Showing papers in "Siam Review in 2017"
TL;DR: The Julia programming language as mentioned in this paper combines expertise from the diverse fields of computer science and computational science to create a new approach to numerical computing, which is designed to be easy and fast and questions notions generally held to be “laws of nature" by practitioners of numerical computing.
Abstract: Bridging cultures that have often been distant, Julia combines expertise from the diverse fields of computer science and computational science to create a new approach to numerical computing. Julia is designed to be easy and fast and questions notions generally held to be “laws of nature" by practitioners of numerical computing: \beginlist \item High-level dynamic programs have to be slow. \item One must prototype in one language and then rewrite in another language for speed or deployment. \item There are parts of a system appropriate for the programmer, and other parts that are best left untouched as they have been built by the experts. \endlist We introduce the Julia programming language and its design---a dance between specialization and abstraction. Specialization allows for custom treatment. Multiple dispatch, a technique from computer science, picks the right algorithm for the right circumstance. Abstraction, which is what good computation is really about, recognizes what remains the same after dif...
3,348 citations
TL;DR: JuMP as mentioned in this paper is an open-source modeling language that allows users to express a wide range of optimization problems (linear, mixed-integer, quadratic, conic-quadratic, semidefinite, and nonlinear) in a high-level, algebraic syntax.
Abstract: JuMP is an open-source modeling language that allows users to express a wide range of optimization problems (linear, mixed-integer, quadratic, conic-quadratic, semidefinite, and nonlinear) in a high-level, algebraic syntax. JuMP takes advantage of advanced features of the Julia programming language to offer unique functionality while achieving performance on par with commercial modeling tools for standard tasks. In this work we will provide benchmarks, present the novel aspects of the implementation, and discuss how JuMP can be extended to new problem classes and composed with state-of-the-art tools for visualization and interactivity.
1,056 citations
TL;DR: This paper is an introductory tutorial for numerical trajectory optimization with a focus on direct collocation methods, and provides an electronic supplement that contains well-documented MATLAB code for all examples and methods presented.
Abstract: This paper is an introductory tutorial for numerical trajectory optimization with a focus on direct collocation methods. These methods are relatively simple to understand and effectively solve a wide variety of trajectory optimization problems. Throughout the paper we illustrate each new set of concepts by working through a sequence of four example problems. We start by using trapezoidal collocation to solve a simple one-dimensional toy problem and work up to using Hermite--Simpson collocation to compute the optimal gait for a bipedal walking robot. Along the way, we cover basic debugging strategies and guidelines for posing well-behaved optimization problems. The paper concludes with a short overview of other methods for trajectory optimization. We also provide an electronic supplement that contains well-documented MATLAB code for all examples and methods presented. Our primary goal is to provide the reader with the resources necessary to understand and successfully implement their own direct collocation...
401 citations
TL;DR: Several approaches for improving the discrete mass balance or even for computing divergence-free solutions will be discussed: grad-div stabilization, higher order mixed methods derived on the basis of an exact de Rham complex, $H(div)$-conforming finite ...
Abstract: The divergence constraint of the incompressible Navier--Stokes equations is revisited in the mixed finite element framework. While many stable and convergent mixed elements have been developed throughout the past four decades, most classical methods relax the divergence constraint and only enforce the condition discretely. As a result, these methods introduce a pressure-dependent consistency error which can potentially pollute the computed velocity. These methods are not robust in the sense that a contribution from the right-hand side, which influences only the pressure in the continuous equations, impacts both velocity and pressure in the discrete equations. This article reviews the theory and practical implications of relaxing the divergence constraint. Several approaches for improving the discrete mass balance or even for computing divergence-free solutions will be discussed: grad-div stabilization, higher order mixed methods derived on the basis of an exact de Rham complex, $H(div)$-conforming finite ...
355 citations
TL;DR: In this article, a near-linear complexity (geometric and meshless/algebraic) multigrid/multiresolution method for PDEs with rough coefficients with rigorous a priori accuracy and performance estimates is introduced.
Abstract: We introduce a near-linear complexity (geometric and meshless/algebraic) multigrid/multiresolution method for PDEs with rough ($L^\infty$) coefficients with rigorous a priori accuracy and performance estimates. The method is discovered through a decision/game theory formulation of the problems of (1) identifying restriction and interpolation operators, (2) recovering a signal from incomplete measurements based on norm constraints on its image under a linear operator, and (3) gambling on the value of the solution of the PDE based on a hierarchy of nested measurements of its solution or source term. The resulting elementary gambles form a hierarchy of (deterministic) basis functions of $H^1_0(\Omega)$ (gamblets) that (1) are orthogonal across subscales/subbands with respect to the scalar product induced by the energy norm of the PDE, (2) enable sparse compression of the solution space in $H^1_0(\Omega)$, and (3) induce an orthogonal multiresolution operator decomposition. The operating diagram of the multig...
154 citations
TL;DR: This paper develops a new method to investigate the meso-scale feature known as core-periphery structure, which entails identifying densely connected core nodes and sparsely connected peripheral nodes in a network.
Abstract: Intermediate-scale (or “meso-scale'') structures in networks have received considerable attention, as the algorithmic detection of such structures makes it possible to discover network features that are not apparent either at the local scale of nodes and edges or at the global scale of summary statistics. Numerous types of meso-scale structures can occur in networks, but investigations of such features have focused predominantly on the identification and study of community structure. In this paper, we develop a new method to investigate the meso-scale feature known as core-periphery structure, which entails identifying densely connected core nodes and sparsely connected peripheral nodes. In contrast to communities, the nodes in a core are also reasonably well-connected to those in a network's periphery. Our new method of computing core-periphery structure can identify multiple cores in a network and takes into account different possible core structures. We illustrate the differences between our method and...
141 citations
TL;DR: This study investigates the desirability of applying a truncated Newton method to FWI and suggests that the inverse Hessian operator plays a crucial role in the parameter reconstruction, as it should help to mitigate finite-frequency effects and to better remove artifacts arising from multiscattered waves.
Abstract: Full waveform inversion (FWI) is a powerful method for reconstructing subsurface parameters from local measurements of the seismic wavefield. This method consists in minimizing the distance between predicted and recorded data. The predicted data are computed as the solution of a wave-propagation problem. Conventional numerical methods for the solution of FWI problems are gradient-based methods, such as the preconditioned steepest descent, the nonlinear conjugate gradient, or more recently the $l$-BFGS quasi-Newton algorithm. In this study, we investigate the desirability of applying a truncated Newton method to FWI. The inverse Hessian operator plays a crucial role in the parameter reconstruction, as it should help to mitigate finite-frequency effects and to better remove artifacts arising from multiscattered waves. For multiparameter reconstruction, the inverse Hessian operator also offers the possibility of better removing trade-offs due to coupling effects between parameter classes. The truncated Newto...
138 citations
TL;DR: In this paper, the authors define a variety of flavors of splines that can be characterized in terms of some differential operator, and the simplest piecewise-constant model corresponds to the derivative operator.
Abstract: Splines come in a variety of flavors that can be characterized in terms of some differential operator $L$. The simplest piecewise-constant model corresponds to the derivative operator. Likewise, on...
95 citations
TL;DR: In this paper, two physically and mathematically distinct regularization mechanisms of scalar hyperbolic conservation laws are considered. And when the flux is convex, the combination of diffusion and dispersion is know.
Abstract: We consider two physically and mathematically distinct regularization mechanisms of scalar hyperbolic conservation laws. When the flux is convex, the combination of diffusion and dispersion is know...
73 citations
TL;DR: The limiting stationary distribution of the Markov chain represents the fraction of the time it takes for a random walk as discussed by the authors to complete a given set of random walks, which is a fundamental model in applied mathematics.
Abstract: Random walks are a fundamental model in applied mathematics and are a common example of a Markov chain. The limiting stationary distribution of the Markov chain represents the fraction of the time ...
72 citations
TL;DR: The exponentially pronounced computational consequences of the anisotropy of the hypercube are described, notably its mismatch with the isotopy of the set of multivariate polynomials of a fixed degree on which some cubature formulas are based.
Abstract: Algorithms that combat the curse of dimensionality take advantage of nonuniformity properties of the underlying functions, which may be rotational (e.g., grid alignment) or translational (e.g., nea...
TL;DR: The dramatic decline in oil prices, from around $110 per barrel in June 2014 to less than $40 in March 2016, highlights the importance of competition between different energy sources.
Abstract: The dramatic decline in oil prices, from around $110 per barrel in June 2014 to less than $40 in March 2016, highlights the importance of competition between different energy sources. Indeed, the s...
TL;DR: In this article, the authors review the development of lower semicontinuity from the beginning of the 20th century to the present day, focusing on signed integrands and their applications in continuum mechanics of solids.
Abstract: Minimization is a recurring theme in many mathematical disciplines ranging from pure to applied. Of particular importance is the minimization of integral functionals, which is studied within the calculus of variations. Proofs of the existence of minimizers usually rely on a fine property of the functional called weak lower semicontinuity. While early studies of lower semicontinuity go back to the beginning of the 20th century, the milestones of the modern theory were established by C. B. Morrey, Jr. [Pacific J. Math., 2 (1952), pp. 25--53] in 1952 and N. G. Meyers [Trans. Amer. Math. Soc., 119 (1965), pp. 125--149] in 1965. We recapitulate the development of this topic from these papers onwards. Special attention is paid to signed integrands and to applications in continuum mechanics of solids. In particular, we review the concept of polyconvexity and special properties of (sub-)determinants with respect to weak lower semicontinuity. In addition, we emphasize some recent progress in lower semicontinuity o...
TL;DR: In this article, a survey of the literature on circle maps and quasi-contractions is presented, and sufficient conditions for the occurrence of two types of bifurcation scenarios involving rich dynamics are provided.
Abstract: This survey article is concerned with the study of bifurcations of discontinuous piecewise-smooth maps, with a special focus on the one-dimensional case. We review the literature on circle maps and quasi-contractions and provide paths through this literature to prove sufficient conditions for the occurrence of two types of bifurcation scenarios involving rich dynamics. The first scenario consists of the appearance of periodic orbits whose symbolic sequences and “rotation” numbers follow a Farey tree structure; the periods of the periodic orbits are given by consecutive addition. This is called the period adding bifurcation, and the proof of its existence relies on results for maps on the circle. In the second scenario, symbolic sequences are obtained by consecutive attachment of a given symbolic block, and the periods of periodic orbits are incremented by a constant term. This is called the period incrementing bifurcation, and its proof relies on results for maps on the interval. We also discuss the expan...
TL;DR: It is shown here how the explicit consideration of block structures at the continuous level can be a useful tool for solving differential and integral equation boundary-value problems in one space dimension by the rectangular differentiation, identity, and integration matrices introduced recently by Driscoll and Hale.
Abstract: Every student of numerical linear algebra is familiar with block matrices and vectors. The same ideas can be applied to the continuous analogues of operators, functions, and functionals. It is show...
TL;DR: In this article, the authors present a new class of randomized iterative algorithms based on diffusion Monte Carlo (DM) for solving linear systems, eigenvalue problems, and matrix exponentiation in dimensions beyond the present limits of numerical linear algebra.
Abstract: We review the basic outline of the highly successful diffusion Monte Carlo technique commonly used in contexts ranging from electronic structure calculations to rare event simulation and data assimilation, and propose a new class of randomized iterative algorithms based on similar principles to address a variety of common tasks in numerical linear algebra. From the point of view of numerical linear algebra, the main novelty of the fast randomized iteration schemes described in this article is that they have dramatically reduced operations and storage cost per iteration (as low as constant under appropriate conditions) and are rather versatile: we will show how they apply to the solution of linear systems, eigenvalue problems, and matrix exponentiation, in dimensions far beyond the present limits of numerical linear algebra. While traditional iterative methods in numerical linear algebra were created in part to deal with instances where a matrix (of size $\mathcal{O}(n^2)$) is too big to store, the algorit...
TL;DR: In this paper, Jiang and Ravikumar developed a technique that leads to solutions of two widely known problems on nonnegative matrices, i.e., the nonnegative rank of a matrix is NP-hard to compute.
Abstract: Using elementary linear algebra, we develop a technique that leads to solutions of two widely known problems on nonnegative matrices. First, we give a short proof of the result by Vavasis stating that the nonnegative rank of a matrix is NP-hard to compute. This proof is essentially contained in the paper by Jiang and Ravikumar, who discussed this topic in different terms fifteen years before the work of Vavasis. Second, we present a solution of the Cohen--Rothblum problem on rational nonnegative factorizations, which was posed in 1993 and remained open until now.
TL;DR: Rader's FFT is derived, Rader's zero-padding technique is described, and the performance of the unpadded and the zero-padded approaches is examined.
Abstract: This note provides a self-contained introduction to Rader's fast Fourier transform (FFT). We start by explaining the need for an additional type of FFT. The properties of the multiplicative group o...
TL;DR: In this article, the authors present a method of constructing examples of such systems when the matrices $A(t)$ have positive off-diagonal entries (strongly cooperative systems), and illustrate those examples both with interactive animations and analytically.
Abstract: It is well known that, contrary to the autonomous case, the stability/instability of solutions of nonautonomous linear ordinary differential equations $x' = A(t) x$ bears no relation to the sign of the real parts of the eigenvalues of $A(t)$. In particular, the real parts of all eigenvalues can be negative and bounded away from zero, but nonetheless there is a solution of magnitude growing to infinity. In this paper we present a method of constructing examples of such systems when the matrices $A(t)$ have positive off-diagonal entries (strongly cooperative systems). We illustrate those examples both with interactive animations and analytically. The paper is written in such a way that it can be accessible to students with diverse mathematical backgrounds/skills.
TL;DR: It is perhaps unknown that mechanical cylinder locks possess a number of important design constraints that uniquely distinguish them from their electronic counterparts.
Abstract: Keys and locks are an omnipresent fixture in our daily life, limiting physical access to privileged resources or spaces. While most of us may have marveled at the intricate shape of a key, the usually hidden mechanical complexity within a cylinder lock is even more awe-inspiring, containing a multitude of tiny movable parts such as springs and pins that have been precision-manufactured from highly customized materials using specialized fabrication techniques. \indent It is perhaps unknown that mechanical cylinder locks possess a number of important design constraints that uniquely distinguish them from their electronic counterparts. Aside from manufacturing costs, a cylinder's most significant limitations are the upper bound on its outer dimensions as well as the lower bound on the size of its internal mechanical security features (pins). Cylinders cannot be very large so that they still fit into doors and avoid the need for large keys. Pins cannot be too small in order to withstand wear and tear and prov...
TL;DR: A quadratic program is proposed for the purpose of stratifying points in a data cloud based on proximity to the boundary of the convex hull and it is shown that the weight vector encodes geometric information concerning the point's relationship to the Boundary of the Convex hull.
Abstract: The convex hull of a set of points, $C$, serves to expose extremal properties of $C$ and can help identify elements in $C$ of high interest. For many problems, particularly in the presence of noise, the true vertex set (and facets) may be difficult to determine. One solution is to expand the list of high interest candidates to points lying near the boundary of the convex hull. We propose a quadratic program for the purpose of stratifying points in a data cloud based on proximity to the boundary of the convex hull. For each data point, a quadratic program is solved to determine an associated weight vector. We show that the weight vector encodes geometric information concerning the point's relationship to the boundary of the convex hull. The computation of the weight vectors can be carried out in parallel, and for a fixed number of points and fixed neighborhood size, the overall computational complexity of the algorithm grows linearly with dimension. As a consequence, meaningful computations can be complete...
TL;DR: The primary goal of this paper is to define and study the interactive information complexity of functions.
Abstract: The primary goal of this paper is to define and study the interactive information complexity of functions. Let $f(x,y)$ be a function, and suppose Alice is given $x$ and Bob is given $y$. Informall...
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TL;DR: Dynamics over Signed Networks, by Guodong Shi, Claudio Altafini, and John S. Baras, is the Survey and Review article in the present issue.
Abstract: Our Survey and Review paper, “The Why, How, and When of Representations for Complex Systems,” by Leo Torres, Ann S. Blevins, Danielle Bassett, and Tina Eliassi-Rad, lists 233 references. Some of th...
TL;DR: This work shows how to construct a mathematical model for sloshing in the very similar problem of a mug on a smooth horizontal table forced to oscillate in one dimension via a spring connection and finds that analyzing this problem using quite simple ideas of mathematical modeling and analysis gives good physical understanding of how to reduce everydaySloshing.
Abstract: Walking across a room carrying a mug of coffee can often lead to spillage. Everyday experience tells us that it is better to walk slowly or not have the mug too full, but it is also well known that carrying coffee in a bucket with a pivoted handle is much less dangerous. Here we show how to construct a mathematical model for sloshing in the very similar problem of a mug on a smooth horizontal table forced to oscillate in one dimension via a spring connection. We find that analyzing this problem using quite simple ideas of mathematical modeling and analysis gives good physical understanding of how to reduce everyday sloshing.
TL;DR: It turns out that the problem on a fixed energy level reduces to the study of a monotone twist map of an annulus, which leads to existence proofs for orbits which do not precess or else precess in the wrong direction.
Abstract: A Foucault pendulum is supposed to precess in a direction opposite to the earth's rotation, but nonlinear terms in the equations of motion can also produce precession. The goal of this paper is to study the motion of a nonlinear, spherical pendulum on a rotating planet. It turns out that the problem on a fixed energy level reduces to the study of a monotone twist map of an annulus. For certain values of the parameters, this leads to existence proofs for orbits which do not precess or else precess in the wrong direction. In fact, there will be nonprecessing periodic solutions which return to their initial state after swinging back and forth just once. For pendulums of modest size, these nonprecessing periodic solutions can be very nearly planar.
TL;DR: Key findings of the SIAM report on undergraduate education in applied mathematics are summarized, focusing on curricular requirements, the role of industry, undergraduate research, student recruitment, and starting a new program.
Abstract: The SIAM Education Committee has released a report called Undergraduate Degree Programs in Applied Mathematics. The report describes the general and specific features of undergraduate education in applied mathematics, based on interviews with 12 diverse but representative programs, and offers commentary based on the experience of the committee. This article summarizes the key findings of the SIAM report, focusing on curricular requirements, the role of industry, undergraduate research, student recruitment, and starting a new program. The goal of the report and this article is to provide guidance to new programs, existing programs, and the development of policy.