Showing papers in "Siam Review in 2019"
TL;DR: A new numerical technique to deal with nonlinear terms in gradient flows by introducing a scalar auxiliary variable (SAV), which can construct unconditionally second-order energy stable schemes and can easily construct even third or fourth order BDF schemes, although not unconditionally stable, which are very robust in practice.
Abstract: We propose a new numerical technique to deal with nonlinear terms in gradient flows. By introducing a scalar auxiliary variable (SAV), we construct efficient and robust energy stable schemes for a ...
364 citations
TL;DR: This article provides a very brief introduction to the basic ideas that underlie deep learning from an applied mathematics perspective and illustrates the ideas with a short MATLAB code that sets up and trains a network.
Abstract: Multilayered artificial neural networks are becoming a pervasive tool in a host of application fields. At the heart of this deep learning revolution are familiar concepts from applied and computati...
162 citations
TL;DR: In this paper, average-case error was proposed in the applied mathematics literature as an alternative criterion with which to assess numerical methods, in contrast to worst case error, this criterion was proposed as a new criterion for numerical methods.
Abstract: Over forty years ago average-case error was proposed in the applied mathematics literature as an alternative criterion with which to assess numerical methods. In contrast to worst-case error, this ...
134 citations
TL;DR: In this paper, the authors proposed a method for solving time-harmonic wave propagation problems by iterative methods, and over the last two decades an important research effort has been devoted to developing preconditioners for the wave propagation problem.
Abstract: Solving time-harmonic wave propagation problems by iterative methods is a difficult task, and over the last two decades an important research effort has gone into developing preconditioners for the...
128 citations
TL;DR: The analysis suggests that frames are a natural generalization of bases in which to develop numerical approximation, and even in the presence of severe ill-conditioning, frames impose sufficient mathematical structure so as to give rise to good accuracy in finite precision calculations.
Abstract: Functions of one or more variables are usually approximated with a basis: a complete, linearly independent system of functions that spans a suitable function space. The topic of this paper is the n...
65 citations
TL;DR: A signed network is a network in which each link is associated with a positive or negative sign as discussed by the authors, and models for nodes interacting over such signed networks arise from various biological, social, politi...
Abstract: A signed network is a network in which each link is associated with a positive or negative sign. Models for nodes interacting over such signed networks arise from various biological, social, politi...
62 citations
TL;DR: This paper presents a new generation of particle filters designed to solve a large class of nonlinear filtering problems with importance weights and shows how these filters have changed in the last decade.
Abstract: Particle filters are a popular and flexible class of numerical algorithms to solve a large class of nonlinear filtering problems. However, standard particle filters with importance weights have bee...
40 citations
TL;DR: A parallel version of the LaSalle–Bouchut–Boyaval–Seiden cellular automaton aimed at simulating the dynamics of weakly perturbed limit cycle oscillators is described.
Abstract: While phase reduction is a tremendously useful tool for understanding the dynamics of weakly perturbed limit cycle oscillators, its assumptions break down as perturbations become larger, limiting i...
36 citations
TL;DR: This paper uses an extremal problem involving rational functio... to study displacement structure of matrices with displacement structure with a variety of applications.
Abstract: Matrices with displacement structure, such as Pick, Vandermonde, and Hankel matrices, appear in a diverse range of applications. In this paper, we use an extremal problem involving rational functio...
34 citations
TL;DR: The usual way in which mathematicians work with randomness is by a rigorous formulation of the idea of Brownian motion, which is the limit of a random walk as the step length goes to zero.
Abstract: The usual way in which mathematicians work with randomness is by a rigorous formulation of the idea of Brownian motion, which is the limit of a random walk as the step length goes to zero. A Browni...
32 citations
TL;DR: This paper addresses the problem of the prohibitively large computational cost of existing Markov chain Monte Carlo methods for large-scale applications with high-dimensional parameter spaces, and proposes a new approach to solve the problem.
Abstract: In this paper we address the problem of the prohibitively large computational cost of existing Markov chain Monte Carlo methods for large-scale applications with high-dimensional parameter spaces, ...
TL;DR: Multistate dynamical processes on networks, where nodes can occupy one of a multitude of discrete states, are gaining widespread use because of their ability to recreate realistic, complex behavior as discussed by the authors.
Abstract: Multistate dynamical processes on networks, where nodes can occupy one of a multitude of discrete states, are gaining widespread use because of their ability to recreate realistic, complex behavior...
TL;DR: The mathematics behind the quality- and efficiency-driven (QED) regime is reviewed, which lets the system operate close to full utilization, while the number of servers grows simultaneously large and delays remain manageable.
Abstract: Multiserver queueing systems describe situations in which users require service from multiple parallel servers. Examples include check-in lines at airports, waiting rooms in hospitals, queues in contact centers, data buffers in wireless networks, and delayed service in cloud data centers. These are all situations with jobs (clients, patients, tasks) and servers (agents, beds, processors) that have large capacity levels, ranging from the order of tens (checkouts) to thousands (processors). This survey investigates how to design such systems to exploit resource pooling and economies-of-scale. In particular, we review the mathematics behind the quality- and efficiency-driven (QED) regime, which lets the system operate close to full utilization, while the number of servers grows simultaneously large and delays remain manageable. Aimed at a broad audience, we describe in detail the mathematical concepts for the basic Markovian many-server system, and we provide only sketches or references for more advanced settings related to, e.g., load balancing, overdispersion, parameter uncertainty, general service requirements, and queueing networks. While serving as a partial survey of a massive body of work, the tutorial is not meant to be exhaustive.
TL;DR: Multiplicative and additive stability, diagonal stability, Schur stability, and $H$-stability are classical concepts which arise in studying linear dynamical systems and are unified in this article.
Abstract: Multiplicative and additive $D$-stability, diagonal stability, Schur $D$-stability, and $H$-stability are classical concepts which arise in studying linear dynamical systems. We unify these types o...
TL;DR: Whether the three-dimensional incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics.
Abstract: Whether the three-dimensional incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics...
TL;DR: It turns out that synchrony breaking often occurs via remarkable anomalous bifurcation scenarios, and it has been noted that homogeneous networks can be realized as quotien...
Abstract: Many systems in science and technology are networks: they consist of nodes with connections between them. Examples include electronic circuits, power grids, neuronal networks, and metabolic systems...
TL;DR: This work characterize the large-sample properties of network modularity in the presence of covariates, under a natural and flexible nonparametric null model, and provides for the first time an objective measure of whether or not a particular value of modularity is meaningful.
Abstract: We characterize the large-sample properties of network modularity in the presence of covariates, under a natural and flexible null model. This provides for the first time an objective measure of wh...
TL;DR: Semidefinite programs (SDPs) are often pathological: the optimal values of the primal and dual problems may be diametric as discussed by the authors, and the optimal solution of the dual problem may be pathological.
Abstract: Semidefinite programs (SDPs)---some of the most useful and versatile optimization problems of the last few decades---are often pathological: the optimal values of the primal and dual problems may d...
TL;DR: Techniques of univariate Newton interpolating polynomials are extended to multivariate data points by different generalizations and practical algorithms.
Abstract: Techniques of univariate Newton interpolating polynomials are extended to multivariate data points by different generalizations and practical algorithms. The Newton basis format, with divided-diffe...
TL;DR: This work decomposes the problem into three steps, each of which are independently useful computational primitives: (i) enumerating non-decreasing sequences, (ii) unranking multiset permutations, and (iii) decoding and encoding z-curve and Morton codes and permutations.
Abstract: Common models for random graphs, such as Erdos--Renyi and Kronecker graphs, correspond to generating random adjacency matrices where each entry is nonzero based on a large matrix of probabilities. ...
TL;DR: Obligate mutualist interactions appear to be ubiquitous in nature but cannot be described by the simple models that have been so effective for developing the theory of other population interactions.
Abstract: Obligate mutualist interactions appear to be ubiquitous in nature but cannot be described by the simple models that have been so effective for developing the theory of other population interactions...
TL;DR: A complete theory of dimensional and scaling analysis is presented and the vector-matrix exponentiation is introduced to simplify notation and calculus.
Abstract: A complete theory of dimensional and scaling analysis is presented and its power is demonstrated through a series of examples. A vector-matrix exponentiation is introduced to simplify notation and calculus.
TL;DR: A new integral representation for the unsteady, incompressible Stokes or Navier--Stokes equations is presented, based on a linear combination of heat and harmonic potentials.
Abstract: We present a new integral representation for the unsteady, incompressible Stokes or Navier--Stokes equations, based on a linear combination of heat and harmonic potentials. For velocity boundary co...
TL;DR: This education article presents a case for first courses on Markov chain modeling to include the topic “sojourn time cycle chart" (STC chart) through a teaching module that co...
Abstract: This education article presents a case for first courses on Markov chain modeling to include the topic “sojourn time cycle chart" (STC chart). This article does so through a teaching module that co...