The convergence of the restarted GMRES method can be significantly improved, for some problems, by using a weighted inner product that changes at each restart. How does this weighting affect convergence, and when is it useful? We show that weighted inner products can help in two distinct ways: when the coefficient matrix has localized eigenvectors, weighting can allow restarted GMRES to focus o...

We introduce a preconditioner that can be both constructed and applied using only the ability to apply the underlying operator. Such a preconditioner can be very attractive in scenarios where one has a highly efficient parallel code for applying the operator. Our method constructs a polynomial preconditioner using a nonlinear least squares (NLLS) algorithm. We show that this polynomial-based NL...

We consider the solution of a linear system of equations using the GMRES iterative method. In [3], a strategy to relax the accuracy of the matrix-vector product is proposed for general systems and illustrated on a large set of numerical experiments. This work is based on some heuristic considerations and proposes a strategy that often enables a convergence of the GMRES iterates xk within a rela...

We study the roundoff error propagation in an algorithm which computes the orthonormal basis of a Krylov subspace with Householder orthonormal matrices . Moreover, we analyze special implementations of the classical GMRES algorithm, and of the Full Orthogonalization Method . These techniques approximate the solution of a large sparse linear system of equations on a sequence of Krylov subspaces ...

None of the existing methods for computing the oscillatory integral ∫ b a f(x)e iωg(x) dx achieve all of the following properties: high asymptotic order, stability, avoiding the computation of the path of steepest descent and insensitivity to oscillations in f . We present a new method that satisfies these properties, based on applying the gmres algorithm to a preconditioned differential operator.

We study the convergence of GMRES/FOM and QMR/BiCG methods for solving nonsymmetric Ax = b. We prove that given the results of a BiCG computation on Ax = b, we can obtain a matrix B with the same eigenvalues as A and a vector c such that the residual norms generated by a FOM computation on Bx = c are identical to those generated by the BiCG computations. Using a unitary equivalence for each of ...

The GMRES method is a popular iterative method for the solution of linear systems of equations with a large nonsymmetric nonsingular matrix. However, little is known about the performance of the GMRES method when the matrix of the linear system is of ill-determined rank, i.e., when the matrix has many singular values of different orders of magnitude close to the origin. Linear systems with such...

In the 27 km circumference Large Hadron Collider, the temperature of more than 1600 main superconducting magnets is stabilized below 2 K by the Superfluid Helium Cryogenic Circuit. The key component of the circuit’s Standard Cell is an over 100 m long bayonet heat exchanger with two phase flow of superfluid helium, He II, that is integrated into the magnets submerged in a static bath of He II. ...

None of the existing methods for computing the oscillatory integral ∫ b a f(x)e iωg(x) dx achieve all of the following properties: high asymptotic order, stability, avoiding deformation into the complex plane and insensitivity to oscillations in f . We present a new method that satisfies these properties, based on applying the gmres algorithm to a shifted linear differential operator.

In recent years competitive domain decomposed preconditioned iterative tech niques have been developed for nonsymmetric elliptic problems In these tech niques a large problem is divided into many smaller problems whose requirements for coordination can be controlled to allow e ective solution on parallel machines A central question is how to choose these small problems and how to arrange the or...