Showing papers in "Siam Review in 2014"
TL;DR: Smart grid goals include a commitment to large penetration of highly fluctuating renewables, thus calling to reconsider current practices, in particular the use of standard OPF, which can lead to frequent conditions where power line flow ratings are significantly exceeded.
Abstract: When uncontrollable resources fluctuate, optimal power flow (OPF), routinely used by the electric power industry to redispatch hourly controllable generation (coal, gas, and hydro plants) over control areas of transmission networks, can result in grid instability and, potentially, cascading outages. This risk arises because OPF dispatch is computed without awareness of major uncertainty, in particular fluctuations in renewable output. As a result, grid operation under OPF with renewable variability can lead to frequent conditions where power line flow ratings are significantly exceeded. Such a condition, which is borne by our simulations of real grids, is considered undesirable in power engineering practice. Possibly, it can lead to a risky outcome that compromises grid stability---line tripping. Smart grid goals include a commitment to large penetration of highly fluctuating renewables, thus calling to reconsider current practices, in particular the use of standard OPF. Our chance-constrained (CC) OPF co...
504 citations
TL;DR: In this paper, a general class of models for self-organized dynamics based on alignment is reviewed, and a natural question which arises in this context is to ask when and how clusters emerge through the self-alignment of agents and what types of "rules of engagement" influence the formation of such clusters.
Abstract: We review a general class of models for self-organized dynamics based on alignment. The dynamics of such systems is governed solely by interactions among individuals or “agents,” with the tendency to adjust to their “environmental averages.” This, in turn, leads to the formation of clusters, e.g., colonies of ants, flocks of birds, parties of people, rendezvous in mobile networks, etc. A natural question which arises in this context is to ask when and how clusters emerge through the self-alignment of agents, and what types of “rules of engagement” influence the formation of such clusters. Of particular interest to us are cases in which the self-organized behavior tends to concentrate into one cluster, reflecting a consensus of opinions, flocking of birds, fish, or cells, rendezvous of mobile agents, and, in general, concentration of other traits intrinsic to the dynamics. Many standard models for self-organized dynamics in social, biological, and physical sciences assume that the intensity of alignment in...
482 citations
TL;DR: It is shown that far from being a curiosity, the trapezoidal rule is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators.
Abstract: It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed, and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators.
481 citations
TL;DR: Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance as mentioned in this paper, which is useful in several applications where the input data consist of an incomplete set of distances and the output is a set of points in Euclidian space realizing those given distances.
Abstract: Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the input data consist of an incomplete set of distances and the output is a set of points in Euclidean space realizing those given distances. We survey the theory of Euclidean distance geometry and its most important applications, with special emphasis on molecular conformation problems.
345 citations
TL;DR: Recent developments in the non-standard asymptotics of the narrow escape problem are reviewed, which are based on several ingredients: a better resolution of the singularity of Neumann's function,resolution of the boundary layer near the small target by conformal mappings of domains with bottlenecks, and the breakup of composite domains into simpler components.
Abstract: The narrow escape problem in diffusion theory is to calculate the mean first passage time of a diffusion process to a small target on the reflecting boundary of a bounded domain. The problem is equivalent to solving the mixed Dirichlet--Neumann boundary value problem for the Poisson equation with small Dirichlet and large Neumann parts. The mixed boundary value problem, which goes back to Lord Rayleigh, originates in the theory of sound and is closely connected to the eigenvalue problem for the mixed problem and for the Neumann problem in domains with bottlenecks. We review here recent developments in the non-standard asymptotics of the problem, which are based on several ingredients: a better resolution of the singularity of Neumann's function, resolution of the boundary layer near the small target by conformal mappings of domains with bottlenecks, and the breakup of composite domains into simpler components. The new methodology applies to two- and higher-dimensional problems. Selected applications are r...
175 citations
TL;DR: A method introduced by Fokas is reviewed, which contains the classical methods as special cases but also allows for the equally explicit solution of problems for which no classical approach exists.
Abstract: The classical methods for solving initial-boundary-value problems for linear partial differential equations with constant coefficients rely on separation of variables and specific integral transforms. As such, they are limited to specific equations, with special boundary conditions. Here we review a method introduced by Fokas, which contains the classical methods as special cases. However, this method also allows for the equally explicit solution of problems for which no classical approach exists. In addition, it is possible to elucidate which boundary-value problems are well posed and which are not. We provide examples of problems posed on the positive half-line and on the finite interval. Some of these examples have solutions obtainable using classical methods, and others do not. For the former, it is illustrated how the classical methods may be recovered from the more general approach of Fokas.
109 citations
TL;DR: A flavor of the code's main features is given and its applicability is illustrated using several case studies to show that \ifiss can be a valuable tool in both teaching and research.
Abstract: The Incompressible Flow & Iterative Solver Software (\ifiss) package contains software which can be run with MATLAB or Octave to create a computational laboratory for the interactive numerical study of incompressible flow problems. It includes algorithms for discretization by mixed finite element methods and a posteriori error estimation of the computed solutions, together with state-of-the-art preconditioned iterative solvers for the resulting discrete linear equation systems. In this paper we give a flavor of the code's main features and illustrate its applicability using several case studies. We aim to show that \ifiss can be a valuable tool in both teaching and research.
99 citations
TL;DR: New sign-definite formulations of the Helmholtz equation posed in either the interior of a star-shaped domain with impedance boundary conditions or the exterior of aStar-shape domain, with implications for both the analysis and the practical implementation of finite element methods are introduced.
Abstract: The usual variational (or weak) formulations of the Helmholtz equation are sign-indefinite in the sense that the bilinear forms cannot be bounded below by a positive multiple of the appropriate norm squared. This is often for a good reason, since in bounded domains under certain boundary conditions the solution of the Helmholtz equation is not unique at wavenumbers that correspond to eigenvalues of the Laplacian, and thus the variational problem cannot be sign-definite. However, even in cases where the solution is unique for all wavenumbers, the standard variational formulations of the Helmholtz equation are still indefinite when the wavenumber is large. This indefiniteness has implications for both the analysis and the practical implementation of finite element methods. In this paper we introduce new sign-definite (also called coercive or elliptic) formulations of the Helmholtz equation posed in either the interior of a star-shaped domain with impedance boundary conditions or the exterior of a star-shape...
84 citations
TL;DR: An expository account of the computational proof that every position of the Rubik's Cube can be solved in 20 moves or fewer, where a move is defined as any twist of any face.
Abstract: We give an expository account of our computational proof that every position of the Rubik's Cube can be solved in 20 moves or fewer, where a move is defined as any twist of any face. The roughly $4.3 \times 10^{19}$ positions are partitioned into about two billion cosets of a specially chosen subgroup, and the count of cosets required to be treated is reduced by considering symmetry. The reduced space is searched with a program capable of solving one billion positions per second, using about one billion seconds of CPU time donated by Google. As a byproduct of determining that the diameter is 20, we also find the exact count of cube positions at distance 15.
55 citations
TL;DR: It is proved that every graph has a spectral sparsifier with a number of edges linear in its number of vertices, and an elementary deterministic polynomial time algorithm is given for constructing $H.
Abstract: A sparsifier of a graph is a sparse graph that approximates it. A spectral sparsifier is one that approximates it spectrally, which means that their Laplacian matrices have similar quadratic forms. We prove that every graph has a spectral sparsifier with a number of edges linear in its number of vertices. In particular, we prove that for every $\epsilon \in (0,1)$ and every undirected, weighted graph $G = (V,E,w)$ on $n$ vertices, there exists a weighted graph $H=(V,F,\tilde{w})$ with at most $\lceil (n-1)/\epsilon^2\rceil$ edges such that for every $x \in \R^{V}$, $ (1-\epsilon)^2 \cdot x^T L_G x \leq x^{T} L_{H} x \leq (1+\epsilon)^2 \cdot x^{T} L_{G} x$, where $L_{G}$ and $L_{H}$ are the Laplacian matrices of $G$ and $H$, respectively. We give an elementary deterministic polynomial time algorithm for constructing $H$. This result is a special case of a significantly more general theorem which provides sparse approximations of general positive semidefinite matrices: given any real matrix $B_{n\times m}$...
48 citations
TL;DR: The purpose of this paper is to recall and popularize a simple graphical interpretation of the Krein signature well known in the spectral theory of polynomial operator pencils.
Abstract: Two concepts, evidently very different in nature, have proved to be useful in analytical and numerical studies of spectral stability in nonlinear wave theory: (i) the Krein signature of an eigenvalue, a quantity usually defined in terms of the relative orientation of certain subspaces that is capable of detecting the structural instability of imaginary eigenvalues and, hence, their potential for moving into the right half-plane leading to dynamical instability under perturbation of the system; and (ii) the Evans function, an analytic function detecting the location of eigenvalues. One might expect these two concepts to be related, but unfortunately examples demonstrate that there is no way in general to deduce the Krein signature of an eigenvalue from the Evans function, for example, by studying derivatives of the latter. The purpose of this paper is to recall and popularize a simple graphical interpretation of the Krein signature well known in the spectral theory of polynomial operator pencils. Once esta...
TL;DR: In this article, the heat equation and wave equation are approximated by diagonalizing symmetric matrices, replacing sums by integrals, and computing Fourier coefficients, and they find accurate approximations with a variety of boundary conditions.
Abstract: When the heat equation and wave equation are approximated by $\bm{u}_t = -\bm{K} \bm{u}$ and $\bm{u}_{tt} = -\bm{K} \bm{u}$ (discrete in space), the solution operators involve $e^{-\bm{K}t}$, $\sqrt{\bm{K}}$, $\cos(\sqrt{\bm{K}}t)$, and $\mathrm{sinc}(\sqrt{\bm{K}}t)$. We compute these four matrices and find accurate approximations with a variety of boundary conditions. The second difference matrix $\bm{K}$ is Toeplitz (shift-invariant) for Dirichlet boundary conditions, but we show why $e^{\bm{-Kt}}$ also has a Hankel (anti-shift-invariant) part. Any symmetric choice of the four corner entries of $\bm{K}$ leads to Toeplitz plus Hankel in all functions $f(\bm{K})$. Overall, this article is based on diagonalizing symmetric matrices, replacing sums by integrals, and computing Fourier coefficients.
TL;DR: This work describes an analytical study of a 4-dimensional HIV antioxidant-therapy model which exhibits viral blips, showing that an increasing, saturating infectivity function contributes to the recurrent behavior of the model.
Abstract: Recurrent infection is characterized by short episodes of high viral reproduction, separated by long periods of relative quiescence. This recurrent pattern is observed in many persistent infections, including the “viral blips” observed during chronic infection with the human immunodeficiency virus (HIV). Although in-host models which incorporate forcing functions or stochastic elements have been shown to display viral blips, simple deterministic models also exhibit this phenomenon. We describe an analytical study of a 4-dimensional HIV antioxidant-therapy model which exhibits viral blips, showing that an increasing, saturating infectivity function contributes to the recurrent behavior of the model. Using dynamical systems theory, we hypothesize four conditions for the existence of viral blips in a deterministic in-host infection model. In particular, we explain how the blips are generated, which is not due to homoclinic bifurcation since no homoclinic orbits exist. These conditions allow us to develop ver...
Journal Article•
TL;DR: This paper will investigate a typical collective behavior of a self-propelled particle system modeled by the nearest neighbor rules and show that the smallest possible interaction radius approximately equals $\sqrt{\log n/(\pi n)}$, with $n$ being the population size, w...
Abstract: A central and fundamental issue in the theory of complex systems is to understand how local rules lead to collective behavior of the whole system. This paper will investigate a typical collective behavior (synchronization) of a self-propelled particle system modeled by the nearest neighbor rules. While connectivity of the dynamic neighbor graphs associated with the underlying systems is crucial for synchronization, it is widely known that the verification of such dynamical connectivity is at the core of theoretical analysis. Ideally, conditions used for synchronization should be imposed on the model parameters and the initial states of the particles. One crucial model parameter is the interaction radius, and we are interested in the following natural and basic question: What is the smallest interaction radius for synchronization? In this paper, we will show that, in a certain sense, the smallest possible interaction radius approximately equals $\sqrt{\log n/(\pi n)}$, with $n$ being the population size, w...
TL;DR: The approach described here is suitable to accommodate new Z-integral representations including Stirling numbers of the first kind, central Delannoy, Euler, Fibonacci, and Genocchi numbers, and the special polynomials of Bell, generalized Bell, Hermite, and Laguerre.
Abstract: We show that the ordinary derivative of a real analytic function of one variable can be realized as a Grassmann--Berezin-type integration over the Zeon algebra, the Z-integral. As a by-product of this representation, we give new proofs of the Faa di Bruno formula and Spivey's identity [M. Z. Spivey, J. Integer Seq., 11 (2008), 08.2.5], and we recover the representation of the Stirling numbers of the second kind and the Bell numbers of Staples and Schott [European J. Combin., 29 (2008), pp. 1133--1138]. The approach described here is suitable to accommodate new Z-integral representations including Stirling numbers of the first kind, central Delannoy, Euler, Fibonacci, and Genocchi numbers, and the special polynomials of Bell, generalized Bell, Hermite, and Laguerre.
TL;DR: The necessary and sufficient conditions for linear and high-order convergence of fixed point and Newton's methods are revisited and Schroder's process of the first kind is extended to increase the order of convergence of the fixed point method.
Abstract: In this paper we revisit the necessary and sufficient conditions for linear and high-order convergence of fixed point and Newton's methods. Based on these conditions, we extend Schroder's process of the first kind to increase the order of convergence of the fixed point method. We also obtain two processes to increase the order of convergence of Newton's method, one of which is Schroder's process of the second kind, for which several forms are also presented. A link between Schroder's two processes is given. Examples and numerical experiments are included.
TL;DR: It is shown that high-order accurate Nystrom discretization leads to well-conditioned finite-dimensional linear systems if and only if the discretized is both norm-preserving in a correctly chosen $L^p$ space and adaptively refined in the internal layer.
Abstract: We investigate the behavior of integral formulations of variable coefficient elliptic partial differential equations in the presence of steep internal layers. In one dimension, the equations that arise can be solved analytically and the condition numbers estimated in various $L^p$ norms. We show that high-order accurate Nystrom discretization leads to well-conditioned finite-dimensional linear systems if and only if the discretization is both norm-preserving in a correctly chosen $L^p$ space and adaptively refined in the internal layer.
TL;DR: In this article, the authors consider an issue of great interest to all students: fairness in grading and represent each grade as a student's intrinsic (overall) aptitude minus a correction penalty.
Abstract: In this paper, we consider an issue of great interest to all students: fairness in grading. Specifically, we represent each grade as a student's intrinsic (overall) aptitude minus a correction repr...
TL;DR: This work starts from the model suggested by Hughes over 30 years ago, on which all other models used until now have been based, and analyzes the situations in which these models are not applicable.
Abstract: In the game of rugby, attempting to score a conversion following a try gives rise to a situation which may be modeled mathematically as an optimization problem, namely, that of choosing the best spot from which to kick the ball. Due to its appeal and simplicity, this problem has been widely used as an example of an application of mathematics to sports. Starting from the model suggested by Hughes over 30 years ago, on which all other models used until now have been based, we analyze the situations in which these models are not applicable. From this we will see how the quantity to be optimized should be changed in order to overcome these problems and also to obtain results which are compatible with the data quoted in the literature for professional players, for the whole range of possible scenarios. To obtain more realistic results, we then incorporate the initial speed and elevation angle of the ball into the model.