This paper is concerned with the new approach to the numerical solution of diffusion equations with high contrast coefficients. The classical P1 finite element system of equations is replaced by the equivalent system ...
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This paper is concerned with the new approach to the numerical solution of diffusion equations with high contrast coefficients. The classical P1 finite element system of equations is replaced by the equivalent system of the saddlepoint type. For the saddlepoint problem a new recently invented asymptotic expansion with respect to a small parameter is discussed and the estimates for the convergence factor are given. Numerical results completely confirm the theoretical accuracy and convergence results for the considered method. (C) 2019 Elsevier B.V. All rights reserved.
In this paper we consider the application of direct methods for solving a sequence of saddle-point systems. Our goal is to design a method that reuses information from one factorization and applies it to the next one....
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In this paper we consider the application of direct methods for solving a sequence of saddle-point systems. Our goal is to design a method that reuses information from one factorization and applies it to the next one. In more detail, when we compute the pivoted LDLT factorization we speed up computation by reusing already computed pivots and permutations. We develop our method in the frame of dynamical systems optimization. Experiments show that the method improves efficiency over Bunch-Parlett and Bunch-Kaufman while delivering the same results. (C) 2019 Elsevier Inc. All rights reserved.
Optimization problems with partial differential equations as constraints arise widely in many areas of science and engineering, in particular in problems of the design. The solution of such class of PDE-constrained op...
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Optimization problems with partial differential equations as constraints arise widely in many areas of science and engineering, in particular in problems of the design. The solution of such class of PDE-constrained optimization problems is usually a major computational task. Because of the complexion for directly seeking the solution of PDE-constrained op- timization problem, we transform it into a system of linear equations of the saddle-point form by using the Galerkin finite-element discretization. For the discretized linear system, in this paper we construct a block-symmetric and a block-lower-triangular preconditioner, for solving the PDE-constrained optimization problem. Both preconditioners exploit the structure of the coefficient matrix. The explicit expressions for the eigenvalues and eigen- vectors of the corresponding preconditioned matrices are derived. Numerical implementa- tions show that these block preconditioners can lead to satisfactory experimental results for the preconditioned GMRES methods when the regularization parameter is suitably small.
This paper applies a simple constrained ordering for the solution of the equality constrained state estimation problem. By low-rank perturbations in the semidefinite (1,1) block of the coefficient matrix, while mainta...
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This paper applies a simple constrained ordering for the solution of the equality constrained state estimation problem. By low-rank perturbations in the semidefinite (1,1) block of the coefficient matrix, while maintaining sparsity, a saddlepointmatrix is formed. The vectors used for generating the perturbations are rows of the matrix associated with the equality constraints that represent the zero injections. The proposed algorithm make use of the Bridson's ordering constraint for saddle-point systems, which is sufficient to guarantee the existence of a signed Cholesky factorization for the perturbed indefinite coefficient matrix, with separate symbolic and numerical phases. The need for numerical pivoting during factorization is avoided, with clear benefits for performance. Two alternative implementations are provided, either modifying a fill-reducing ordering algorithm to incorporate this constraint or modifying an existing fill-reducing ordering to respect the constraint. The proposed method is compared with existing methods in terms of computational time and convergence robustness. The IEEE 300-bus and the FRCC 3949-bus systems are used as test beds for this study.
This paper presents a robust method for the solution of the power system state estimation with equality constraints. The existing methods formulate coefficient matrices which are symmetric and indefinite and require f...
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This paper presents a robust method for the solution of the power system state estimation with equality constraints. The existing methods formulate coefficient matrices which are symmetric and indefinite and require factorization routines with additional logic to process the zero pivots. In this paper the formulated coefficient matrix has unique triangular factorization which can be accomplished symbolically using only the sparsity criterion. The method is illustrated with the IEEE 14-bus system. Test results are given with the FRCC 3949-bus system.
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