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Iterative Methods for Parallel Computing Systems


Non-Stationary Iterative Decomposition Methods for MIMD-type Parallel Computing Systems: Theory and Applications

Tech Area / Field

  • INF-COM/High Performance Computing and Networking/Information and Communications
  • INF-SOF/Software/Information and Communications

3 Approved without Funding

Registration date

Leading Institute
VNIITF, Russia, Chelyabinsk reg., Snezhinsk

Project summary

Mathematical modeling natural phenomena often requires solving large systems of linear and nonlinear algebraic equations. Similar systems arise from approximation of equations with partial derivatives in 2-D and especially 3-D case by methods of finite differences or finite elements.

For solving high-order systems of linear algebraic equations (SLAE) iterative methods are widely used. There are many iterative algorithms described in scientific publications. However, every algorithm has own characteristics and a range limited of application. Besides, parallel computing systems (PCS) are intensively developed last years. Two basic types of PCS, SIMD and MIMD, differing in processors control are existed. Both types possess advantages and lacks, but more flexible MIMD-type has more wide prevalence. A choice of an effective iterative method for solving a concrete problem depends essentially on PCS characteristics and architecture. Due to PCS development of iterative algorithms widely applied and easy adapted to various architectures is actual problem.

The main aim of the project is to develop effective stationary and non-stationary iterative decomposition methods for solving systems of linear and nonlinear algebraic equations on PCS of MIMD-type. The project has theoretical and practical aspects.

The theoretical aspect is to construct a family of stationary and non-stationary iterative solvers for SLAE (in general with block unsymmetric complex-valued matrices) with a high level of parallelism and to investigate its convergence on some classes of matrices. The construction will base on iterations in subspaces in general with arbitrary overlapping. Domain decomposition methods belong to this family. For 2-D and 3-D model diffusion equation is supposed to obtain optimal domain decomposition in dependence on parameters of a difference task and PCS characteristics. In the project convergence rate of alternating Schwarz-type method will be investigated with a view to find optimal domain decomposition in the case of simple domains. The idea of SLAE solvers construction is planed to generalize on nonlinear systems by means of combination with the Newton’s method. H-mappings (some nonlinear generalization of block H-matrices) is supposed to develop. Convergence of nonlinear solvers is designed to investigate in the case of H-mappings. All theoretical investigations will be supported by appropriate computational experiments.

The applied aspect of the project is to develop a parallel program for solving 2-D equations of ideal MHD on regular and nonregular grids by using the solvers constructed. In addition a parallel program for solving complex-valued SLAE arising from calculation of spectral lines profiles and absorption spectral coefficients of multi-electron ions in plasma is supposed to be worked out on the base of these solvers.

Thus, in the course of the project realization next tasks will be solved:

· Development and convergence analysis of arbitrary group iterative (AGI) method with a high level of parallelism for solving SLAE;

· Optimal parallel implementation of the AGI method for numerical solving 2-D and 3-D model diffusion equation;

· Convergence analysis of alternating Schwarz-type method for 2-D and 3-D model diffusion equation, obtaining an effective upper estimation for its convergence rate;

· Generalization and convergence analysis of the AGI method for solving systems of nonlinear algebraic equations;

· Development of a parallel program for solving 2-D equations of ideal MHD on regular grids by using the AGI method;

· Development of a program for solving SLAE arising from approximation of differential equations on nonregular grid consisting of Voronoy’s cells by using the AGI method;

· Development of a parallel program for calculating spectral lines profiles and absorption spectral coefficients of multi-electron ions in plasma.

Scientific researches being conducted within the framework of the project will be based on the theory of iterative methods of SLAE solution, the theory of fixed points of nonlinear mappings, the theory of elliptic and parabolic differential equations, will use mathematical models of ideal MHD and high-temperature dense plasma. For development of the parallel programs high-level languages (Fortran-77, Fortran-90, C) and standard libraries of parallel means (MPI, PVM) will be applied.

The solution of actual tasks mentioned above in the field of parallel iterative methods and mathematical modeling physical processes will allow to direct activities of VNIITF specialists group possessing the experience in this field to civil themes, will further peaceful uses of fundamental and applied investigations and extension of international contacts.


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