Uzawa Algorithms for Fully Fuzzy Linear Systems

An associated linear system has been previously employed to solve Fully Fuzzy Linear System (FFLS) with trapezoidal or triangular fuzzy numbers (TFNs). This paper proposes a modified associated linear system for solving FFLS with a positive hexagonal fuzzy number (HFN). FFLS is proved to be an equivalent linear system to the modified associated linear system. Subsequently, the solution can be represented and obtained using any classical linear system method. The required theorems are derived to inspect a sufficient and necessary condition in order to solve the system. An example is given to illustrate our work.

The growing significance, technical development and employment of electromagnetic (EM) methods in exploration geophysics have led to the increasing need for reliable and fast techniques of interpretation of 3-D EM data sets acquired in complex geological environments. The first and most important step to creating an inversion method is the development of a solver for the forward problem. In order to create an efficient, reliable and practical 3-D EM inversion, it is necessary to have a 3-D EM modelling code that is highly accurate, robust and very fast. This thesis focuses precisely on this crucial and very demanding step to building a 3-D EM interpretation method. The thesis presents as its main contribution a highly accurate, robust, very fast and extremely scalable numerical method for 3-D EM modelling in geophysics that is based on finite elements (FE) and designed to run on massively parallel computing platforms. Thanks to the fact that the FE approach supports completely unstructured tetrahedral meshes as well as local mesh refinements, the presented solver is able to represent complex geometries of subsurface structures very precisely and thus improve the solution accuracy and avoid misleading artefacts in images. Consequently, it can be successfully used in geological environments of arbitrary geometrical complexities. The parallel implementation of the method, which is based on the domain decomposition and a hybrid MPI-OpenMP scheme, has proved to be highly scalable - the achieved speed-up is close to the linear for more than a thousand processors. Thanks to this, the code is able to deal with extremely large problems, which may have hundreds of millions of degrees of freedom, in a very efficient way. The importance of having this forward-problem solver lies in the fact that it is now possible to create a 3-D EM inversion that can deal with data obtained in extremely complex geological environments in a way that is realistic for practical use in industry. So far, such imaging tool has not been proposed due to a lack of efficient, parallel FE solutions as well as the limitations of efficient solvers based on finite differences. In addition, the thesis discusses physical, mathematical and numerical aspects and challenges of 3-D EM modelling, which have been studied during my research in order to properly design the presented software for EM field simulations on 3-D areas of the Earth. Through this work, a physical problem formulation based on the secondary Coulomb-gauged EM potentials has been validated, proving that it can be successfully used with the standard nodal FE method to give highly accurate numerical solutions. Also, this work has shown that Krylov subspace iterative methods are the best solution for solving linear systems that arise after FE discretisation of the problem under consideration. More precisely, it has been discovered empirically that the best iterative method for this kind of problems is biconjugate gradient stabilised with an elaborate preconditioner. Since most commonly used preconditioners proved to be either unable to improve the convergence of the implemented solvers to the desired extent, or impractical in the parallel context, I have proposed a preconditioning technique for Krylov methods that is based on algebraic multigrid. Tests for various problems with different conductivity structures and characteristics have shown that the new preconditioner greatly improves the convergence of different Krylov subspace methods, which significantly reduces the total execution time of the program and improves the solution quality. Furthermore, the preconditioner is very practical for parallel implementation. Finally, it has been concluded that there are not any restrictions in employing classical parallel programming models, MPI and OpenMP, for parallelisation of the presented FE solver. Moreover, they have proved to be enough to provide an excellent scalability for it.

In this paper, a fully fuzzy linear system (FFLS) is considered. By defuzzifying, the (n × n) FFLS can be replaced by three (n × n) crisp linear systems, and consequently its homomorphic solution in canonical trapezoidal form based on three (n × n) crisp linear solutions associated with three parameters, value, ambiguity, and fuzziness, is calculated.

Krylov subspace methods are crucial techniques for solving linear systems. To effectively tackle large-scale linear systems, parallelism techniques are frequently utilized. However, the use of parallelism can increase the non-associativity of floating-point operations, potentially resulting in non-reproducibility of the computations. In order to address the issue of non-reproducibility in parallel preconditioned Krylov subspace iterative methods, we investigated the sources of irreproducibility and solutions to achieve reproducibility. The main reason for the irreproducibility in iterative methods is attributed to distributed dot products. To tackle this, we investigated three reproducible Basic Linear Algebra Subprograms (BLAS) as replacements for standard distributed dot products. We utilized OzBLAS, ExBLAS, and ReproBLAS to ensure reproducibility in iterative methods. We combined Jacobi preconditioners and tested and verified on three iterative methods: CG, BiCGSTAB and GMRES. Numerical experiments demonstrate that the iterative method based on OzBLAS is reproducible in a multi-threaded environment. The iterative method based on ExBLAS and ReproBLAS achieves reproducibility in a multi-process environment. Our evaluation indicated that ReproBLAS performed well in terms of performance, robustness, and parallelism. Therefore, we employ a reproducible iterative method based on ReproBLAS to solve the large-scale algebraic equations resulting from the discretization of the two-dimensional three-temperature heat conduction equation, which arises from practical application problems in the field of laser fusion radiation hydrodynamics. Numerical experiments demonstrate the reproducible solution of the three-temperature heat conduction algebraic equations.

An efficient finite-difference method for solving Helmholtz equation depends on two points: one is discrete scheme, the other is efficient algorithm. In this paper, we adopt the average-derivative scheme, which owns three advantages : Firstly, it can be applied to unequal directional sampling intervals for Helmholtz equation. Secondly, the scheme is pointwise consistent with Helmholtz equation in a perfect matched layer. And thirdly, it requires less than 4 grid points sampling per wavelength. To solve the discrete Helmholtz equation, which is extremely large and indefinite, direct methods cannot resolve well, and the Krylov subspace iterative methods, such as Bi-CGSTAB and GMRES combining a multigrid-based preconditioner, are good choices. However, the standard multigrid algorithm fails to converge when it encounters unequal directional sampling intervals, which is called anisotropy in multigrid. We analyze the most important three parts of standard multigrid : full weighting restriction operator, point relaxation methods and bilinear interpolation operator, and then we replace them with semi-coarsening, line relaxation and operator-dependent interpolation to make it convergent in anisotropic problems. Consequently, we obtain a satisfactory convergence speed for low and moderate frequency iterative problems in heterogeneous media.

This paper discusses on the choice of surface integral equation formulations in electromagnetic scattering and radiation problems. The accuracy of the solution and convergence of iterative (Krylov subspace) methods are investigated. It is shown that the solution accuracy and iteration convergence are almost opposite properties. Reasons and remedies to this phenomenon are discussed. (5 pages)

The relaxed nonlinear PHSS-like iteration method for absolute value equations

Krylov subspace iterative methods have recently received considerable attention as regularizing techniques for solving linear systems with a coefficient matrix of ill-determined rank and a right-hand side vector perturbed by noise. For many of them little is known from this point of view. In this paper, the regularizing properties of some methods of Krylov type (CGLS, GMRES, QMR, CGS, BiCG, Bi-CGSTAB) are examined. CGLS, for which a theoretical analysis is available, is taken as a reference method. Tools for measuring the regularization efficiency and the consistency with the discrepancy principle are introduced. An extensive experimentation validates the proposed measures for the studied methods.

This paper presents a method for solving a complex fully fuzzy linear system (FFLS), particularly applicable to electric circuits. The constructed method enables solving arbitrary coefficient of FFLS in the form of LR complex fuzzy numbers by applying associated complex linear system. Then, the solution of this system is obtained by using the inverse method. Consequently, a numerical example illustrates the application of a simple electric circuit in a fuzzy controller. Finally, the solution is verified by employing modified fuzzy arithmetic multiplication operations.

In this paper, an unambiguous graphical interpretation of fully fuzzy linear equation in two variables where all the parameters are symmetric triangular fuzzy number (STFN) is presented. Several new terminologies such as disassembled fuzzy straight line and assembled fuzzy straight line are proposed and theorems concerning those are established. This interpretation is extended to \(2\times 2\) fully fuzzy linear system (FFLS) with STFN which enables to comment on the existence of solution of the system. Then an application of FFLS is focused in a real-life circuit analysis problem where the solving procedure using available method in the literature leads to weak solution. Why this happens is explained using the proposed geometrical interpretation. Finally, a modification to the definition of TFN is proposed to manage the situation easily where weak solution arises.

In this paper, we propose a two-parameter preconditioned variant of the deteriorated PSS iteration method (J. Comput. Appl. Math., 273, 41–60 (2015)) for solving singular saddle point problems. Semi-convergence analysis shows that the new iteration method is convergent unconditionally. The new iteration method can also be regarded as a preconditioner to accelerate the convergence of Krylov subspace methods. Eigenvalue distribution of the corresponding preconditioned matrix is presented, which is instructive for the Krylov subspace acceleration. Note that, when the leading block of the saddle point matrix is symmetric, the new iteration method will reduce to the preconditioned accelerated HSS iteration method (Numer. Algor., 63 (3), 521–535 2013), the semi-convergence conditions of which can be simplified by the results in this paper. To further improve the effectiveness of the new iteration method, a relaxed variant is given, which has much better convergence and spectral properties. Numerical experiments are presented to investigate the performance of the new iteration methods for solving singular saddle point problems.

We propose a method to approximate the solutions of fully fuzzy linear system (FFLS), the so-calledgeneral solutions. So, we firstly solve the 1-cut position of a system, then some unknown spreads are allocated to each row of an FFLS. Using this methodology, we obtain some general solutions which are placed in the well-known solution sets like Tolerable solution set (TSS) and Controllable solution set (CSS). Finally, we solved two examples in order to demonstrate the ability of the proposed method.

In the present paper, we propose a separate approach as a new strategy to solve the saddle point problem arising from the stochastic Galerkin finite element discretization of Stokes problems. The preconditioner is obtained by replacing the (1,1) and (1,2) blocks in the RHSS preconditioner by others well chosen and the parameter α in (2,2) −block of the RHSS preconditioner by another parameter β. The proposed preconditioner can be used as a preconditioner corresponding to the stationary itearative method or to accelerate the convergence of the generalized minimal residual method (GMRES). The convergence properties of the GMRHSS iteration method are derived. Meanwhile, we analyzed the eigenvalue distribution and the eigenvectors of the preconditioned matrix. Finally, numerical results show the effectiveness of the proposed preconditioner as compared with other preconditioners.

One-way domain decomposition method with exact radiation condition and fast GMRES solver for the solution of Maxwell's equations

<abstract><p>The strong <italic>P</italic>-regular splitting is put forward and defined for iterative methods of non-Hermitian linear systems in the paper. The strong <italic>P</italic>-regular splitting combining SOR iterative methods and relaxed SOR iterative methods are established, and conditions guaranteeing the convergence are presented. Furthermore, two numerical experiments are done to illustrate the convergence and effectiveness of our iterative methods.</p></abstract>