo, David; Dalcín, Lisandro; Calo, Victor M.

doi:10.1016/j.cma.2014.11.020

Cited by 17 publications

(10 citation statements)

References 32 publications

(41 reference statements)

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“…Moreover we plane to execute the whole system on a parallel distributed memory Linux cluster, using efficient isogeometric multifrontal direct solvers [14,11], as well as porting this algorithm to a GPU environment, using our fast GPU solver for linear algebraic systems [13].…”

Section: Discussionmentioning

confidence: 99%

A New Time Integration Scheme for Cahn-hilliard Equations

Schaefer¹,

Smoka²,

Dalcín³

et al. 2015

Procedia Computer Science

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In this paper we present a new integration scheme that can be applied to solving difficult non-stationary non-linear problems. It is obtained by a successive linearization of the CrankNicolson scheme, that is unconditionally stable, but requires solving non-linear equation at each time step. We applied our linearized scheme for the time integration of the challenging Cahn-Hilliard equation, modeling the phase separation in fluids. At each time step the resulting variational equation is solved using higher-order isogeometric finite element method, with Bspline basis functions. The method was implemented in the PETIGA framework interfaced via the PETSc toolkit. The GMRES iterative solver was utilized for the solution of a resulting linear system at every time step. We also apply a simple adaptivity rule, which increases the time step size when the number of GMRES iterations is lower than 30. We compared our method with a non-linear, two stage predictor-multicorrector scheme, utilizing a sophisticated step length adaptivity. We controlled the stability of our simulations by monitoring the Ginzburg-Landau free energy functional. The proposed integration scheme outperforms the two-stage competitor in terms of the execution time, at the same time having a similar evolution of the free energy functional.

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Section: Discussionmentioning

confidence: 99%

A New Time Integration Scheme for Cahn-hilliard Equations

Schaefer¹,

Smoka²,

Dalcín³

et al. 2015

Procedia Computer Science

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“…In the final part of the paper we conclude with performance measures of a shared-memory implementation of the algorithm (see also [10,12]) and useful remarks for the potential implementations of the solvers.…”

Section: Background and Motivationmentioning

confidence: 99%

Comparison of the Structure of Equation Systems and the GPU Multifrontal Solver for Finite Difference, Collocation and Finite Element Method

Lipski

Woźniak

Paszyński

2015

Procedia Computer Science

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The article is an in-depth comparison of numerical solvers and corresponding solution processes of the systems of algebraic equations resulting from finite difference, collocation, and finite element approximations. The paper considers recently developed isogeometric versions of the collocation and finite element methods, employing B-splines for the computations and ensuring C p−1 continuity on the borders of elements for the B-splines of the order p. For solving the systems, we use our GPU implementation of the state-of-the-art parallel multifrontal solver, which leverages modern GPU architectures and allows to reduce the complexity. We analyze the structures of linear equation systems resulting from each of the methods and how different matrix structures lead to different multifrontal solver elimination trees. The paper also considers the flows of multifrontal solver depending on the originally employed method. Background and motivationEssentially, each of the analyzed methods (finite difference method (FDM), collocation, finite element method (FEM)) yields a system of linear algebraic equations. The system is then passed to a solver -either sequential or parallel. The paper focuses on our GPU implementation of the state-of-the-art parallel multifrontal solver as described in [5,6,7,8,9], featuring logarithmic computational complexity with respect to the size of the matrix.The paper compares the structure of the matrices generated by 1D FDM, collocation method and FEM and the corresponding forward elimination trees built within the solver. We have not found such a holistic comparison in the existing literature. We believe such a comparison may be of benefit to the researchers who deal with direct solvers and isogeometric analysis. We firstly derive the system of linear algebraic equations and multifrontal solver trees for FDM and FEM with hierarchical basis functions [3] and then switch to isogeometric bases.

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“…We analyzed the parallel scalability of highly-continuous IGA systems in [35], where we showed that the direct solution of IGA systems has similar parallel scalability properties to standard C 0 FE systems. Although the performance of the direct solution of IGA systems is significantly worse than that of FE systems for a fixed number of degrees of freedom.…”

Section: Solution and Communication Cost Estimatesmentioning

confidence: 99%

Parallel Refined Isogeometric Analysis in 3D

Siwik

Woźniak

Trujillo

et al. 2019

IEEE Trans. Parallel Distrib. Syst.

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We study three-dimensional isogeometric analysis (IGA) and the solution of the resulting system of linear equations via a direct solver. IGA uses highly continuous C p−1 basis functions, which provide multiple benefits in terms of stability and convergence properties. However, smooth basis significantly deteriorate the direct solver performance and its parallel scalability. As a partial remedy for this, refined Isogeometric Analysis (rIGA) method improves the sequential execution of direct solvers. The refinement strategy enriches traditional highly-continuous C p−1 IGA spaces by introducing low-continuity C 0-hyperplanes along the boundaries of certain pre-defined macro-elements. In this work, propose a solution strategy for rIGA for parallel distributed memory machines and compare the computational costs of solving rIGA vs IGA discretizations. We verify our estimates with parallel numerical experiments. Results show that the weak parallel scalability of the direct solver improves approximately by a factor of p 2 when considering rIGA discretizations rather

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Computational cost of isogeometric multi-frontal solvers on parallel distributed memory machines

Cited by 17 publications

References 32 publications

A New Time Integration Scheme for Cahn-hilliard Equations

A New Time Integration Scheme for Cahn-hilliard Equations

Comparison of the Structure of Equation Systems and the GPU Multifrontal Solver for Finite Difference, Collocation and Finite Element Method

Parallel Refined Isogeometric Analysis in 3D

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