Nonlinear eigenvalue problems arise in a variety of science and engineering applications, and in the past ten years there have been numerous breakthroughs in the development of numerical methods. This article surveys nonlinear eigenvalue problems associated with matrix-valued functions which depend nonlinearly on a single scalar parameter, with a particular emphasis on their mathematical properties and available numerical solution techniques. Solvers based on Newton’s method, contour integration and sampling via rational interpolation are reviewed. Problems of selecting the appropriate parameters for each of the solver classes are discussed and illustrated with numerical examples. This survey also contains numerous MATLAB code snippets that can be used for interactive exploration of the discussed methods.

This survey provides an introduction to the use of randomization in the design of fast algorithms for numerical linear algebra. These algorithms typically examine only a subset of the input to solve basic problems approximately, including matrix multiplication, regression and low-rank approximation. The survey describes the key ideas and gives complete proofs of the main results in the field. A central unifying idea is sampling the columns (or rows) of a matrix according to their squared lengths.

We review the construction and analysis of numerical methods for strongly nonlinear PDEs, with an emphasis on convex and non-convex fully nonlinear equations and the convergence to viscosity solutions. We begin by describing a fundamental result in this area which states that stable, consistent and monotone schemes converge as the discretization parameter tends to zero. We review methodologies to construct finite difference, finite element and semi-Lagrangian schemes that satisfy these criteria, and, in addition, discuss some rather novel tools that have paved the way to derive rates of convergence within this framework.

The structure from motion (SfM) problem in computer vision is to recover the three-dimensional (3D) structure of a stationary scene from a set of projective measurements, represented as a collection of two-dimensional (2D) images, via estimation of motion of the cameras corresponding to these images. In essence, SfM involves the three main stages of (i) extracting features in images (e.g. points of interest, lines, etc.) and matching these features between images, (ii) camera motion estimation (e.g. using relative pairwise camera positions estimated from the extracted features), and (iii) recovery of the 3D structure using the estimated motion and features (e.g. by minimizing the so-called reprojection error). This survey mainly focuses on relatively recent developments in the literature pertaining to stages (ii) and (iii). More specifically, after touching upon the early factorization-based techniques for motion and structure estimation, we provide a detailed account of some of the recent camera location estimation methods in the literature, followed by discussion of notable techniques for 3D structure recovery. We also cover the basics of the simultaneous localization and mapping (SLAM) problem, which can be viewed as a specific case of the SfM problem. Further, our survey includes a review of the fundamentals of feature extraction and matching (i.e. stage (i) above), various recent methods for handling ambiguities in 3D scenes, SfM techniques involving relatively uncommon camera models and image features, and popular sources of data and SfM software.

Mathematical and numerical modelling of the cardiovascular system is a research topic that has attracted remarkable interest from the mathematical community because of its intrinsic mathematical difficulty and the increasing impact of cardiovascular diseases worldwide. In this review article we will address the two principal components of the cardiovascular system: arterial circulation and heart function. We will systematically describe all aspects of the problem, ranging from data imaging acquisition, stating the basic physical principles, analysing the associated mathematical models that comprise PDE and ODE systems, proposing sound and efficient numerical methods for their approximation, and simulating both benchmark problems and clinically inspired problems. Mathematical modelling itself imposes tremendous challenges, due to the amazing complexity of the cardiocirculatory system, the multiscale nature of the physiological processes involved, and the need to devise computational methods that are stable, reliable and efficient. Critical issues involve filtering the data, identifying the parameters of mathematical models, devising optimal treatments and accounting for uncertainties. For this reason, we will devote the last part of the paper to control and inverse problems, including parameter estimation, uncertainty quantification and the development of reduced-order models that are of paramount importance when solving problems with high complexity, which would otherwise be out of reach.

This paper provides an overview of AMG methods for solving large-scale systems of equations, such as those from discretizations of partial differential equations. AMG is often understood as the acronym of ‘algebraic multigrid’, but it can also be understood as ‘abstract multigrid’. Indeed, we demonstrate in this paper how and why an algebraic multigrid method can be better understood at a more abstract level. In the literature, there are many different algebraic multigrid methods that have been developed from different perspectives. In this paper we try to develop a unified framework and theory that can be used to derive and analyse different algebraic multigrid methods in a coherent manner. Given a smoother $R$ for a matrix $A$, such as Gauss–Seidel or Jacobi, we prove that the optimal coarse space of dimension $n_{c}$ is the span of the eigenvectors corresponding to the first $n_{c}$ eigenvectors $\bar{R}A$ (with $\bar{R}=R+R^{T}-R^{T}AR$). We also prove that this optimal coarse space can be obtained via a constrained trace-minimization problem for a matrix associated with $\bar{R}A$, and demonstrate that coarse spaces of most existing AMG methods can be viewed as approximate solutions of this trace-minimization problem. Furthermore, we provide a general approach to the construction of quasi-optimal coarse spaces, and we prove that under appropriate assumptions the resulting two-level AMG method for the underlying linear system converges uniformly with respect to the size of the problem, the coefficient variation and the anisotropy. Our theory applies to most existing multigrid methods, including the standard geometric multigrid method, classical AMG, energy-minimization AMG, unsmoothed and smoothed aggregation AMG and spectral AMGe.