Dynamic Programming Algorithm for Generation of Optimal Elimination Trees for Multi-frontal Direct Solver Over H-refined Grids

AbouEisha, Hassan; Moshkov, Mikhail; Calo, Victor M.; Paszyński, Maciej; Goik, Damian; Jopek, Konrad

doi:10.1016/j.procs.2014.05.085

Cited by 17 publications

(6 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The method of how we order the tasks in a sequential manner over an arbitrary mesh is prescribed by the bisections weighted by an element-size algorithm [1]. It is worth mentioning that this ordering is possible to obtain with the hypergraph model but not with the CP-graph model due to the structure of the graphs.…”

Section: From the Comparison Presented Inmentioning

confidence: 99%

Hypergraph Grammar Based Multi-Thread Multi-Frontal Direct Solver With Galois Scheduler

Paszyński

Jopek²,

Paszyńska³

et al. 2019

csci

Self Cite

View full text Add to dashboard Cite

In this paper, we analyze two-dimensional grids with point and edge singularities in order to develop an efficient parallel hypergraph grammar-based multifrontal direct solver algorithm. We express these grids by a hypergraph. For these meshes, we define a sequence of hypergraph grammar productions expressing the construction of frontal matrices, eliminating fully assembled nodes, merging the resulting Schur complements, and repeating the process of elimination and merging until a single frontal matrix remains. The dependency relationship between hypergraph grammar productions is analyzed, and a dependency graph is plotted (which is equivalent to the elimination tree of a multifrontal solver algorithm). We utilize a classical multi-frontal solver algorithm; the hypergraph grammar productions allow us to construct an efficient elimination tree based on the graph representation of the computational mesh (not the global matrix itself). The hypergraph grammar productions are assigned to nodes on a dependency graph, and they are implemented as tasks in the GALOIS parallel environment and scheduled according to the developed dependency graph over the shared memory parallel machine. We show that our hypergraph grammar-based solver outperforms the parallel MUMPS solver.

show abstract

Section: From the Comparison Presented Inmentioning

confidence: 99%

Hypergraph Grammar Based Multi-Thread Multi-Frontal Direct Solver With Galois Scheduler

Paszyński

Jopek²,

Paszyńska³

et al. 2019

csci

Self Cite

View full text Add to dashboard Cite

show abstract

“…These results are an extension of the conference presentations (AbouEisha et al, 2015;AbouEisha et al, 2014), where we focused on one representative tree, while in this paper we look at the whole class of trees.…”

Section: 1mentioning

confidence: 99%

Element Partition Trees For H-Refined Meshes to Optimize Direct Solver Performance. Part I: Dynamic Programming

AbouEisha

Calo

Jopek

et al. 2017

International Journal of Applied Mathematics and Computer Science

Self Cite

View full text Add to dashboard Cite

We consider a class of two- and three-dimensional h-refined meshes generated by an adaptive finite element method. We introduce an element partition tree, which controls the execution of the multi-frontal solver algorithm over these refined grids. We propose and study algorithms with polynomial computational cost for the optimization of these element partition trees. The trees provide an ordering for the elimination of unknowns. The algorithms automatically optimize the element partition trees using extensions of dynamic programming. The construction of the trees by the dynamic programming approach is expensive. These generated trees cannot be used in practice, but rather utilized as a learning tool to propose fast heuristic algorithms. In this first part of our paper we focus on the dynamic programming approach, and draw a sketch of the heuristic algorithm. The second part will be devoted to a more detailed analysis of the heuristic algorithm extended for the case of hp-adaptive grids.

show abstract

“…Second, in addition to the convergence problems, iterative solvers may be slower than direct solvers when a problem with multiple right-hand-side needs to be solved, as it occurs in the case of gradient-based inverse methods in order to compute the Jacobian and Hessian matrices. Iterative solvers may also be slower than direct solvers when several matrices with a common set of rows and columns need to be solved, as it occurs in mesh-based methods when local grid-refinements are performed [30,21,1]. Moreover, direct solvers are a main building block of most iterative solvers.…”

Section: Introductionmentioning

confidence: 99%

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

Woźniak

Paszyński

Pardo

et al. 2015

Computer Methods in Applied Mechanics and Engineering

View full text Add to dashboard Cite

This paper derives theoretical estimates of the computational cost for isogeometric multi-frontal direct solver executed on parallel distributed memory machines. We show theoretically that for the C p−1 global continuity of the isogeometric solution, both the computational cost and the communication cost of a direct solver is of order O(log(N )p 2 ) for the one dimensional (1D) case, O(N p 2 ) for the two dimensional (2D) case, and O(N 4/3 p 2 ) for the three dimensional (3D) case, where N is the number of degrees of freedom and p is the polynomial order of the B-spline basis functions. The theoretical estimates are verified by numerical experiments performed with three parallel multi-frontal direct solvers: MUMPS, PaStiX and SuperLU, available through PETIGA toolkit built on top of PETSc. Numerical results confirm these theoretical estimates both in terms of p and N . For a given problem size, the strong efficiency rapidly decreases as the number of processors increases, becoming about 20 percent for 256 processors for a 3D example with Preprint submitted to Computer Methods in Applied Mechanics and EngineeringNovember 23, 2014 128 3 unknowns and linear B-splines with C 0 global continuity, and 15 percent for a 3D example with 64 3 unknowns and quartic B-splines with C 3 global continuity. At the same time, one cannot arbitrarily increase the problem size, since the memory required by higher order continuity spaces is large, quickly consuming all the available memory resources even in the parallel distributed memory version. Numerical results also suggest that the use of distributed parallel machines is highly beneficial when solving higher order continuity spaces, although the number of processors that one can efficiently employ is somehow limited.

show abstract

Dynamic Programming Algorithm for Generation of Optimal Elimination Trees for Multi-frontal Direct Solver Over H-refined Grids

Cited by 17 publications

References 6 publications

Hypergraph Grammar Based Multi-Thread Multi-Frontal Direct Solver With Galois Scheduler

Hypergraph Grammar Based Multi-Thread Multi-Frontal Direct Solver With Galois Scheduler

Element Partition Trees For H-Refined Meshes to Optimize Direct Solver Performance. Part I: Dynamic Programming

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

Contact Info

Product

Resources

About