Showing papers in "Siam Review in 2021"
TL;DR: Deep learning has achieved remarkable success in diverse applications; however, its use in solving partial differential equations (PDEs) has emerged only recently as discussed by the authors, and a comprehensive overview of deep learning for PDEs can be found in Section 2.1.
Abstract: Deep learning has achieved remarkable success in diverse applications; however, its use in solving partial differential equations (PDEs) has emerged only recently. Here, we present an overview of p...
760 citations
TL;DR: The ubiquity of semilinear parabolic equations is clear from their numerous applications ranging from physics and biology to materials and social sciences as discussed by the authors, and they have been used in many applications.
Abstract: The ubiquity of semilinear parabolic equations is clear from their numerous applications ranging from physics and biology to materials and social sciences. In this paper, we consider a practically ...
88 citations
TL;DR: Schrodinger's problem represents an early example of a fundame... as discussed by the authors, which is an example of large deviations of the empirical distribution of a hot gas Gedanken experiment.
Abstract: In 1931--1932, Erwin Schrodinger studied a hot gas Gedankenexperiment (an instance of large deviations of the empirical distribution). Schrodinger's problem represents an early example of a fundame...
71 citations
TL;DR: Complex systems, composed at the most basic level of units and their interactions, describe phenomena in a wide variety of domains, from neuroscience to computer science and economics as discussed by the authors, from computer science to economics.
Abstract: Complex systems, composed at the most basic level of units and their interactions, describe phenomena in a wide variety of domains, from neuroscience to computer science and economics. The wide var...
65 citations
TL;DR: In this paper, the authors propose a graph analytics approach for better understanding and control of complex networks, but traditional methods suffer from the high computational cost and excessive memory requirements asso...
Abstract: Graph analytics can lead to better quantitative understanding and control of complex networks, but traditional methods suffer from the high computational cost and excessive memory requirements asso...
30 citations
TL;DR: In this paper, the transformation of data into an appropriate topological object (which, in our case, is a simplicial complex) is considered, which is a crucial step in the analysis of persistent homology.
Abstract: A crucial step in the analysis of persistent homology is the transformation of data into an appropriate topological object (which, in our case, is a simplicial complex) Software packages for compu
21 citations
TL;DR: In this article, the authors developed an algorithm for computing smoothed approximations of spectral measures associated with self-adjoint operators, which can achieve arbitrarily high O(n) accuracy.
Abstract: Using the resolvent operator, we develop an algorithm for computing smoothed approximations of spectral measures associated with self-adjoint operators. The algorithm can achieve arbitrarily high o...
19 citations
TL;DR: In this paper, the authors provide a strong law of large numbers for integration on digital nets randomized by a nested uniform scramble. The motivating problem is optimization over some variables of an integra...
Abstract: This article provides a strong law of large numbers for integration on digital nets randomized by a nested uniform scramble. The motivating problem is optimization over some variables of an integra...
17 citations
TL;DR: In this paper, the stationary distribution of a continuous-time Markov chain (CTMC) is computed by solving a set of linear equations, where the number of equations is either infinite or too large.
Abstract: Computing the stationary distributions of a continuous-time Markov chain (CTMC) involves solving a set of linear equations. In most cases of interest, the number of equations is infinite or too lar...
14 citations
TL;DR: In this paper, the authors present a successful deployment of high-fidelity large-eddy simulation (LES) technologies based on spectral/$hp$ element methods to industrial flow problems, which are characterized by high Reyn...
Abstract: We present a successful deployment of high-fidelity large-eddy simulation (LES) technologies based on spectral/$hp$ element methods to industrial flow problems, which are characterized by high Reyn...
12 citations
TL;DR: The trapezoidal rule as discussed by the authors uses function values at equispaced nodes, which is very accurate for integrals over periodic intervals, but is usually quite inaccurate in nonperiodic cases.
Abstract: The trapezoidal rule uses function values at equispaced nodes. It is very accurate for integrals over periodic intervals, but is usually quite inaccurate in nonperiodic cases. Commonly used improve...
TL;DR: In this paper, the authors show that the Vandermonde matrices are exponentially ill-conditioned, rendering the familiar polyval(polyfit) algorithm for polynomial interpolation and least-squares fitting ineffective at higher degrees.
Abstract: Vandermonde matrices are exponentially ill-conditioned, rendering the familiar “polyval(polyfit)” algorithm for polynomial interpolation and least-squares fitting ineffective at higher degrees. We ...
TL;DR: It has been found advantageous for finite-volume discretizations of flow equations to possess additional (secondary) invariants, besides the primary invariants from the constituting conservation as mentioned in this paper.
Abstract: It has been found advantageous for finite-volume discretizations of flow equations to possess additional (secondary) invariants, besides the (primary) invariants from the constituting conservation ...
TL;DR: In this article, the authors propose a class of models called "compartment models" that are useful when considering the flow of objects, people, or energy between different labeled states, referred to as compartments.
Abstract: Compartment models are a widely used class of models that are useful when considering the flow of objects, people, or energy between different labeled states, referred to as compartments. Classic e...
TL;DR: While students may find spline interpolation quite digestible based on their familiarity with the continuity of a function and its derivatives, some of its inherent value may be missed when they on...
Abstract: While students may find spline interpolation quite digestible based on their familiarity with the continuity of a function and its derivatives, some of its inherent value may be missed when they on...
TL;DR: The determination of the mean first passage time (MFPT) for a Brownian particle in a bounded 2D domain containing small absorbing traps is a fundamental problem with biophysical applications.
Abstract: The determination of the mean first passage time (MFPT) for a Brownian particle in a bounded 2-D domain containing small absorbing traps is a fundamental problem with biophysical applications. The ...
TL;DR: Mathieu functions of period $\pi$ or $2\pi$, also called elliptic cylinder functions, were introduced in 1868 by Emile Mathieu together with modified Mathieu functions, in order to help u...
Abstract: Mathieu functions of period $\pi$ or $2\pi$, also called elliptic cylinder functions, were introduced in 1868 by Emile Mathieu together with so-called modified Mathieu functions, in order to help u...
TL;DR: In this article, the authors analyze the dynamics of a bilateral model that depends on the simultaneous comparison between odour profiles of humans and animals, and show that it can be used to localize sources and follow odour trails.
Abstract: Animals use stereo sampling of odor concentration to localize sources and follow odor trails. We analyze the dynamics of a bilateral model that depends on the simultaneous comparison between odor c...
TL;DR: In this paper, the high speed of $x_{k}\rightarrow x^\ast\in{\mathbb R}$ was measured using the $C$-, $Q$-, or $R$-orders.
Abstract: The high speed of $x_{k}\rightarrow x^\ast\in{\mathbb R}$ is usually measured using the $C$-, $Q$-, or $R$-orders: \[ lim \tfrac |x^\ast - x_k+1||x^\ast - x_k|^p_0\in(0,+\infty), lim \tfrac ln |x^\...
TL;DR: The Rodrigues formula for the $3\times 3$ rotation matrix is hardly ever derived from first principles in a simple and intuitive way that is accessible to undergraduate students as discussed by the authors.
Abstract: The Rodrigues formula for the $3\times 3$ rotation matrix is hardly ever derived from first principles in a simple and intuitive way that is accessible to undergraduate students. Two different deri...
TL;DR: In this paper, a randomized method for computing the min-plus product (a.k.a. tropical product) of two $n \times n$ matrices was presented, yielding a faster algorithm for solving the all-pairs shortest path problem.
Abstract: We present a randomized method for computing the min-plus product (a.k.a. tropical product) of two $n \times n$ matrices, yielding a faster algorithm for solving the all-pairs shortest path problem...
TL;DR: The multivariable chain rule is often challenging to students because it is usually presented with ambiguities and other defects that hamper systematic and reliable application as discussed by the authors, but it is a very simple formu...
Abstract: The multivariable chain rule is often challenging to students because it is usually presented with ambiguities and other defects that hamper systematic and reliable application. A very simple formu...
TL;DR: In this paper, the authors consider a class of infinite-dimensional optimization problems in which a distributed vector-valued variable should pointwise almost everywhere take values from a given finite set.
Abstract: We consider a class of infinite-dimensional optimization problems in which a distributed vector-valued variable should pointwise almost everywhere take values from a given finite set $\mathcal{M}\s...
TL;DR: For students in applied mathematics courses, the phenomenon of delay-induced stability and instability offers exciting educational opportunities as mentioned in this paper, where the authors explore the onset of instability in delay-dependent systems.
Abstract: For students in applied mathematics courses, the phenomenon of delay-induced stability and instability offers exciting educational opportunities. Exploration of the onset of instability in delay di...