Computing the extremal eigenvalue bounds of interval matrices is non-deterministic polynomial-time (NP)-hard. We investigate bounds on real eigenvalues of real symmetric tridiagonal interval matrices and prove that for a given real symmetric tridiagonal interval matrices, we can achieve its exact range of the smallest and largest eigenvalues just by computing extremal eigenvalues of four symmetric tridiagonal matrices.

Pseudospectra and structured pseudospectra are important tools for the analysis of matrices. Their computation, however, can be very demanding for all but small matrices. A new approach to compute approximations of pseudospectra and structured pseudospectra, based on determining the spectra of many suitably chosen rank-one or projected rank-one perturbations of the given matrix is proposed. The choice of rank-one or projected rank-one perturbations is inspired by Wilkinson's analysis of eigenvalue sensitivity. Numerical examples illustrate that the proposed approach gives much better insight into the pseudospectra and structured pseudospectra than random or structured random rank-one perturbations with lower computational burden. The latter approach is presently commonly used for the determination of structured pseudospectra.

Adaptive principal component analysis is prohibitively expensive when a large-scale data matrix must be updated frequently. Therefore, we consider the truncated URV decomposition that allows faster updates to its approximation to the singular value decomposition while still producing a good enough approximation to recover principal components. Specifically, we suggest an efficient algorithm for the truncated URV decomposition when a rank 1 matrix updates the data matrix. After the algorithm development, the truncated URV decomposition is successfully applied to the template tracking problem in a video sequence proposed by Matthews et al. [IEEE Trans. Pattern Anal. Mach. Intell., 26:810-815 2004], which requires computation of the principal components of the augmented image matrix at every iteration. From the template tracking experiments, we show that, in adaptive applications, the truncated URV decomposition maintains a good approximation to the principal component subspace more efficiently than other procedures.

Coarsening is a crucial component of algebraic multigrid (AMG) methods for iteratively solving sparse linear systems arising from scientific and engineering applications. Its application largely determines the complexity of the AMG iteration operator. Usually, high operator complexities lead to fast convergence of the AMG method; however, they require additional memory and as such do not scale as well in parallel computation. In contrast, although low operator complexities improve parallel scalability, they often lead to deterioration in convergence. This study introduces a new type of coarsening strategy called *algebraic interface-based coarsening* that yields a better balance between convergence and complexity for a class of multi-scale sparse matrices. Numerical results for various model-type problems and a radiation hydrodynamics practical application are provided to show the effectiveness of the proposed AMG solver.

The purpose of this article is to propose algebraic techniques to solve symmetric complex-valued systems. In particular, we focus on the systems which arise from the modeling of electromagnetic fields in the frequency domain. These systems usually present a very ill-conditioned number, so specific preconditioning techniques are supposed to be used. Finally, both analytical and numerical reference tests are proposed on the simple case of the magnetic field produced by a solenoid containing conductive or non-conducting materials.

We propose a residual based sparse approximate inverse (RSAI) preconditioning procedure, for the large sparse linear system *A**x*=*b*. Different from the SParse Approximate Inverse (SPAI) algorithm proposed by Grote and Huckle (SIAM Journal on Scientific Computing, 18 (1997), pp. 838–853.), RSAI uses only the *dominant* other than *all* the information on the current residual and augments sparsity patterns adaptively during loops. In order to control the sparsity of *M*, we develop two practical algorithms RSAI(*f**i**x*) and RSAI(*t**o**l*). RSAI(*f**i**x*) retains the prescribed number of large nonzero entries and adjusts their positions in each column of *M* among *all* available ones, in which the number of large entries is increased by a fixed number at each loop. In contrast, the existing indices of *M* by SPAI are untouched in subsequent loops and a few most profitable indices are added to each column of *M* from the new candidates in the next loop. RSAI(*t**o**l*) is a tolerance based dropping algorithm and retains all large entries by dynamically dropping small ones below some tolerances, and it better suits for the problem where the numbers of large entries in the columns of *A*^{−1} differ greatly. When the two preconditioners *M* have almost the same or comparable numbers of nonzero entries, the numerical experiments on real-world problems demonstrate that RSAI(*f**i**x*) is highly competitive with SPAI and can outperform the latter for some problems. We also make comparisons of RSAI(*f**i**x*), RSAI(*t**o**l*), and power sparse approximate inverse(*t**o**l*) proposed Jia and Zhu (Numerical Linear Algebra with Applications, 16 (2009), pp.259–299.) and incomplete LU factorization type methods and draw some general conclusions.

Large-scale scientific computing models are needed for the simulation of wave propagation especially for multiple frequency and high-frequency models in complex heterogeneous media. Multigrid methods provide efficient iterative solvers for many large sign-definite systems of equations resulting from physical models. Time-harmonic wave propagation models lead to sign-indefinite systems with eigenvalues in the left half of the complex plane. Thus standard multigrid approaches applied in conjunction with a low-order finite difference or finite element method are not sufficient. In this work, we describe a high-order finite element method model for multiple (low to high) frequency time-harmonic acoustic wave propagation on general curved, non-convex, and non-smooth domains with heterogeneous media using a multigrid approximation of the shifted Laplacian operator as a preconditioner. We implement the model using an efficient geometric multigrid approach with parallel grid transfer operator calculations to simulate the model using the BiCGStab iterative solver. We demonstrate the efficiency and parallel performance of the computational model with multiple low (5 wavelength) to high-frequency (100 wavelength) input incident waves. Copyright © 2016 John Wiley & Sons, Ltd.

The incompressible. Stokes equations are a widely used model of viscous or tightly confined flow in which convection effects are negligible. In order to strongly enforce the conservation of mass at the element scale, special discretization techniques must be employed. In this paper, we consider a discontinuous Galerkin approximation in which the velocity field is *H*(div,Ω)-conforming and divergence-free, based on the Brezzi, Douglas, and Marini finite-element space, with complementary space (*P*_{0}) for the pressure. Because of the saddle-point structure and the nature of the resulting variational formulation, the linear systems can be difficult to solve. Therefore, specialized preconditioning strategies are required in order to efficiently solve these systems. We compare the effectiveness of two families of preconditioners for saddle-point systems when applied to the resulting matrix problem. Specifically, we consider block-factorization techniques, in which the velocity block is preconditioned using geometric multigrid, as well as fully coupled monolithic multigrid methods. We present parameter study data and a serial timing comparison, and we show that a monolithic multigrid preconditioner using Braess–Sarazin style relaxation provides the fastest time to solution for the test problem considered. Copyright © 2016 John Wiley & Sons, Ltd.

This paper proposes a new, low-communication algorithm for solving PDEs on massively parallel computers. The range decomposition (RD) algorithm exposes coarse-grain parallelism by applying nested iteration and adaptive mesh refinement locally before performing a global communication step. Just a few such steps are observed to be sufficient to obtain accuracy within a small multiple of discretization error. The target applications are petascale and exascale machines, where hierarchical parallelism is required and traditional parallel numerical PDE communication patterns are costly because of message latency. The RD algorithm uses a partition of unity to equally distribute the error, and thus, the work. The computational advantages of this approach are that the decomposed problems can be solved in parallel without any communication until the partitioned solutions are summed. This offers potential advantages in the paradigm of expensive communication but very cheap computation. This paper introduces the method and explains the details of the communication step. Two performance models are developed, showing that the latency cost associated with a traditional parallel implementation of nested iteration is proportional to *l**o**g*(*P*)^{2}, whereas the RD method reduces the communication latency to *l**o**g*(*P*), while maintaining similar bandwidth costs. Numerical results for two problems, Laplace and advection diffusion, demonstrate the enhanced performance, and a heuristic argument explains why the method converges quickly. Copyright © 2016 John Wiley & Sons, Ltd.

The following paper discusses the application of a multigrid-in-time scheme to Least Squares Shadowing (LSS), a novel sensitivity analysis method for chaotic dynamical systems. While traditional sensitivity analysis methods break down for chaotic dynamical systems, LSS is able to compute accurate gradients. Multigrid is used because LSS requires solving a very large Karush–Kuhn–Tucker system constructed from the solution of the dynamical system over the entire time interval of interest. Several different multigrid-in-time schemes are examined, and a number of factors were found to heavily influence the convergence rate of multigrid-in-time for LSS. These include the iterative method used for the smoother, how the coarse grid system is formed and how the least squares objective function at the center of LSS is weighted. Copyright © 2014 John Wiley & Sons, Ltd.

No abstract is available for this article.

]]>A poroelastic saturated medium can be modeled by means of Biot's theory of consolidation. It describes the time-dependent interaction between the deformation of porous material and the fluid flow inside of it. Here, for the efficient solution of the poroelastic equations, a multigrid method is employed with an Uzawa-type iteration as the smoother. The Uzawa smoother is an equation-wise procedure. It shall be interpreted as a combination of the symmetric Gauss-Seidel smoothing for displacements, together with a Richardson iteration for the Schur complement in the pressure field. The Richardson iteration involves a relaxation parameter which affects the convergence speed, and has to be carefully determined. The analysis of the smoother is based on the framework of local Fourier analysis (LFA) and it allows us to provide an analytic bound of the smoothing factor of the Uzawa smoother as well as an optimal value of the relaxation parameter. Numerical experiments show that our upper bound provides a satisfactory estimate of the exact smoothing factor, and the selected relaxation parameter is optimal. In order to improve the convergence performance, the acceleration of multigrid by iterant recombination is taken into account. Numerical results confirm the efficiency and robustness of the acceleration scheme.

In data science, data are often represented by using an undirected graph where vertices represent objects and edges describe a relationship between two objects. In many applications, there can be many relations arising from different sources and/or different types of models. Clustering of multiple undirected graphs over the same set of vertices can be studied. Existing clustering methods of multiple graphs involve costly optimization and/or tensor computation. In this paper, we study block spectral clustering methods for these multiple graphs. The main contribution of this paper is to propose and construct block Laplacian matrices for clustering of multiple graphs. We present a novel variant of the Laplacian matrix called the block intra-normalized Laplacian and prove the conditions required for zero eigenvalues in this variant. We also show that eigenvectors of the constructed block Laplacian matrix can be shown to be solutions of the relaxation of multiple graphs cut problems, and the lower and upper bounds of the optimal solutions of multiple graphs cut problems can also be established. Experimental results are given to demonstrate that the clustering accuracy and the computational time of the proposed method are better than those of tested clustering methods for multiple graphs.

A variant of the simpler GMRES method is developed for solving shifted linear systems (SGMRES-Sh), exhibiting almost the same advantage of the simpler GMRES method over the regular GMRES method. Because the remedy adapted by GMRES-Sh is no longer feasible for SGMRES-Sh due to the differences between simpler GMRES and GMRES for constructing the residual vectors of linear systems, we take an alternative strategy to force the residual vectors of the add system also be orthogonal to the subspaces, to which the residual vectors of the seed system are orthogonal when the seed system is solved with the simpler GMRES method. In addition, a seed selection strategy is also employed for solving the rest non-converged linear systems. Furthermore, an adaptive version of SGMRES-Sh is presented for the purpose of improving the stability of SGMRES-Sh based on the technique of the adaptive choice of the Krylov subspace basis developed for the adaptive simpler GMRES. Numerical experiments demonstrate the benefits of the presented methods.

In the present paper, we describe an adaptive modified rational global Lanczos algorithm for model-order reduction problems using multipoint moment matching-based methods. The major problem of these methods is the selection of some interpolation points. We first propose a modified rational global Lanczos process and then we derive Lanczos-like equations for the global case. Next, we propose adaptive techniques for choosing the interpolation points. Second-order dynamical systems are also considered in this paper, and the adaptive modified rational global Lanczos algorithm is applied to an equivalent state space model. Finally, some numerical examples will be given.