I will describe the development of novel, multi-tensor generalizations of the singular value decomposition, and their use in the comparisons of brain, lung, ovarian, and uterine cancer and normal genomes, to uncover patterns of DNA copy-number alterations that predict survival and response to treatment, statistically better than, and independent of, the best indicators in clinical use and existing laboratory tests. Recurring alterations have been recognized as a hallmark of cancer for over a century, and observed in these cancers' genomes for decades; however, copy-number subtypes predictive of patients' outcomes were not identified before. The data had been publicly available, but the patterns remained unknown until the data were modeled by using the multi-tensor decompositions, illustrating the universal ability of these decompositions --- generalizations of the frameworks that underlie the theoretical description of the physical world --- to find what other methods miss.

Nonlinear eigenvalue problems pose intriguing challenges for analysis and computation. For starters, a finite-dimensional problem can give infinitely many eigenvalues, quantities that reveal the asymptotic behavior of associated dynamical systems. Delay differential equations provide an especially rich source of such problems. We describe several approaches for approximating solutions to nonlinear eigenvalue problems, based on ideas originally deployed for model-order reduction: a data-driven (Loewner) rational interpolation technique, and a structure-preserving interpolatory projection method. While eigenvalues describe asymptotic dynamics, the behavior of systems on transient time scales can be more subtle. We will describe how careful use of pseudospectra can give insight about the transient behavior of solutions to delay differential equations. This talk describes collaborative work with Michael Brennan, Alex Grimm, and Serkan Gugercin.

The need to iteratively solve large and sparse saddle-point systems continues to be a challenging task in numerical linear algebra. In particular, it is important to design preconditioning techniques that take into account the numerical properties of the underlying discrete operators. In this talk we consider saddle-point matrices whose leading block has a low rank. We show that under specific assumptions on the rank and a few additional mild assumptions, the inverse has unique mathematical properties. It is possible to utilize null spaces of the leading block or the off-diagonal blocks as an alternative to Schur complements. Consequently, a family of indefinite block preconditioners can be developed. We also introduce a new minimum residual short-recurrence method for solving saddle-point systems, which is capable of handling singularity of the leading block.

This talk discusses robust preconditioners and iterative methods for solving large sparse linear systems of equations. The issues addressed include robustness as well as communication reduction for increasing the scalability of linear solvers on large scale computers. The focus is in particular on enlarged Krylov subspace methods and preconditioners based on low rank corrections, as well as associated computational kernels as computing a low rank approximation of a sparse matrix. The efficiency of the proposed methods is tested on matrices arising from linear elasticity problems as well as convection diffusion problems with highly heterogeneous coefficients.

Kaczmarz's and Cimmino's methods are examples of algebraic iterative reconstruction methods, primarily used to solve discretized inverse problems in computed tomography. They are very flexible because the underlying system $A x = b$ requires no assumption about the scanning geometry, and it is easy to incorporate convex constraints (e.g., box constraints). Their success in computing regularized solutions is due to a mechanism called semi-convergence.

While the asymptotic convergence for noise-free data is well understood, there are surprising few theoretical results related to the convergence for real-world problems with noisy data and model errors. For the same reason, we lack efficient and robust stopping rules that terminate the iterations at the point of semi-convergence.

In this talk I will survey some recent results related to the convergence and semi-convergence of various algebraic iterative reconstruction methods, both (block) row and (block) column versions, and I will illustrate the results with numerical results.

The development of efficient numerical algorithms for solving large-scale linear systems is one of the success stories of numerical linear algebra that has had a tremendous impact on our ability to perform complex numerical simulations and large-scale statistical computations. Many of these developments are based on multilevel and domain decomposition techniques, which are intimately linked to Schur complements and low-rank updates of matrices. These tools do not carry over in a direct manner to other important linear algebra problems, including matrix functions and matrix equations. In this talk, we describe a new framework for performing low-rank updates of matrix functions. This allows to address a wide variety of matrix functions and matrix structures, including sparse matrices as well as matrices with hierarchical low rank and Toeplitz-like structures. The versality of this framework will be demonstrated with several applications and extensions. This talk is primarily based on joint work with Bernhard Beckermann and Marcel Schweitzer.

Linear matrix equations arise in an amazingly growing number of applications. Classically, they have been extensively encountered in Control theory and eigenvalue problems. More recently they have been shown to provide a natural platform in the discretization of certain partial differential equations (PDEs), both in the deterministic setting, and in the presence of uncertainty in the data.

We first review some numerical techniques for solving various classes of large scale linear matrix equations commonly occurring in applications. Then we focus on recent developments in the solution of (systems of) linear matrix equations associated with the numerical treatment of various stochastic PDE problems.

The 2D eigenvalue problem (2dEVP) is a class of the double eigenvalue problems first studied by Blum and Chang in 1970s. The 2dEVP seeks scalars \lambda, \mu, and a corresponding vector x satisfying the following equations
Ax = \lambda x + \mu Cx,
x^HCx=0,
x^Hx=1,
where A and C are Hermitian and C is indefinite. We show the connections between 2dEVP with well-known numerical linear algebra and optimization problems such as quadratic programming, the distance to instability and H_\infty-norm. We will discuss (1) fundamental properties of 2dEVP including well-posedness, types and regularity, (2) perturbation theory and (3) numerical algorithms with backward error analysis.

Algebraic multigrid (AMG) is an efficient solver for large-scale scientific computing and an essential component of many simulation codes. However, with single-core speeds plateauing, future increases in computing performance have to rely on more complicated, often heterogenous computer architectures, which provide new challenges for efficient implementations of AMG. How one views the linear system, e.g. in terms of structured grids and stencils, or a traditional matrix-vector system, can significantly affect the design and performance of AMG. Structured AMG can take advantage of additional information in the structured matrix data structures, potentially leading to more efficient implementations but is confined to structured problems, whereas unstructured AMG can be applied to more general problems. We will discuss these methods, their implementation and performance and introduce a new semi-structured multigrid method that can take advantage of the structured parts of a problem, but is capable to solve more general problems.

Interpolative decomposition is a simple and yet powerful tool for approximating low-rank matrices. After discussing the theory and algorithm, we will present a few new applications of interpolative decomposition in numerical linear algebra, partial differential equations, quantum chemistry, and machine learning.