Many ill-posed problems are solved using a discretization that results in a least squares problem or a linear system involving a Toeplitz matrix. The exact solution to such problems is often hopelessly contaminated by noise, since the discretized problem is quite ill conditioned, and noise components in the approximate null-space dominate the solution vector. Therefore we seek an approximate so...

We consider applying the preconditioned conjugate gradient (PCG) method to solve linear systems Ax = b where the matrix A comes from the discretization of second-order elliptic operators. Let (L +)) ?1 (L t +) denote the block Cholesky factorization of A with lower block triangular matrix L and diagonal block matrix. We propose a preconditioner M = (^ L +)) ?1 (^ L t +) with block diagonal matr...

We describe a block matrix iterative algorithm for solving a linearquadratic parabolic optimal control problem (OCP) on a finite time interval. We derive a reduced symmetric indefinite linear system involving the control variables and auxiliary variables, and solve it using a preconditioned MINRES iteration, with a symmetric positive definite block diagonal preconditioner based on the parareal ...

The parameterized Uzawa preconditioners for saddle point problems are studied in this paper. The eigenvalues of the preconditioned matrix are located in (0, 2) by choosing the suitable parameters. Furthermore, we give two strategies to optimize the rate of convergence by finding the suitable values of parameters. Numerical computations show that the parameterized Uzawa preconditioners can lead ...

A preconditioned Jacobi (PJ) method is provided for solving fuzzy linear systems whose coefficient matrices are crisp M matrices and the right-hand side columns are arbitrary fuzzy number vectors. The iterative algorithm is given for the preconditioned Jacobi method. The convergence is analyzed with convergence theorems. Numerical examples are given to illustrate the procedure and show the effe...

In this paper we propose a parallel preconditioner for the Newton method in the computation of the leftmost eigenpairs of large and sparse symmetric positive definite matrices. A sequence of preconditioners starting from an enhanced approximate inverse RFSAI [13] and enriched by a BFGS-like update formula is proposed to accelerate the Preconditioned Conjugate Gradient solution of the linearized...