Ivan Lirkov

Circulant block-factorization preconditioners for elliptic problems

New circulant block-factorization preconditioners are introduced and studied. The general approach is first formulated for the case of block tridiagonal sparse matrices. Then estimates of the relative condition number for a model Dirichlet boundary value problem are derived. In the case ofy-periodic problems the circulant block-factorization preconditioner is shown to give an optimal convergence rate. Finally, using a proper imbedding of the original Dirichlet boundary value problem to ay-periodic one a preconditioner of optimal convergence rate for the general case is obtained. The total computational cost of the preconditioner isO (N logN) (based on FFT), whereN is the number of unknowns. That is, the algorithm is nearly optimal. Various numerical tests that demonstrate the features of the circulant block-factorization preconditioners are presented.ZusammenfassungNeue zyklische Matrixzerlegungen werden eingeführt und untersucht. Der allgemeine Ansatz wird für den Fall blocktridiagonaler schwachbesetzter Matrizen formuliert. Danach werden Abschätzungen der relativen Konditionszahl für ein Dirichlet-Modellproblem abgeleitet. Es wird gezeigt, daß die zyklische Matrixzerlegung im Falley-periodischer Aufgaben optimale Konvergenzraten liefert. Nach Einbettung des ursprünglichen Dirichlet-Problems in einey-periodische Aufgabe erhält man den allgemeinen Fall. Der Gesamtaufwand des Präkonditionierers beträgtO (N logN) gemäß des FFT-Aufwandes, wobeiN die Zahl der Unbekannten ist. Damit ist der Algorithmus fast optimal. Verschiedene numerische Tests zeigen die Eigenschaften der zyklischen Matrixzerlegung.

Ant colony system approach for protein folding

The protein folding problem is a fundamental problem in computational molecular biology and biochemical physics. The high resolution 3D structure of a protein is the key to the understanding and manipulating of its biochemical and cellular functions. All information necessary to fold a protein to its native structure is contained in its amino-acid sequence. Even under simplified models, the problem is NP-hard and the standard computational approach are not powerful enough to search for the correct structure in the huge conformation space. Due to the complexity of the protein folding problem simplified models such as hydrophobic-polar (HP) model have become one of the major tools for studying protein structure. Various optimization methods have been applied on folding problem including Monte Carlo methods, evolutionary algorithm, ant colony optimization algorithm. In this work we develop an ant algorithm for 3D HP protein folding problem. It is based on very simple design choices in particular with respect to the solution components reinforced in the pheromone matrix. The achieved results are compared favorably with specialized state-of-the-art methods for this problem. Our empirical results indicate that our rather simple ant algorithm outperforms the existing results for standard benchmark instances from the literature. Furthermore, we compare our folding results with proteins with known folding.

Agents as Resource Brokers in Grids -- Forming Agent Teams

Recently we have proposed an approach to utilizing agent teams as resource brokers and managers in the Grid. Thus far we have discussed the general overview of the proposed system, how to efficiently implement matchmaking services, as well as proposed a way by which agents select a team that will execute their job. In this paper we focus our attention on processes involved in agents joining a team.

Trust Management in an Agent‐Based Grid Resource Brokering System‐Preliminary Considerations

It has been suggested that utilization of autonomous software agents in computational Grids may deliver the needed functionality to speed‐up Grid adoption. I our recent work we have outlined an approach in which agent teams facilitate Grid resource brokering and management. One of the interesting questions is how to manage trust in such a system. The aim of this paper is to outline our proposed solution.

Large-Scale Scientific Computing

The paper concerns porous media flow in rigid or deformable matrix. It starts with stationary Darcy flow, but the main interest is in extending Darcy problem to involve time dependent behaviour and deformation of the matrix. The considered problems are discretized by mixed FEM in space and stable time discretization methods as backward Euler and second order Radau methods. The discretization leads to time stepping methods which involve solution of a linear system within each time step. The main focus of the paper is then devoted to the construction of suitable preconditioners for these Euler and Radau systems. The paper presents also numerical experiments for illustration of efficiency of the suggested numerical algorithms.

Circulant block-factorization preconditioning of anisotropic elliptic problems

The recently introduced circulant block-factorization preconditioners are studied in this paper. The general approach is first formulated for the case of block-tridiagonal sparse matrices. Then an estimate of the condition number of the preconditioned matrix for a model anisotropic Dirichlet boundary value problem is derived in the formκ<√2ε(n+1)+2, whereN=n2 is the size of the discrete problem, andε stands for the ratio of the anisotropy. Various numerical tests demonstrating the behavior of the circulant block-factorization preconditioners for anisotropic problems are presented.ZusammenfassungIn dieser Arbeit werden die kürzlich eingeführten zirkulanten Präkonditionierer untersucht, die aus Block-Faktorisierungen herrühren. Der allgemeine Zugang ist für schwach besetzte Block-Tridiagonalmatrizen formuliert. Es wird die Kondition der präkonditionierten Systemmatrix für ein Modellproblem, eine anisotrope Dirichlet-Randwertaufgabe, durch die Größeκ<√2ε(n+1)+2 abgeschätzt. Dabei stelltN=n2 die Größe des diskretisierten Problems undε das Anisotropieverhältnis dar. Verschiedene numerische Experimente werden vorgestellt, die das Verhalten der Präkonditionierung beim Modellproblem aufzeigen.

A parallel algorithm for systems of convection-diffusion equations

The numerical solution of systems of convection-diffusion equations is considered. The problem is described by a system of second order partial differential equations (PDEs). This system is discretized by Courant-elements. The preconditioned conjugate gradient method is used for the iterative solution of the large-scale linear algebraic systems arising after the finite element discretization of the problem. Discrete Helmholtz preconditioners are applied to obtain a mesh independent superlinear convergence of the iterative method. A parallel algorithm is derived for the proposed preconditioner. A portable parallel code using Message Passing Interface (MPI) is developed. Numerical tests well illustrate the performance of the proposed method on a parallel computer architecture.

MPI solver for 3D elasticity problems

The numerical solution of 3D linear elasticity equations is considered. The problem is described by a coupled system of second-order elliptic partial differential equations. This system is discretized by trilinear parallelepipedal finite elements.The preconditioned conjugate gradient iterative method is used for solving of the large-scale linear algebraic systems arising after the finite element method (FEM) discretization of the problem. Displacement decomposition technique is applied at the first step to construct a preconditioner using the decoupled block-diagonal part of the original matrix. Then circulant block-factorization is used for preconditioning of the obtained block-diagonal matrix. Both techniques, displacement decomposition and circulant block-factorization, are highly parallelizable.A parallel algorithm is invented for the proposed preconditioner. The theoretical analysis of the execution time shows that the algorithm is highly efficient for coarse-grain parallel computer systems.A portable MPI parallel FEM code is developed. Numerical tests for real-life engineering problems of the geomechanics in geosciences on a number of modern parallel computers are presented. The reported speed-up and parallel efficiency well illustrate the parallel features of the proposed method and its implementation.

MPI parallel implementation of CBF preconditioning for 3D elasticity problems

The numerical solution of 3D linear elasticity equations is considered. The problem is described by a coupled system of second-order elliptic partial differential equations. This system is discretized by trilinear brick finite elements. The PCG iterative method is used for solving the large-scale linear algebraic systems arising after the FEM discretization of the problem. Displacement decomposition technique is applied at the first step to construct a preconditioner using the decoupled block-diagonal part of the original matrix. Then circulant block-factorization is used for preconditioning of the obtained block-diagonal matrix.

Applying Saaty's Multicriterial Decision Making Approach in Grid Resource Management

In this paper we consider combining ontologically represented information with Saaty’s Analytic Hierarchy Process (AHP) to facilitate decision support for Grid users. The context for the proposal is provided by the Agents in Grid project (AiG), which aims at development of an agent-based infrastructure for efficient resource management in the Grid. In the AiG project, agents form teams, managers of which negotiate with clients and workers terms of potential collaboration. Here, we focus on the scenario, in which the user is searching for resources to execute a task, while the resources and the expert domain knowledge are organized in an ontology. Taking into account the complex nature of resource description and domain knowledge, multicriterial assessment of how accurate is the user description of her needs, and how it can be extended/refined, plays a crucial role. For instance, it should help the user to choose optimal algorithm and/or resource to solve her problem. Furthermore, ontologically described contract proposals, that are the results of autonomous negotiations, require multicriterial assessment. The AHP method is based on pairwise comparisons of criteria, and relies on the judgment of a panel of experts. In the context of the AiG project, we show on the example of the AHP method, how multicriteria group decision making can be used to support user in resource selection and assessment of contract proposals. DOI: http://dx.doi.org/10.5755/j01.itc.43.1.4587