Showing papers in "Siam Review in 2013"
TL;DR: In this article, the properties of the Laplace operator in bounded Euclidean domains with Dirichlet, Neumann, or Robin boundary conditions are summarized at a level accessible to scientists ranging from mathematics to physics and computer sciences.
Abstract: We summarize the properties of eigenvalues and eigenfunctions of the Laplace operator in bounded Euclidean domains with Dirichlet, Neumann, or Robin boundary condition. We keep the presentation at a level accessible to scientists from various disciplines ranging from mathematics to physics and computer sciences. The main focus is placed onto multiple intricate relations between the shape of a domain and the geometrical structure of eigenfunctions.
275 citations
TL;DR: A spectral method is developed for the direct solution of linear ordinary differential equations with variable coefficients and general boundary conditions that leads to matrices that are LaSalle matrices.
Abstract: A spectral method is developed for the direct solution of linear ordinary differential equations with variable coefficients and general boundary conditions. The method leads to matrices that are al...
220 citations
TL;DR: The theory leads to a rigorous proof of the exponential off-diagonal decay ("nearsightedness") for the density matrix of gapped systems at zero electronic temperature in both orthogonal and non-orthogonal representations, thus providing a firm theoretical basis for the possibility of linear scaling methods in electronic structure calculations for non-metallic systems.
Abstract: Motivated by applications in quantum chemistry and solid state physics, we apply general results from approximation theory and matrix analysis to the study of the decay properties of spectral projectors associated with large and sparse Hermitian matrices. Our theory leads to a rigorous proof of the exponential off-diagonal decay ("nearsightedness") for the density matrix of gapped systems at zero electronic temperature in both orthogonal and nonorthogonal representations, thus providing a firm theoretical basis for the possibility of linear scaling methods in electronic structure calculations for nonmetallic systems. We further discuss the case of density matrices for metallic systems at positive electronic temperature. A few other possible applications are also discussed.
148 citations
TL;DR: This article surveys the recent developments in computational methods for second order fully nonlinear partial differential equations (PDEs) and intends to introduce these current advancements and new results to the SIAM community and generate more interest in numerical methods for fully non linear PDEs.
Abstract: This article surveys the recent developments in computational methods for second order fully nonlinear partial differential equations (PDEs), a relatively new subarea within numerical PDEs Due to their ever increasing importance in mathematics itself (eg, differential geometry and PDEs) and in many scientific and engineering fields (eg, astrophysics, geostrophic fluid dynamics, grid generation, image processing, optimal transport, meteorology, mathematical finance, and optimal control), numerical solutions to fully nonlinear second order PDEs have garnered a great deal of interest from the numerical PDE and scientific communities Significant progress has been made for this class of problems in the past few years, but many problems still remain open This article intends to introduce these current advancements and new results to the SIAM community and generate more interest in numerical methods for fully nonlinear PDEs
140 citations
TL;DR: The success of this algorithm suggests that there might be variants of Pade approximation that are pointwise convergent as the degrees of the numerator and denominator increase to $\infty$, unlike traditional Pade approximants, which converge only in measure or capacity.
Abstract: Pade approximation is considered from the point of view of robust methods of numerical linear algebra, in particular, the singular value decomposition. This leads to an algorithm for practical computation that bypasses most problems of solution of nearly-singular systems and spurious pole-zero pairs caused by rounding errors, for which a MATLAB code is provided. The success of this algorithm suggests that there might be variants of Pade approximation that are pointwise convergent as the degrees of the numerator and denominator increase to $\infty$, unlike traditional Pade approximants, which converge only in measure or capacity.
120 citations
TL;DR: ConGradU is a simplified version of the old and well-known conditional gradient algorithm with unit step size that yields a generic algorithm which either is given by an analytic formula or requires a very low computational complexity.
Abstract: The sparsity constrained rank-one matrix approximation problem is a difficult mathematical optimization problem which arises in a wide array of useful applications in engineering, machine learning, and statistics, and the design of algorithms for this problem has attracted intensive research activities. We introduce an algorithmic framework, called ConGradU, that unifies a variety of seemingly different algorithms that have been derived from disparate approaches, and that allows for deriving new schemes. Building on the old and well-known conditional gradient algorithm, ConGradU is a simplified version with unit step size that yields a generic algorithm which either is given by an analytic formula or requires a very low computational complexity. Mathematical properties are systematically developed and numerical experiments are given.
112 citations
TL;DR: The types of radial basis functions that fit in this analysis show global convergence to first-order critical points for the ORBIT algorithm and the use of ORBIT in finding local minima on a computationally expensive environmental engineering problem involving remediation of contaminated groundwater.
Abstract: We analyze globally convergent, derivative-free trust-region algorithms relying on radial basis function interpolation models. Our results extend the recent work of Conn, Scheinberg, and Vicente [SIAM J. Optim., 20 (2009), pp. 387--415] to fully linear models that have a nonlinear term. We characterize the types of radial basis functions that fit in our analysis and thus show global convergence to first-order critical points for the ORBIT algorithm of Wild, Regis, and Shoemaker [SIAM J. Sci. Comput., 30 (2008), pp. 3197--3219]. Using ORBIT, we present numerical results for different types of radial basis functions on a series of test problems. We also demonstrate the use of ORBIT in finding local minima on a computationally expensive environmental engineering problem involving remediation of contaminated groundwater.
88 citations
TL;DR: The proposed algorithm computes a dynamic measure of how well pairs of nodes can communicate by taking account of routes through the network that respect the arrow of time, and takes the conventional approach of downweighting for length and the novel feature of down Weighting for age.
Abstract: We propose a new algorithm for summarizing properties of large-scale time-evolving networks. This type of data, recording connections that come and go over time, is generated in many modern applications, including telecommunications and online human social behavior. The algorithm computes a dynamic measure of how well pairs of nodes can communicate by taking account of routes through the network that respect the arrow of time. We take the conventional approach of downweighting for length (messages become corrupted as they are passed along) and add the novel feature of downweighting for age (messages go out of date). This allows us to generalize widely used Katz-style centrality measures that have proved popular in network science to the case of dynamic networks sampled at nonuniform points in time. We illustrate the new approach on synthetic and real data.
80 citations
TL;DR: The optimal uncertainty quantification (OUQ) framework as mentioned in this paper is based on the observation that, given a set of assumptions and information about the problem, there exist optimal bounds on uncertainties: these are obtained as values of well-defined optimization problems corresponding to extremizing probabilities of failure, or of deviations, subject to the constraints imposed by the scenarios compatible with the assumptions.
Abstract: We propose a rigorous framework for uncertainty quantification (UQ) in which the UQ objectives and its assumptions/information set are brought to the forefront. This framework, which we call optimal uncertainty quantification (OUQ), is based on the observation that, given a set of assumptions and information about the problem, there exist optimal bounds on uncertainties: these are obtained as values of well-defined optimization problems corresponding to extremizing probabilities of failure, or of deviations, subject to the constraints imposed by the scenarios compatible with the assumptions and information. In particular, this framework does not implicitly impose inappropriate assumptions, nor does it repudiate relevant information. Although OUQ optimization problems are extremely large, we show that under general conditions they have finite-dimensional reductions. As an application, we develop optimal concentration inequalities (OCI) of Hoeffding and McDiarmid type. Surprisingly, these results show that uncertainties in input parameters, which propagate to output uncertainties in the classical sensitivity analysis paradigm, may fail to do so if the transfer functions (or probability distributions) are imperfectly known. We show how, for hierarchical structures, this phenomenon may lead to the nonpropagation of uncertainties or information across scales. In addition, a general algorithmic framework is developed for OUQ and is tested on the Caltech surrogate model for hypervelocity impact and on the seismic safety assessment of truss structures, suggesting the feasibility of the framework for important complex systems. The introduction of this paper provides both an overview of the paper and a self-contained minitutorial on the basic concepts and issues of UQ.
79 citations
TL;DR: This work derives a nonlocal partial differential equation describing the evolving population density of a continuum aggregation and finds exact analytical expressions for the equilibria.
Abstract: Biological aggregations such as fish schools, bird flocks, bacterial colonies, and insect swarms have characteristic morphologies governed by the group members' intrinsic social interactions with e...
68 citations
TL;DR: This article builds up the mathematical framework of compressed sensing to show how combining efficient sampling methods with elementary ideas from linear algebra and a bit of approximation theory, optimization, and probability allows the estimation of unknown quantities with far less sampling of data than traditional methods.
Abstract: This article offers an accessible but rigorous and essentially self-contained account of some of the central ideas in compressed sensing, aimed at nonspecialists and undergraduates who have had linear algebra and some probability. The basic premise is first illustrated by considering the problem of detecting a few defective items in a large set. We then build up the mathematical framework of compressed sensing to show how combining efficient sampling methods with elementary ideas from linear algebra and a bit of approximation theory, optimization, and probability allows the estimation of unknown quantities with far less sampling of data than traditional methods.
TL;DR: In this paper, a simple geometric result suffices to derive the form of the optimal solution in a large class of finite and infinite-dimensional maximum entropy problems concerning probability distributions, spectral densities, and covariance matrices.
Abstract: We show that a simple geometric result suffices to derive the form of the optimal solution in a large class of finite- and infinite-dimensional maximum entropy problems concerning probability distributions, spectral densities, and covariance matrices. These include Burg's spectral estimation method and Dempster's covariance completion, as well as various recent generalizations of the above. We then apply this orthogonality principle to the new problem of completing a block-circulant covariance matrix when an a priori estimate is available.
TL;DR: A simple, step-by-step example to explain the use of Bayesian techniques for estimating the coefficients of gravity and air friction in the equations describing a falling body is provided.
Abstract: All mathematical models of real-world phenomena contain parameters that need to be estimated from measurements, either for realistic predictions or simply to understand the characteristics of the model. Bayesian statistics provides a framework for parameter estimation in which uncertainties about models and measurements are translated into uncertainties in estimates of parameters. This paper provides a simple, step-by-step example---starting from a physical experiment and going through all of the mathematics---to explain the use of Bayesian techniques for estimating the coefficients of gravity and air friction in the equations describing a falling body. In the experiment we dropped an object from a known height and recorded the free fall using a video camera. The video recording was analyzed frame by frame to obtain the distance the body had fallen as a function of time, including measures of uncertainty in our data that we describe as probability densities. We explain the decisions behind the various cho...
TL;DR: The roots on the interval of a function f(x) is holomorphic on an interval x in [a, b] can be computed by the following three-step procedure.
Abstract: When a function $f(x)$ is holomorphic on an interval $x \in [a, b]$, its roots on the interval can be computed by the following three-step procedure. First, approximate $f(x)$ on $[a, b]$ by a poly...
TL;DR: A scavenger species that scavenges the predator and is also a predator of the common prey is introduced, analytically proving that all trajectories are bounded in forward time, and numerically demonstrating persistent bounded paired cascades of period-doubling orbits over a wide range of parameter values.
Abstract: The dynamics of the classic planar two-species Lotka--Volterra predator-prey model are well understood. We introduce a scavenger species that scavenges the predator and is also a predator of the common prey. For this model, we analytically prove that all trajectories are bounded in forward time, and numerically demonstrate persistent bounded paired cascades of period-doubling orbits over a wide range of parameter values. Standard analytical and numerical techniques are used in the analysis of this model, making it an ideal pedagogical tool. We include exercises and an open-ended project to promote mastery of these techniques.
TL;DR: In this article, the problem of packing ellipsoids of different sizes and shapes into an ellipseidal container so as to minimize a measure of overlap between ellipssoids is considered.
Abstract: The problem of packing ellipsoids of different sizes and shapes into an ellipsoidal container so as to minimize a measure of overlap between ellipsoids is considered. A bilevel optimization formulation is given, together with an algorithm for the general case and a simpler algorithm for the special case in which all ellipsoids are in fact spheres. Convergence results are proved and computational experience is described and illustrated. The motivating application---chromosome organization in the human cell nucleus---is discussed briefly, and some illustrative results are presented.
TL;DR: In this article, the classical one-equation (or Johnson-Nedelec) coupling of finite and boundary elements can be applied with a Lipschitz coupling interface.
Abstract: In this short article we prove that the classical one-equation (or Johnson--Nedelec) coupling of finite and boundary elements can be applied with a Lipschitz coupling interface. Because of the way it was originally approached from the analytical standpoint, this BEM--FEM scheme has generally required smooth boundaries and hence produced a consistency error in the finite element part. With a variational argument, we prove that this requirement is not needed and that stability holds for all pairs of discrete space, as it inherits the underlying ellipticity of the problem
TL;DR: The inference-for-prediction method exactly replicates the prediction results of either truncated singular value decomposition, Tikhonov-regularized, or Gaussian statistical inverse problem formulations independent of data; it sacrifices accuracy in parameter estimate for online efficiency.
Abstract: Inference of model parameters is one step in an engineering process often ending in predictions that support decision in the form of design or control. Incorporation of end goals into the inference process leads to more efficient goal-oriented algorithms that automatically target the most relevant parameters for prediction. In the linear setting the control-theoretic concepts underlying balanced truncation model reduction can be exploited in inference through a dimensionally optimal subspace regularizer. The inference-for-prediction method exactly replicates the prediction results of either truncated singular value decomposition, Tikhonov-regularized, or Gaussian statistical inverse problem formulations independent of data; it sacrifices accuracy in parameter estimate for online efficiency. The new method leads to low-dimensional parameterization of the inverse problem enabling solution on smartphones or laptops in the field.
TL;DR: The author proposes an explicit, constructive, and analytical definition of the donating region based on two characteristic curves of the flow field, viz. streaklines and timelines, and shows that, within a time interval of finite size, each particle in the proposed donating region has a net effect of going across the fixed curve once.
Abstract: Lagrangian flux through a fixed curve segment naturally gives rise to the notion of a donating region, a compact point set in which each particle will pass through the curve and contribute to the flux. A precise geometric determination of the donating region has not been available until now. The author proposes an explicit, constructive, and analytical definition of the donating region based on two characteristic curves of the flow field, viz. streaklines and timelines. It is also shown that, within a time interval of finite size, each particle in the proposed donating region has a net effect of going across the fixed curve once. The presented analysis might be potentially useful for flow topology studies, Lagrangian flux calculation, and explicit interface tracking.
TL;DR: The notion of angle of repose for granular materials is explored and the free surface problems for centrifuged granular piles are solved.
Abstract: We explore the notion of angle of repose for granular materials. As an illustration, we solve free surface problems for centrifuged granular piles. These have recently been considered as an affordable and simple way to experiment with powders and dust in reduced gravity environments such as on the Moon or on Mars.
Journal Article•
TL;DR: In this article, a nonlocal PDE, known as the aggregation equation, was derived to describe evolving population density in locust swarms, which can exhibit a variety of behaviors including spreading without bound, concentrating into δ-functions and formation of compactly supported equilibria.
Abstract: Biological aggregations (swarms) exhibit morphologies governed by social interactions and responses to environment. Starting from a particle model we derive a nonlocal PDE, known as the aggregation equation, which describes evolving population density. The solutions to the aggregation equation can exhibit a variety of behaviors including spreading without bound, concentrating into δ-functions and formation of compactly supported equilibria. We describe some tools for investigating the asymptotic behavior of solutions. We also study equilibria and their stability via the calculus of variations which yields analytical solutions. Finally we present a case study about how these methods can be used to construct a model of locust swarms.
TL;DR: This module, suitable for inclusion in an advanced undergraduate or graduate linear algebra course, explores the theoretical properties of the matrices in the mathematical system and should provide a good physical motivation for the theoretical explorations in such a course.
Abstract: In order to overcome loss in optical fibers, experimentalists are interested in employing parametric amplifiers using four-wave mixing. Upon linearizing the nonlinear Schrodinger equation typically used as a model for such amplifiers, a system of ODEs results for the complex amplitude. The solution can also be expressed as the product of transfer matrices and the initial condition and its conjugate. Physical insight about the fiber-optic system can be gained by examining the theoretical properties of the matrices in the mathematical system. This module, suitable for inclusion in an advanced undergraduate or graduate linear algebra course, explores these properties and should provide a good physical motivation for the theoretical explorations in such a course.
Journal Article•
TL;DR: This work considers the problem of securing data using linear and nonlinear codes over the binary numbers and develops a conservation law for codes, which shows that binary quadratic codes suffer from the same deficiencies as linear codes.
Abstract: We consider the problem of securing data using linear and nonlinear codes over the binary numbers. We start by developing a conservation law for codes. Then we explain why linear codes, which are easy to understand and implement, are useful when protecting data from rarely occurring random errors. By a simple argument, we demonstrate that linear codes are not a good way to secure data against an attacker. Having ruled out linear codes for this purpose, we take up nonlinear codes. We explain what a finite field is and how data can be represented by elements of a finite field. We then consider codes that are nonlinear functions of the data---the elements of the finite field. We show that binary quadratic codes suffer from the same deficiencies as linear codes. Next we consider cubic codes. First, we show that cubic codes do a good job of detecting changes made by an attacker. Then we demonstrate that certain cubic codes provide a large measure of protection against attackers and some protection against cert...