Konrad Jopek

Dynamic Programming Algorithm for Generation of Optimal Elimination Trees for Multi-Frontal Direct Solver over h-Refined Grids

In this paper we present a dynamic programming algorithm for finding optimal elimination trees for computational grids refined towards point or edge singularities. The elimination tree is utilized to guide the multi-frontal direct solver algorithm. Thus, the criterion for the optimization of the elimination tree is the computational cost associated with the multi-frontal solver algorithm executed over such tree. We illustrate the paper with several examples of optimal trees found for grids with point, isotropic edge and anisotropic edge mixed with point singularity. We show the comparison of the execution time of the multi-frontal solver algorithm with results of MUMPS solver with METIS library, implementing the nested dissection algorithm.

Graph grammar based multi-thread multi-frontal direct solver with Galois scheduler

In this paper, we present a multi-frontal solver algorithm for the adaptive finite element method expressed by graph grammar productions. The graph grammar productions construct first the binary elimination tree, and then process frontal matrices stored in distributed manner in nodes of the elimination tree. The solver is specialized for a class of one, two and three dimensional h refined meshes whose elimination tree has a regular structure. In particular, this class contains all one dimensional grids, two and three dimensional grids refined towards point singularities, two dimensional grids refined in an anisotropic way towards edge singularity as well as three dimensional grids refined in an anisotropic way towards edge or face singularities. In all these cases, the structure of the elimination tree and the structure of the frontal matrices are similar. The solver is implemented within the Galois environment, which allows parallel execution of graph grammar productions. We also compare the performance of the Galois implementation of our graph grammar based solver with the MUMPS solver.

Quasi-Optimal elimination trees for 2D grids with singularities

We construct quasi-optimal elimination trees for 2D finite element meshes with singularities. These trees minimize the complexity of the solution of the discrete system. The computational cost estimates of the elimination process model the execution of the multifrontal algorithms in serial and in parallel shared-memory executions. Since the meshes considered are a subspace of all possible mesh partitions, we call these minimizers quasi-optimal. We minimize the cost functionals using dynamic programming. Finding these minimizers is more computationally expensive than solving the original algebraic system. Nevertheless, from the insights provided by the analysis of the dynamic programming minima, we propose a heuristic construction of the elimination trees that has cost O(Ne log(Ne)), where Ne is the number of elements in the mesh. We show that this heuristic ordering has similar computational cost to the quasi-optimal elimination trees found with dynamic programming and outperforms state-of-the-art alternatives in our numerical experiments.

Porting HPC applications to the cloud: A multi-frontal solver case study

Abstract In this paper we argue that scientific applications traditionally considered as representing typical HPC workloads can be successfully and efficiently ported to a cloud infrastructure. We propose a porting methodology that enables parallelization of communication – and memory-intensive applications while achieving a good communication to computation ratio and a satisfactory performance in a cloud infrastructure. This methodology comprises several aspects: (1) task agglomeration heuristic enabling increasing granularity of tasks while ensuring they will fit in memory; (2) task scheduling heuristic increasing data locality; and (3) two-level storage architecture enabling in-memory storage of intermediate data. We implement this methodology in a scientific workflow system and use it to parallelize a multi-frontal solver for finite-element meshes, deploy it in a cloud, and execute it as a workflow. The results obtained from the experiments confirm that the proposed porting methodology leads to a significant reduction of communication costs and achievement of a satisfactory performance. We believe that these results constitute a valuable step toward a wider adoption of cloud infrastructures for computational science applications.

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

Abstract 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.

Hybrid Direct and Iterative Solver with Library of Multi-criteria Optimal Orderings for h Adaptive Finite Element Method Computations

In this paper we present a multi-criteria optimization of element partition trees and resulting orderings for multi-frontal solver algorithms executed for two dimensional h adaptive finite element method. In particular, the problem of optimal ordering of elimination of rows in the sparse matrices resulting from adaptive finite element method computations is reduced to the problem of finding of optimal element partition trees. Given a two dimensional h refined mesh, we find all optimal element partition trees by using the dynamic programming approach. An element partition tree defines a prescribed order of elimination of degrees of freedom over the mesh. We utilize three different metrics to estimate the quality of the element partition tree. As the first criterion we consider the number of floating point operations(FLOPs) performed by the multi-frontal solver.As the second criterion we consider the number of memory transfers (MEMOPS) performed by the multi-frontal solver algorithm. As the third criterion we consider memory usage (NONZEROS) of the multi-frontal direct solver. We show the optimization results for FLOPs vs MEMOPS as well as for the execution time estimated as FLOPs+100*MEMOPS vs NONZEROS. We obtain Pareto fronts with multiple optimal trees, for each mesh, and for each refinement level. We generate a library of optimal elimination trees for small grids with local singularities. We also propose an algorithm that for a given large mesh with identified local sub-grids, each one with local singularity. We compute Schur complements over the sub-grids using the optimal trees from the library, and we submit the sequence of Schur complements into the iterative solver ILUPCG.

An Ontology-based Approach to Performance Monitoring of MUSCLE-bound Multi-scale Applications

Abstract In this paper we present an evolved and extended approach to the monitoring of data flow and resources’ usage in multi- scale applications built with the MUSCLE environment. Most multi-scale platforms provide the ability of running complex computations, but do not sufficiently support monitoring features if any. Combining the monitored multi-scale application with components that support gathering low-level monitoring data into high abstraction level performance information and visualizing it at runtime is the main objective of our approach. Performance monitoring data should be presented to allow the user to easily observe and interpret computation progress. Data access and processing based on ontologies enables easy reconfiguration of monitored resources and application parameters. Moreover, ontologies facilitate automatic reasoning on performance flaws.

Towards Green Multi-frontal Solver for Adaptive Finite Element Method☆

Abstract In this paper we present the optimization of the energy consumption for the multi-frontal solver algorithm executed over two dimensional grids with point singularities. The multi-frontal solver algorithm is controlled by so-called elimination tree, defining the order of elimination of rows from particular frontal matrices, as well as order of memory transfers for Schur complement matrices. For a given mesh there are many possible elimination trees resulting in different number of floating point operations (FLOPs) of the solver or different amount of data transferred via memory transfers. In this paper we utilize the dynamic programming optimization procedure and we compare elimination trees optimized with respect to FLOPs with elimination trees optimized with respect to energy consumption.

Telescopic Hybrid Fast Solver for 3D Elliptic Problems with Point Singularities

Abstract This paper describes a telescopic solver for two dimensional h adaptive grids with point singularities. The input for the telescopic solver is an h refined two dimensional computational mesh with rectangular finite elements. The candidates for point singularities are first localized over the mesh by using a greedy algorithm. Having the candidates for point singularities, we execute either a direct solver, that performs multiple refinements towards selected point singularities and executes a parallel direct solver algorithm which has logarithmic cost with respect to refinement level. The direct solvers executed over each candidate for point singularity return local Schur complement matrices that can be merged together and submitted to iterative solver. In this paper we utilize a parallel multi-thread GALOIS solver as a direct solver. We use Incomplete LU Preconditioned Conjugated Gradients (ILUPCG) as an iterative solver. We also show that elimination of point singularities from the refined mesh reduces significantly the number of iterations to be performed by the ILUPCG iterative solver.

Performance Monitoring and Analysis System for MUSCLE-based Applications in PL-Grid

In this paper we present a system for the monitoring of data flow and resources usage in applications running in the MUSCLE environment. While MUSCLE provides the ability of running complex experiments, it does not support any monitoring features. By combining the monitoring functionality supported by Sem Mon and Nagios, we are able to design and implement a system for gathering and visualizing important run-time data relating to application performance. Fluent experiment execution is highly dependable on real-time collecting and presenting essential information connected to task processing. Of particular importance for monitoring system users is that the use of the system should be as easy as possible with regard to storing, observing and interpreting the monitoring data. These features are enabled by introducing ontologies into the operation of the monitoring system. In addition to the conventional monitoring activities, using ontologies makes it possible to automate the process of reasoning on performance flaws and to easily change the focus of monitoring. In the paper we will focus on the concept and some implementation details of our monitoring system, assuming that an infrastructure to support the transport and storage of performance data on the usage of resources in MUSCLE-based applications should be transparent and lightweight.