# Showing papers in "Siam Review in 2003"

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TL;DR: Developments in this field are reviewed, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.

Abstract: Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.

16,520 citations

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TL;DR: Methods involv- ing approximation theory, dierential equations, the matrix eigenvalues, and the matrix characteristic polynomial have been proposed, indicating that some of the methods are preferable to others, but that none are completely satisfactory.

Abstract: In principle, the exponential of a matrix could be computed in many ways. Methods involving approximation theory, differential equations, the matrix eigenvalues, and the matrix characteristic polyn...

2,051 citations

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TL;DR: This review begins by briefly summarizing the history of direct search methods and considering the special properties of problems for which they are well suited, then turns to a broad class of methods for which the underlying principles allow general-ization to handle bound constraints and linear constraints.

Abstract: Direct search methods are best known as unconstrained optimization techniques that do not explicitly use derivatives. Direct search methods were formally proposed and widely applied in the 1960s but fell out of favor with the mathematical optimization community by the early 1970s because they lacked coherent mathematical analysis. Nonetheless, users remained loyal to these methods, most of which were easy to program, some of which were reliable. In the past fifteen years, these methods have seen a revival due, in part, to the appearance of mathematical analysis, as well as to interest in parallel and distributed com- puting. This review begins by briefly summarizing the history of direct search methods and considering the special properties of problems for which they are well suited. Our focus then turns to a broad class of methods for which we provide a unifying framework that lends itself to a variety of convergence results. The underlying principles allow general- ization to handle bound constraints and linear constraints. We also discuss extensions to problems with nonlinear constraints.

1,530 citations

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TL;DR: This work presents a general overview of the common features of all spread options by discussing in detail their roles as speculation devices and risk management tools, and describes the mathematical framework used to model them.

Abstract: We survey theoretical and computational problems associated with the pricing and hedging of spread options. These options are ubiquitous in the financial markets, whether they be equity, fixed income, foreign exchange, commodities, or energy markets. As a matter of introduction, we present a general overview of the common features of all spread options by discussing in detail their roles as speculation devices and risk management tools. We describe the mathematical framework used to model them, and we review the numerical algorithms actually used to price and hedge them. There is already extensive literature on the pricing of spread options in the equity and fixed income markets, and our contribution is mostly to put together material scattered across a wide spectrum of recent textbooks and journal articles. On the other hand, information about the various numerical procedures that can be used to price and hedge spread options on physical commodities is more difficult to find. For this reason, we make a systematic effort to choose examples from the energy markets in order to illustrate the numerical challenges associated with these instruments. This gives us a chance to discuss an interesting application of spread options to an asset valuation problem after it is recast in the framework of real options. This approach is currently the object of intense mathematical research. In this spirit, we review the two major avenues to modeling energy price dynamics. We explain how the pricing and hedging algorithms can be implemented in the framework of models for both the spot price dynamics and the forward curve dynamics.

327 citations

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TL;DR: A high-order formulation for solving hyperbolic conservation laws using the discontinuous Galerkin method (DGM) is presented and an orthogonal basis for the spatial discretization is introduced and use explicit Runge--Kutta time discretized.

Abstract: We present a high-order formulation for solving hyperbolic conservation laws using the discontinuous Galerkin method (DGM). We introduce an orthogonal basis for the spatial discretization and use explicit Runge--Kutta time discretization. Some results of higher order adaptive refinement calculations are presented for inviscid Rayleigh--Taylor flow instability and shock reflection problems. The adaptive procedure uses an error indicator that concentrates the computational effort near discontinuities.

186 citations

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TL;DR: This paper presents theoretical, computational, and experimental aspects of the instability development in the flow of thin fluid films, and derivation of the thin film equation using lubrication approximation is presented.

Abstract: This paper presents theoretical, computational, and experimental aspects of the instability development in the flow of thin fluid films. The theoretical part involves basic fluid me- chanics and presents derivation of the thin film equation using lubrication approximation. A simplified version of this equation is then analyzed analytically using linear stability analysis, and also numerically. The results are then compared directly to experiments. The experimental part outlines the setup, as well as data acquisition and analysis. This immediate comparison to experiments is very useful for gaining better insight into the interpretation of various theoretical and computational results.

123 citations

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TL;DR: This work defines and explores the properties of the exchange operator, which maps J-orthogonal matrices to orthogonalMatrices and vice versa, and shows how the exchange operators can be used to obtain a hyperbolic CS decomposition of a J- Orthogonal matrix directly from the usual CS decompositions of an orthogsonal matrix.

Abstract: A real, square matrix Q is J-orthogonal if QTJQ = J, where the signature matrix $J = \diag(\pm 1)$. J-orthogonal matrices arise in the analysis and numerical solution of various matrix problems involving indefinite inner products, including, in particular, the downdating of Cholesky factorizations. We present techniques and tools useful in the analysis, application, and construction of these matrices, giving a self-contained treatment that provides new insights. First, we define and explore the properties of the exchange operator, which maps J-orthogonal matrices to orthogonal matrices and vice versa. Then we show how the exchange operator can be used to obtain a hyperbolic CS decomposition of a J-orthogonal matrix directly from the usual CS decomposition of an orthogonal matrix. We employ the decomposition to derive an algorithm for constructing random J-orthogonal matrices with specified norm and condition number. We also give a short proof of the fact that J-orthogonal matrices are optimally scaled und...

94 citations

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TL;DR: Lame's formulas for the eigenvalues and eigenfunctions of the Laplacian with Dirichlet boundary conditions on an equilateral triangle are derived using direct elementary mathematical techniques and shown to form a complete orthonormal system.

Abstract: Lame's formulas for the eigenvalues and eigenfunctions of the Laplacian with Dirichlet boundary conditions on an equilateral triangle are derived using direct elementary mathematical techniques. They are shown to form a complete orthonormal system. Various properties of the spectrum and nodal lines are explored. Implications for related geometries are considered.

89 citations

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TL;DR: This approach addresses the load balancing problem in a new way, requiring far less communication than current approaches, and allows existing sequential adaptive PDE codes such as PLTMG and MC to run in a parallel environment without a large investment in recoding.

Abstract: We present a new approach to the use of parallel computers with adaptive finite element methods. This approach addresses the load balancing problem in a new way, requiring far less communication than current approaches. It also allows existing sequential adaptive PDE codes such as PLTMG and MC to run in a parallel environment without a large investment in recoding. In this new approach, the load balancing problem is reduced to the numerical solution of a small elliptic problem on a single processor, using a sequential adaptive solver, without requiring any modifications to the sequential solver. The small elliptic problem is used to produce a posteriori error estimates to predict future element densities in the mesh, which are then used in a weighted recursive spectral bisection of the initial mesh. The bulk of the calculation then takes place independently on each processor, with no communication, using possibly the same sequential adaptive solver. Each processor adapts its region of the mesh independent...

88 citations

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TL;DR: A simple computational exercise to compare weak and strong integer programming formulations of the traveling salesman problem, where students can optimally solve instances with up to 70 cities in a few minutes by adding cuts from the stronger formulation to the weaker, but simpler one.

Abstract: We designed a simple computational exercise to compare weak and strong integer programming formulations of the traveling salesman problem. Using commercial IP software, and a short (60 line long) MATLAB code, students can optimally solve instances with up to 70 cities in a few minutes by adding cuts from the stronger formulation to the weaker, but simpler one.

86 citations

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Rice University

^{1}TL;DR: Two simple examples are presented where GM RES(1) converges exactly in three iterations, while GMRES(2) stagnates, revealing that GMRES (1) convergence can be extremely sensitive to small changes in the initial residual.

Abstract: When solving large nonsymmetric systems of linear equations with the restarted GMRES algorithm, one is inclined to select a relatively large restart parameter in the hope of mimicking the full GMRES process. Surprisingly, cases exist where small values of the restart parameter yield convergence in fewer iterations than larger values. Here, two simple examples are presented where GMRES(1) converges exactly in three iterations, while GMRES(2) stagnates. One of these examples reveals that GMRES(1) convergence can be extremely sensitive to small changes in the initial residual.

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TL;DR: In this paper, a computational methodology for multiclass prediction that combines class-specific (one vs. all) binary support vector machines was proposed for the diagnosis of multiple common adult malignancies using DNA microarray data.

Abstract: Modern cancer treatment relies upon microscopic tissue examination to classify tumors according to anatomical site of origin. This approach is effective but subjective and variable even among experienced clinicians and pathologists. Recently, DNA microarray-generated gene expression data has been used to build molecular cancer classifiers. Previous work from our group and others demonstrated methods for solving pairwise classification problems using such global gene expression patterns. However, classification across multiple primary tumor classes poses new methodological and computational challenges. In this paper we describe a computational methodology for multiclass prediction that combines class-specific (one vs. all) binary support vector machines. We apply this methodology to the diagnosis of multiple common adult malignancies using DNA microarray data from a collection of 198 tumor samples, spanning 14 of the most common tumor types. Overall classification accuracy is 78%, far exceeding the expecte...

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TL;DR: The stages through which CSE education is evolving, from initial recognition in the 1980s to present growth, are discussed, as is the emergence of a set of core elements common to different approaches.

Abstract: The multidisciplinary nature of computational science and engineering (CSE) and its rela- tion to other disciplines is described. The stages through which CSE education is evolving, from initial recognition in the 1980s to present growth, are discussed. The challenges and benefits of different approaches to CSE education are discussed, as is the emergence of a set of core elements common to different approaches. The content of courses, curricula, and degrees offered in CSE are reviewed, and a survey is made of all undergraduate de- gree programs. The curricula of different programs are examined for the common "tool set" they define and analyzed for their relative weighting of computing, application, and mathematics. A trend toward a standard curriculum is noted.

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TL;DR: The results demonstrate that constrained nonlinear programming is a worthwhile exercise for GARCH models, especially for the bivariate and trivariate cases, as they offer a significant improvement in the quality of the solution of the optimization problem over the diagonal VECH and the BEKK representations of the multivariate GARCH model.

Abstract: This paper proposes a constrained nonlinear programming view of generalized autoregressive conditional heteroskedasticity (GARCH) volatility estimation models in financial econometrics. These models are usually presented to the reader as unconstrained optimization models with recursive terms in the literature, whereas they actually fall into the domain of nonconvex nonlinear programming. Our results demonstrate that constrained nonlinear programming is a worthwhile exercise for GARCH models, especially for the bivariate and trivariate cases, as they offer a significant improvement in the quality of the solution of the optimization problem over the diagonal VECH and the BEKK representations of the multivariate GARCH model.

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TL;DR: The paper produces explicit extremizers when the drum is a disk, while, for general shapes, existence and necessary conditions are established, and a pair of numerical methods are built and test.

Abstract: Allowed to fasten, say, one-half of a drum's boundary, which half produces the lowest or highest bass note? The answer is a natural limit of solutions to a family of extremal Robin problems for the least eigenvalue of the Laplacian. We produce explicit extremizers when the drum is a disk, while, for general shapes, we establish existence and necessary conditions, and build and test a pair of numerical methods.

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TL;DR: The heat equation posed on the half-line may be used as a simple mathematical model describing the operation of an amperometric ion sensor, used to measure ion concentrations in the laboratory.

Abstract: The heat equation posed on the half-line may be used as a simple mathematical model describing the operation of an amperometric ion sensor. These sensors represent the next generation of sensors that are in routine use today. Such sensors may be used to measure ion concentrations in the laboratory, for clinical analysis, environmental monitoring, process and quality control, biomedical analysis, and physiological applications. Study of the heat equation and its solutions provides insight into the operation of these ion sensors.

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TL;DR: In this paper, the authors studied numerically different cases of bistability between steady, periodic, and quasi-periodic regimes and compared the Hopf bifurcation diagrams of the original laser equations and the slow time amplitude equation.

Abstract: Hopf bifurcation theory for an oscillator subject to a weak feedback but a large delay is investigated for a specific laser system. The problem is motivated by semiconductor laser instabilities which are initiated by undesirable optical feedbacks. Most of these instabilities start from a single Hopf bifurcation. Because of the large delay, a delayed amplitude appears in the slow time bifurcation equation which generates new bifurcations to periodic and quasi-periodic states. We determine analytical expressions for all branches of periodic solutions and show the emergence of secondary bifurcation points from double Hopf bifurcation points. We study numerically different cases of bistability between steady, periodic, and quasi-periodic regimes. Finally, the validity of the Hopf bifurcation approximation is investigated numerically by comparing the bifurcation diagrams of the original laser equations and the slow time amplitude equation.

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TL;DR: This paper shows that a generalization of a popular motion planning puzzle called Lunar Lockout is computationally intractable and proves that the problem is PSPACE-complete, and begins with a review of NP-completeness and polynomial-time reductions, and introduces the class PSPACE.

Abstract: In this paper we show that a generalization of a popular motion planning puzzle called Lunar Lockout is computationally intractable. In particular, we show that the problem is PSPACE-complete. We begin with a review of NP-completeness and polynomial-time reductions, introduce the class PSPACE, and motivate the significance of PSPACE-complete problems. Afterwards, we prove that determining whether a given instance of a generalized Lunar Lockout puzzle is solvable is PSPACE-complete.

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TL;DR: The objective of this paper is to present a simple framework for the derivation and analysis of orthogonal IIR transfer functions, which are directly related to Orthogonal rational functions.

Abstract: Finite impulse response (FIR) models are among the most basic tools in control theory and signal processing and are routinely used in almost all fields of application. The connections to orthogonal...

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TL;DR: This article develops two facets of spread spectrum, showing how to understand the properties of direct sequence spread spectrum and how to design the pseudorandom sequences that spread spectrum transmitters and receivers generally use.

Abstract: Spread spectrum techniques are often employed when transmitting information. They are widely used in wireless and cellular telephony. Spread spectrum techniques allow one to partition bandwidth, to hide transmissions, and to protect one's transmissions from being jammed. In this article, we develop two facets of spread spectrum. We show how to understand the properties of direct sequence spread spectrum. For this purpose we use probabilistic arguments. We also show how to design the pseudorandom sequences that spread spectrum transmitters and receivers generally use. This leads us to consider the properties of recurrence relations (and polynomials) over the integers modulo 2.

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TL;DR: A simple model of welding and clamping of beams which demonstrates that the order in which the clamps and welds are applied influences the final shape of the assembly, and helps explain why certain welding sequence may be preferred.

Abstract: This paper is meant to serve as a case study of mathematical modeling in industry. The problem, which arises in the automotive industry, is to predict the variation in the final assembly given the variation in the parts and tooling. In this paper, we present a simple model of welding and clamping of beams which demonstrates that the order in which the clamps and welds are applied influences the final shape of the assembly. The modeling of the process is done by simple mechanics. To solve the mathematical problem, we use standard ideas from constrained optimization and scientific computation. Additionally, using a statistical simulation we show that clamping and welding from the inside out leads to a smaller standard deviation in the result (as measured by the displacement of the right end of the beams) in response to normal distributions of variations in parts and welding. The findings help explain why certain welding sequence may be preferred.

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TL;DR: A simple optimal control problem is formulated in which one must find the optimal path along which to bike in order to get maximum suntan, using the maximum principle and the Euler equation of the classical calculus of variations.

Abstract: A simple optimal control problem is formulated in which one must find the optimal path along which to bike in order to get maximum suntan. Both the maximum principle and the Euler equation of the classical calculus of variations are used to calculate this optimal path. The interrelationship of the two approaches is elucidated; the adjoint variables in the maximum principle approach (which happen to be constants) are integration constants when solving via the Euler equation. Several slightly different versions of this problem are treated, with some surprising phenomena in the solution.