Piotr Gurgul

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.

Application of multi-agent paradigm to hp-adaptive projection-based interpolation operator

Abstract In this paper we discuss applications and design of the agent-based, hp-adaptive projection-based interpolation (PBI) operator. We describe the use of mesh adaptation process to produce a faithful representation of an input image in the Finite Element (FE) space. This can be used, in turn, to generate from the input bitmap continuous material functions required for further FE computations. We propose an agent-based architecture suitable for localized implementation of the PBI operator. Finally, we show how to apply it to an exemplary problem of austenite–ferrite phase transformation.

A Linear Complexity Direct Solver for H-adaptive Grids with Point Singularities☆

Abstract In this paper we present a theoretical proof of linear computational cost and complexity for a recently developed direct solver driven by hypergraph grammar productions. The solver is specialized for computational meshes with point singularities in two and three dimensions. Linear complexity is achieved due to utilizing the special structure of such grids. We describe the algorithm and estimate the exact computational cost on an example of a two-dimensional mesh containing a single point singularity. We extend this reasoning to the three dimensional meshes.

Two-dimensional HP-adaptive Algorithm for Continuous Approximations of Material Data Using Space Projection

In this paper we utilize the concept of the L 2 and H 1 projections used to adaptively generate a continuous approximation of an input material data in the finite element (FE) base. This approximation, along with a corresponding FE mesh, can be used as material data for FE solvers. We begin with a brief theoretical background, followed by description of the hp-adaptive algorithm adopted here to improve gradually quality of the projections. We investigate also a few distinct sample problems, apply the aforementioned algorithms and conclude with numerical results evaluation.

Graph Grammar Based Direct Solver for hp-adaptive Finite Element Method with Point Singularities

Abstract In this paper we present a graph grammar based direct solver algorithm for hp-adaptive finite element method simulations with point singularities. The solver algorithm is obtained by representing computational mesh as a graph and prescribing the solver algorithm by graph grammar productions. Classical direct solvers deliver O(Np 4 +N 1.5 ) computational cost for regular 2D grids, and O(Np6+N2) for regular 3D grids, where N denotes number of degrees of freedom and p denotes the polynomial order of approximation. The solver presented in this paper delivers linear computational cost for uniform polynomial order of approximation p. For non-uniform polynomial order the computational cost is almost linear.

Agent-based parallel system for numerical computations

Abstract The paper describes a way of applying agent paradigm to hp-adaptive Finite Element Method (hp-FEM). We discuss a choice of classical numerical algorithms suitable for incorporating into an agent-based application, along with an efficient way of adopting them into an agent-based application. We define formally a Computing Multi Agent System (C-MAS) for adaptive 1D FEM based on Smart Solid Agent model and describe tasks executed by hp-FEM agents. Finally, we spare a few paragraphs for numerical experiments performed with an application developed accordingly to the described model.

Agent-oriented image processing with the hp-adaptive projection-based interpolation operator

In this paper we discuss applications and design of the agent-oriented, hp-adaptive projection-based interpolation technique. We describe the use of the mesh adaptation process to produce the most faithful representation of the input image in the Finite Element space. We discuss the advantages of the agent-oriented application model both in general and in terms of the hp-adaptive application properties. Lastly, we describe a sample problem used as a proof of concept.

An Automatic Way of Finding Robust Elimination Trees for a Multi-frontal Sparse Solver for Radical 2D Hierarchical Meshes

In this paper we present a dynamic programming algorithm for finding optimal elimination trees for the multi-frontal direct solver algorithm executed over two dimensional meshes with point singularities. The elimination tree found by the optimization algorithm results in a linear computational cost of sequential direct solver. Based on the optimal elimination tree found by the optimization algorithm we construct heuristic sequential multi-frontal direct solver algorithm resulting in a linear computational cost as well as heuristic parallel multi-frontal direct solver algorithm resulting in a logarithmic computational cost. The resulting parallel algorithm is implemented on NVIDIA CUDA GPU architecture based on our graph-grammar approach.

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.

Hypergrammar Based Parallel Multi-Frontal Solver For Grids With Point Singularities

This paper describes the application of hypergraph grammars to drive a linear computational cost solver for grids with point singularities. Such graph gram- mar productions are the first mathematical formalisms used to describe solver algorithms, and each indicates the smallest atomic task that can be executed in parallel, which is very useful in the case of parallel execution. In particular, the partial order of execution of graph grammar productions can be found, and the sets of independent graph grammar productions can be localized. They can be scheduled set by set into a shared memory parallel machine. The graph- grammar-based solver has been implemented with NVIDIA CUDA for GPU. Graph grammar productions are accompanied by numerical results for a 2D case. We show that our graph-grammar-based solver with a GPU accelerator is, by order of magnitude, faster than the state-of-the-art MUMPS solver.