Igor N. Konshin

ILU Preconditioners for Nonsymmetric Saddle-Point Matrices with Application to the Incompressible Navier--Stokes Equations

Motivated by the numerical solution of the linearized incompressible Navier--Stokes equations, we study threshold incomplete LU factorizations for nonsymmetric saddle-point matrices. The resulting preconditioners are used to accelerate the convergence of a Krylov subspace method applied to finite element discretizations of fluid dynamics problems in three space dimensions. The paper presents and examines an extension for nonsymmetric matrices of the Tismenetsky--Kaporin incomplete factorization. It is shown that in numerically challenging cases of higher Reynolds number flows one benefits from using this two-parameter modification of a standard threshold ILU preconditioner. The performance of the ILU preconditioners is studied numerically for a wide range of flow and discretization parameters, and the efficiency of the approach is shown if threshold parameters are chosen suitably. The practical utility of the method is further demonstrated for the haemodynamic problem of simulating blood flow in a right c...

A parallel block overlap preconditioning with inexact submatrix inversion for linear elasticity problems

We present a parallel preconditioned iterative solver for large sparse symmetric positive definite linear systems. The preconditioner is constructed as a proper combination of advanced preconditioning strategies. It can be formally seen as being of domain decomposition type with algebraically constructed overlap. Similar to the classical domain decomposition technique, inexact subdomain solvers are used, based on incomplete Cholesky factorization. The proper preconditioner is shown to be near optimal in minimizing the so-called K-condition number of the preconditioned matrix. The efficiency of both serial and parallel versions of the solution method is illustrated on a set of benchmark problems in linear elasticity. Copyright

Parallel Solution of Large Sparse SPD Linear Systems Based on Overlapping Domain Decomposition

We present a parallel iterative solver for large sparse symmetric positive definite (SPD) linear systems based on a new theory describing the convergence ofthe Preconditioned Conjugate Gradient (PCG) method and a proper combination ofa dvanced preconditioning strategies. Formally, the preconditioning can be interpreted as a special (nearly optimum from the viewpoint of the new PCG theory) version of overlapping domain decomposition with incomplete Cholesky solutions over subdomains. The estimates ofpa rallel efficiency are given as well as the results ofn umerical experiments for the serial and parallel versions oft he solver.

Application of the Parallel INMOST Platform to Subsurface Flow and Transport Modelling

INMOST (Integrated Numerical Modelling and Object-oriented Supercomputing Technologies) is a tool for supercomputer simulations characterized by a maximum generality of supported computational meshes, distributed data structure flexibility and cost-effectiveness, as well as crossplatform portability. INMOST is a software platform for developing parallel numerical models on general meshes. User guides, online documentation, and the open-source code of the library is available at http://www.inmost.org.

Load Balancing of Parallel Block Overlapped Incomplete Cholesky Preconditioning

A modification of the second order Incomplete Cholesky (IC) factorization with controllable amount of fill-in is described and analyzed. This algorithm is applied to the construction of well balanced coarse-grain parallel preconditioning for the Conjugate Gradient (CG) iterative solution of linear systems with symmetric positive definite matrix. The efficiency of the resulting parallel algorithm is illustrated by a series of numerical experiments using large-scale ill-conditioned test matrices taken from the collection of the University of Florida.

An Algebraic Solver for the Oseen Problem with Application to Hemodynamics

The paper studies an iterative method for algebraic problems arising in numerical simulation of blood flows. Here we focus on a numerical solver for the fluid part of otherwise coupled fluid-structure system of equations which models the hemodynamics in vessels. Application of the finite element method and semi-implicit time discretization leads to the discrete Oseen problem at every time step of the simulation. The problem challenges numerical methods by anisotropic geometry, open boundary conditions, small time steps and transient flow regimes. We review known theoretical results and study the performance of recently proposed preconditioners based on two-parameter threshold ILU factorization of non-symmetric saddle point problems. The preconditioner is applied to the linearized Navier–Stokes equations discretized by the stabilized Petrov–Galerkin finite element (FE) method. Careful consideration is given to the dependence of the solver on the stabilization parameters of the FE method. We model the blood flow in the digitally reconstructed right coronary artery under realistic physiological regimes. The paper discusses what is special in such flows for the iterative algebraic solvers, and shows how the two-parameter ILU preconditioner is able to meet these specifics.

Hierarchical Domain Representation in the AlgoWiki Encyclopedia: From Problems to Implementations

Algorithm description is the basic unit in the AlgoWiki Open Encyclopedia of Algorithmic Features. However, computational algorithms are not objectives in and of themselves: they are needed to address problems encountered in various fields of science and industry. On the other hand, there are many practical problems that can be tackled using various methods. This warrants the introduction of another basic term that fits between the concepts of a problem and an algorithm. Also, any algorithm can have different implementations, whether related to a single computing platform or to different platforms. The “problem–method–algorithm–implementation” chain is the basis for describing any subject area in AlgoWiki. This paper describes the permitted freedom in describing such chains, which arises when studying the approaches to address various practical problems.

Scalable Computations of GeRa Code on the Base of Software Platform INMOST

The hydrogeological modeling code GeRa is based on INMOST software platform, which operates with distributed mesh data and allows to assemble and solve the system of linear equations. The set of groundwater flow models with filtration, transport, and chemical processes are considered. The comparison of parallel efficiency for different linear solvers in the INMOST framework is performed. The analysis of scalability of GeRa code on different computer platforms from multicore laptop to Lomonosov supercomputer is presented.

Ani3D-Extension of Parallel Platform INMOST and Hydrodynamic Applications

The paper is devoted to an extension of the parallel platform INMOST by finite element and meshing libraries of the Ani3D software package. The extension allows us to develop parallel finite element solvers of boundary value problems and, in particular, hydrodynamic problems. The Ani3D package allows one to build, refine, locally adapt and improve the quality of tetrahedral meshes, perform finite element discretizations of partial differential equations for various types of finite elements, solve the appearing algebraic systems, and visualize the discrete solutions. The INMOST software platform provides tools for creating and storing distributed general conformal grids with arbitrary polyhedral cells, parallel assembling and parallel solution of arising distributed linear systems. We present the integration of two libraries from Ani3D into INMOST platform and demonstrate the functionality of the joint software on the solution of two model hydrodynamic problems on multiprocessor systems.

Dynamic Optimization of Linear Solver Parameters in Mathematical Modelling of Unsteady Processes

The optimization of linear solver parameters in unsteady multiphase groundflow modelling is considered. Two strategies of dynamic parameters setting for the linear solver are proposed when the linear systems properties are modified during simulation in the INMOST framework. It is shown that the considered algorithms for dynamic selection of linear solver parameters provide a more efficient solution than any prescribed set of parameters. The results of numerical experiments on the INM RAS cluster are presented.