A black-box rational Arnoldi variant for Cauchy-Stieltjes matrix functions

Rational Arnoldi is a powerful method for approximating functions of large sparse matrices times a vector. The selection of asymptotically optimal parameters for this method is crucial for its fast convergence. We present and investigate a novel strategy for the automated parameter selection when the function to be approximated is of Cauchy-Stieltjes (or Markov) type, such as the matrix square root or the logarithm. The performance of this approach is demonstrated by numerical examples involving symmetric and nonsymmetric matrices. These examples suggest that our black-box method performs at least as well, and typically better, as the standard rational Arnoldi method with parameters being manually optimized for a given matrix

Robust Padé approximation via SVD

Padé approximation is considered from the point of view of robust methods of numerical linear algebra, in particular the singular value decomposition. This leads to an algorithm for practical computation that bypasses most problems of solution of nearly-singular systems and spurious pole-zero pairs caused by rounding errors; a Matlab code is provided. The success of this algorithm suggests that there might be variants of Padé approximation that would be pointwise convergent as the degrees of the numerator and denominator increase to infinity, unlike traditional Padé approximants, which converge only in measure or capacity

Zolotarev Quadrature Rules and Load Balancing for the FEAST Eigensolver

The FEAST method for solving large sparse eigenproblems is equivalent to subspace iteration with an approximate spectral projector and implicit orthogonalization. This relation allows to characterize the convergence of this method in terms of the error of a certain rational approximant to an indicator function. We propose improved rational approximants leading to FEAST variants with faster convergence, in particular, when using rational approximants based on the work of Zolotarev. Numerical experiments demonstrate the possible computational savings especially for pencils whose eigenvalues are not well separated and when the dimension of the search space is only slightly larger than the number of wanted eigenvalues. The new approach improves both convergence robustness and load balancing when FEAST runs on multiple search intervals in parallel.Comment: 22 pages, 8 figure

Efficient high-order rational integration and deferred correction with equispaced data

Stable high-order linear interpolation schemes are well suited for the accurate approximation of antiderivatives and the construction of efficient quadrature rules. In this paper we utilize for this purpose the family of linear barycentric rational interpolants by Floater and Hormann, which are particularly useful for interpolation with equispaced nodes. We analyze the convergence of integrals of these interpolants to those of analytic functions as well as functions with a finite number of continuous derivatives. As a by-product, our convergence analysis leads to an extrapolation scheme for rational quadrature at equispaced nodes. Furthermore, as a main application of our analysis, we present and investigate a new iterated deferred correction method for the solution of initial value problems, which allows to work efficiently even with large numbers of equispaced data. This so-called rational deferred correction (RDC) method turns out to be highly competitive with other methods relying on more involved implementations or non-equispaced node distributions. Extensive numerical experiments are carried out, comparing the RDC method to the well established spectral deferred correction (SDC) method by Dutt, Greengard and Rokhlin

GMRES with randomized sketching and deflated restarting

We present a new Krylov subspace recycling method for solving a linear system of equations, or a sequence of slowly changing linear systems. Our approach is to reduce the computational overhead of recycling techniques while still benefiting from the acceleration afforded by such techniques. As such, this method augments an unprojected Krylov subspace. Furthermore, it combines randomized sketching and deflated restarting in a way that avoids orthogononalizing a full Krylov basis. We call this new method GMRES-SDR (sketched deflated restarting). With this new method, we provide new theory, which initially characterizes unaugmented sketched GMRES as a projection method for which the projectors involve the sketching operator. We demonstrate that sketched GMRES and its sibling method sketched FOM are an MR/OR pairing, just like GMRES and FOM. We furthermore obtain residual convergence estimates. Building on this, we characterize GMRES-SDR also in terms of sketching-based projectors. Compression of the augmented Krylov subspace for recycling is performed using a sketched version of harmonic Ritz vectors. We present results of numerical experiments demonstrating the effectiveness of GMRES-SDR over competitor methods such as GMRES-DR and GCRO-DR

Nationality and Colonial Strategies: Germany and America – How the American Expansion Resonated in Germany

We all tend to see what we want to see — in ourselves, in our friends, in our culture, and in other cultures. In his dissertation, Jens-Uwe Guettel takes a penetrating look at how Germany viewed America over the course of the 19th century, the period of America’s great expansion westward. In the following interview and excerpt, you will find highlights of Prof. Guettel’s wide-ranging consideration of the many authors, themes and images which were part of this cultural “moment.” In the dissertation itself, you will find a deeper look at the novels and writings which reflect the complex attitudes and ideas of the times. Germans certainly noticed what Americans were doing as they expanded the nation westward, but not always the same we saw ourselves. What makes this dissertation so explosive (to me, anyway) is what comes next – what is off-screen, so to speak. When Prof. Guettel brings up the concept of lebensraum, we realize that his thesis is by no means an obscure topic of study: the colonial attitudes of the 19th century can be seen to lead directly to the German nationalism of the modern era and to the rise of the Third Reich. Most certainly, German views of American colonialism formed the roots of the two world wars which dominated the 20th century. Understanding the deeper cultural roots of war is important to all of us. As I write this, our entire nation is at war – two wars, actually – and each and every citizen is part of that decision. We need to understand why these conflicts have happened in the past, are happening today, and may break out again soon

Some observations on weighted GMRES

We investigate the convergence of the weighted GMRES method for solving linear systems. Two different weighting variants are compared with unweighted GMRES for three model problems, giving a phenomenological explanation of cases where weighting improves convergence, and a case where weighting has no effect on the convergence. We also present a new alternative implementation of the weighted Arnoldi algorithm which under known circumstances will be favourable in terms of computational complexity. These implementations of weighted GMRES are compared for a large number of examples. We find that weighted GMRES may outperform unweighted GMRES for some problems, but more often this method is not competitive with other Krylov subspace methods like GMRES with deflated restarting or BICGSTAB, in particular when a preconditioner is used