Sparse Systems Of Linear Algebraic Equations Research Articles

Iterative solver with folded preconditioner for finite element simulation of magnetotelluric fields

We propose an iterative solver with a folded preconditioner for large sparse systems of linear algebraic equations (SLAEs), which are obtained as a result of a vector finite element approximation of electromagnetic fields in magnetotelluric problems. The solver is based on the computational scheme of the Conjugate A-Orthogonal Conjugate Residual method (COCR), diagonal preconditioner, and residual smoothing. To build a finite element approximation, the primary-secondary field approach is used. To construct a preconditioner, we use a matrix, the elements of which are the weights of the representation of gradients of nodal basis functions as a linear combination of edge basis functions. For Dublin Test Model 1 in a wide frequency range from 0.0001 Hz to 10 Hz, we compare the accuracy and computational efficiency of our code with the results of other authors, as well as compare the computational efficiency of the proposed diagonal folded preconditioned (DFP) COCR solver with other solvers. It is shown that the apparent resistivities obtained on the coarsest mesh with our code differ by no more than a few percent from the solution on a finer mesh and from some solutions obtained by other authors using the finite element, finite difference, and integral equation methods. The computational time required for our code is 8 times less than the minimum time presented by other authors (with a corresponding recalculation of the performance of the computers used). It is also shown that the DFP COCR solver practically does not increase the number of iterations at low frequencies and its computational time is about 20 times less than the time required for the COCR solver with a standard diagonal preconditioner. A comparison of the DFP COCR solver with the Parallel Direct Sparse solver (PARDISO) shows that even at low and medium frequencies, the computational time of the DFP COCR solver is about 20 times less than the computational time of the PARDISO, and at high frequencies this difference reaches already 50 times.

Monte Carlo method for solving a parabolic problem

In this paper, we present a numerical method based on random sampling for a parabolic problem. This method combines use of the Crank-Nicolson method and Monte Carlo method. In the numerical algorithm, we first discretize governing equations by Crank-Nicolson method, and obtain a large sparse system of linear algebraic equations, then use Monte Carlo method to solve the linear algebraic equations. To illustrate the usefulness of this technique, we apply it to some test problems.

A new method for solving a class of heat conduction equations

A numerical method for solving a class of heat conduction equations with variable coefficients in one dimensional space is demonstrated. This method combines the Crank-Nicolson and Monte Carlo methods. Using Crank-Nicolson method, the governing equations are discretized into a large sparse system of linear algebraic equations, which are solved by Monte Carlo method. To illustrate the usefulness of this technique, we apply it to two problems. Numerical results show the performance of the present work.

Combined probabilistic algorithm for solving high dimensional problems

The present study establishes an accurate and efficient algorithm based on Monte Carlo (MC) simulation for solving high dimensional linear systems of algebraic equations (LSAEs) and two-dimensional Fredholm integral equations of the second kind (FIESK). This new combined numerical-probabilistic algorithm is based on Jacobi over-relaxation method and MC simulation in conjunction with the iterative refinement technique to find the unique solution of the large sparse LSAEs. It has an excellent accuracy, low cost and simple structure. Theoretical results are established to justify the convergence of the algorithm. To confirm the accuracy and efficiency of the present work, the proposed algorithm is used for solving and LSAEs. Furthermore, the algorithm is coupled with Galerkin's method to illustrate the power and effectiveness of the proposed algorithm for solving two-dimensional FIESK.

A method for the automatic selection and tuning of the parameters of a sparse SLE solver

A technique for the automatic selection of the method for solution of a sparse system of linear algebraic equations (SLE) and selection of the method parameters from the analysis of the SLE matrix of coefficients is presented. The parameters are selected so as to maximize the performance metric. A formal model of the performance of a SLE solver is presented and is used to formulate an algorithm for construction, precedent learning, and application of the proposed method. The method is approved for iterative SLE solvers from the PETSc parallel library of the MVS-100K supercomputer platform at the Interdepartmental Supercomputer Center of the Russian Academy of Sciences.