Hydrodynamic interactions play an important role in the dynamics of macromolecules. The most common way to take into account hydrodynamic effects in molecular simulations is in the context of a Brownian dynamics simulation. However, the calculation of correlated Brownian noise vectors in these simulations is computationally very demanding and alternative methods are desirable. This paper studie...

Consider the linear system Ax = b, where A ∈ CN×N is a singular matrix. In the present work we propose a general framework within which Krylov subspace methods for Drazininverse solution of this system can be derived in a convenient way. The Krylov subspace methods known to us to date treat only the cases in which A is hermitian and its index ind(A) is unity necessarily. In the present work A i...

For a given deBranges space H(E) we investigate deBranges subspaces defined in terms of majorants on the real axis: If ω is a nonnegative function on R, we consider the subspace Rω(E) = ClosH(E) { F ∈ H(E) : ∃C > 0 : |EF | ≤ Cω on R } . We show that Rω(E) is a deBranges subspace and describe all subspaces of this form. Moreover, we give a criterion for the existence of positive minimal majorant...

Communication-Avoiding Krylov Subspace Methods in Theory and Practice

Krylov subspace methods are used for solving systems of linear equations Hx+ c = 0. We present a general Krylov subspace method that can be applied to a symmetric system of linear equations, i.e., for a system in which H = H . In each iteration, we have a choice of scaling of the orthogonal vectors that successively make the Krylov subspaces available. We define an extended representation of ea...

Algebraic solvers based on preconditioned Krylov subspace methods are among the most powerful tools for large scale numerical computations in applied mathematics, sciences, technology, as well as in emerging applications in social sciences. As the name suggests, Krylov subspace methods can be viewed as a sequence of projections onto nested subspaces of increasing dimension. They are therefore b...

We investigate the use of Krylov subspace methods to solve linear, oscillatory ODEs. When we apply a Krylov subspace method to a properly formulated equation, we retain the asymptotic accuracy of the asymptotic expansion whilst converging to the exact solution. We will demonstrate the effectiveness of this method by computing Error and Mathieu functions. Oxford University Computing Laboratory N...

It is well known that the block Krylov subspace solvers work efficiently for some cases of the solution of differential equations with multiple right-hand sides. In lattice QCD calculation of physical quantities on a given configuration demands us to solve the Dirac equation with multiple sources. We show that a new block Krylov subspace algorithm recently proposed by the authors reduces the co...