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 new alternative implementations of the weighted Arnoldi algorithm which may be favorable in terms of computational complexity, and examine stability issues connected with these implementations. Two 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

Solving Eigenproblems with application in collapsible channel flows

Collapsible channel flows have been attracting the interest of many researchers, because of the physiological applications in the cardiovascular system, the respiratory system and urinary system. The linear stability analysis of the collapsible channel flows in the Fluid-Beam Model can be finalized as a large sparse asymmetric generalized eigenvalue problem, where the stiffness matrix is sparse, asymmetric and nonsingular, and the mass matrix is sparse, asymmetric and singular. The dimensions of the both matrices can reach about ten thousand or more, and the traditional QZ Algorithm is so expensive for this size of eigenvalue problem, due to its large requirement of computational resources and the quite long elapsed time. Unlike the traditional direct methods, the projection methods are much more efficient for solving some specified eigenpairs of the large scale eigenvalue problems, because normally a small subspace is made use of, and the original eigenvalue problem is projected to this small subspace. With this projection, the size of the eigenvalue problem is reduced significantly, and then the small dimensional eigenvalue problem can be easily and rapidly worked out by employing a traditional solver. Combined with a restarting strategy, this can be used to solve large dimensional eigenvalue problem much more rapidly and precisely. So far as we know, the Implicitly Restarted Arnoldi iteration(IRA) is considered as one of the most effective asymmetric eigenvalue solvers. In order to improve the efficiency of linear stability analysis in collapsible channel flows, an IRA method is employed to the linear stability analysis of collapsible channel flows in FBM. A Frontal Solver, which is an efficient solver of large sparse linear system, is also used to replace the process of shift-and-invert transformation. After applying these two efficient solvers, the new eigenvalue solver of collapsible channel flows---Arnoldi method with a Frontal Solver(AR-F), not only gets rid of the restriction of memory storage, but also reduces the computational time observably. Some validating and testing work have been done to variety of meshes. The AR-F can solve the eigenvalues with largest real parts very quickly, and can also solve the large scale eigenvalue problems, which cannot be solved by the QZ Algorithm, whose results have been proved to be correct with the unsteady simulations. Compared with the traditional QZ Algorithm, not only a great deal of elapsed time is saved, but also the increasing rate of the operation numbers is dropped to $O(n)$ from $O(n^3)$ of QZ Algorithm. With the powerful AR-F, the stability problems of refined meshes in collapsible channel flows are no long a barrier to the study. So AR-F is used to solve the eigenvalue problems from two refined meshes of the two different boundary conditions(pressure-driven system and flow-driven system), and the two neutral curves obtained are both revised and extended. This is the first time that IRA is made use of in the problem of fluid-structure interaction, and this is also a critical footstone to adopt a three dimensional model over FBM. Recently, the energy analysis and the energetics are the centre of research in collapsible channel flow. Because the linear stability analysis is much more accurate and faster than the unsteady simulation, the energy solutions from eigenpairs are also achieved in this thesis. The energy analysis with eigenpairs has its own advantages: the accuracy, the timing, the division, any mode and any point. In order to analyze the energy from eigenpairs much more clearly, the energy results with different initial solutions are presented first, then the energy solutions with eigenpairs are validated with those presented by Liu et al. in the pressure-driven system. By using the energy analysis with eigenpairs, much more energy results in flow-dirven system are obtained and analyzed

Refined isogeometric analysis for generalized Hermitian eigenproblems

We use refined isogeometric analysis (rIGA) to solve generalized Hermitian eigenproblems (Ku = λMu). rIGA conserves the desirable properties of maximum-continuity isogeometric analysis (IGA) while it reduces the solution cost by adding zero-continuity basis functions, which decrease the matrix connectivity. As a result, rIGA enriches the approximation space and reduces the interconnection between degrees of freedom. We compare computational costs of rIGA versus those of IGA when employing a Lanczos eigensolver with a shift-and-invert spectral transformation. When all eigenpairs within a given interval [λ_s,λ_e] are of interest, we select several shifts σ_k ∈ [λ_s,λ_e] using a spectrum slicing technique. For each shift σ_k, the factorization cost of the spectral transformation matrix K − σ_k M controls the total computational cost of the eigensolution. Several multiplications of the operator matrix (K − σ_k M)^−1 M by vectors follow this factorization. Let p be the polynomial degree of the basis functions and assume that IGA has maximum continuity of p−1. When using rIGA, we introduce C^0 separators at certain element interfaces to minimize the factorization cost. For this setup, our theoretical estimates predict computational savings to compute a fixed number of eigenpairs of up to O(p^2) in the asymptotic regime, that is, large problem sizes. Yet, our numerical tests show that for moderate-size eigenproblems, the total observed computational cost reduction is O(p). In addition, rIGA improves the accuracy of every eigenpair of the first N_0 eigenvalues and eigenfunctions, where N_0 is the total number of modes of the original maximum-continuity IGA discretization

Algorithms for Large Scale Problems in Eigenvalue and Svd Computations and in Big Data Applications

As ”big data” has increasing influence on our daily life and research activities, it poses significant challenges on various research areas. Some applications often demand a fast solution of large, sparse eigenvalue and singular value problems; In other applications, extracting knowledge from large-scale data requires many techniques such as statistical calculations, data mining, and high performance computing. In this dissertation, we develop efficient and robust iterative methods and software for the computation of eigenvalue and singular values. We also develop practical numerical and data mining techniques to estimate the trace of a function of a large, sparse matrix and to detect in real-time blob-filaments in fusion plasma on extremely large parallel computers. In the first work, we propose a hybrid two stage SVD method for efficiently and accurately computing a few extreme singular triplets, especially the ones corresponding to the smallest singular values. The first stage achieves fast convergence while the second achieves the final accuracy. Furthermore, we develop a high-performance preconditioned SVD software based on the proposed method on top of the state-of-the-art eigensolver PRIMME. The method can be used with or without preconditioning, on parallel computers, and is superior to other state-of-the-art SVD methods in both efficiency and robustness. In the second study, we provide insights and develop practical algorithms to accomplish efficient and accurate computation of interior eigenpairs using refined projection techniques in non-Krylov iterative methods. By analyzing different implementations of the refined projection, we propose a new hybrid method to efficiently find interior eigenpairs without compromising accuracy. Our numerical experiments illustrate the efficiency and robustness of the proposed method. In the third work, we present a novel method to estimate the trace of matrix inverse that exploits the pattern correlation between the diagonal of the inverse of the matrix and that of some approximate inverse. We leverage various sampling and fitting techniques to fit the diagonal of the approximation to that of the inverse. Our method may serve as a standalone kernel for providing a fast trace estimate or as a variance reduction method for Monte Carlo in some cases. An extensive set of experiments demonstrate the potential of our method. In the fourth study, we provide first results on applying outlier detection techniques to effectively tackle the fusion blob detection problem on extremely large parallel machines. We present a real-time region outlier detection algorithm to efficiently find and track blobs in fusion experiments and simulations. Our experiments demonstrated we can achieve linear time speedup up to 1024 MPI processes and complete blob detection in two or three milliseconds

Inexact and Nonlinear Extensions of the FEAST Eigenvalue Algorithm

Eigenvalue problems are a basic element of linear algebra that have a wide variety of applications. Common examples include determining the stability of dynamical systems, performing dimensionality reduction on large data sets, and predicting the physical properties of nanoscopic objects. Many applications require solving large dimensional eigenvalue problems, which can be very challenging when the required number of eigenvalues and eigenvectors is also large. The FEAST algorithm is a method of solving eigenvalue problems that allows one to calculate large numbers of eigenvalue/eigenvector pairs by using contour integration in the complex plane to divide the large number of desired pairs into many small groups; these small groups of eigenvalue/eigenvector pairs may then be simultaneously calculated independently of each other. This makes it possible to quickly solve eigenvalue problems that might otherwise be very difficult to solve efficiently. The standard FEAST algorithm can only be used to solve eigenvalue problems that are linear, and whose matrices are small enough to be factorized efficiently (thus allowing linear systems of equations to be solved exactly). This limits the size and the scope of the problems to which the FEAST algorithm may be applied. This dissertation describes extensions of the standard FEAST algorithm that allow it to efficiently solve nonlinear eigenvalue problems, and eigenvalue problems whose matrices are large enough that linear systems of equations can only be solved inexactly