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January 2012
The 20 articles with the most full-text downloads during the month, in descending order.
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John von Neumann's Analysis of Gaussian Elimination and the Origins of Modern Numerical Analysis SIAM Rev. 53, pp. 607-682 (76 pages) Online Publication Date: November 07, 2011
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Just when modern computers (digital, electronic, and programmable) were being invented, John von Neumann and Herman Goldstine wrote a paper to illustrate the mathematical analyses that they believed would be needed to use the new machines effectively and to guide the development of still faster computers. Their foresight and the congruence of historical events made their work the first modern paper in numerical analysis. Von Neumann once remarked that to found a mathematical theory one had to prove the first theorem, which he and Goldstine did for the accuracy of mechanized Gaussian elimination—but their paper was about more than that. Von Neumann and Goldstine described what they surmized would be the significant questions once computers became available for computational science, and they suggested enduring ways to answer them. |
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Time-Consistent Portfolio Management SIAM J. Finan. Math. 3, pp. 1-32 (32 pages) Online Publication Date: January 03, 2012
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This paper considers the portfolio management problem for an investor with finite time horizon who is allowed to consume and take out life insurance. Natural assumptions, such as different discount rates for consumption and life insurance, lead to time inconsistency. This situation can also arise when the investor is in fact a group, the members of which have different utilities and/or different discount rates. As a consequence, the optimal strategies are not implementable. We focus on hyperbolic discounting, which has received much attention lately, especially in the area of behavioral finance. Following [I. Ekeland and T. A. Pirvu, Math. Financ. Econ., 2 (2008), pp. 57–86], we consider the resulting problem as a leader-follower game between successive selves, each of whom can commit for an infinitesimally small amount of time. We then define policies as subgame perfect equilibrium strategies. Policies are characterized by an integral equation which is shown to have a solution in the case of constant relative risk aversion utilities. Our results can be extended for more general preferences as long as the equations admit solutions. Numerical simulations reveal that for the Merton problem with hyperbolic discounting, the consumption increases up to a certain time, after which it decreases; this pattern does not occur in the case of exponential discounting and is therefore known in the literature as the “consumption puzzle.” Other numerical experiments explore the effect of time varying aggregation rate on the insurance premium. |
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The Structure and Function of Complex Networks SIAM Rev. 45, pp. 167-256 (90 pages) Online Publication Date: August 04, 2006
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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. |
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Power-Law Distributions in Empirical Data SIAM Rev. 51, pp. 661-703 (43 pages) Online Publication Date: November 06, 2009
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Power-law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man-made phenomena. Unfortunately, the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution—the part of the distribution representing large but rare events—and by the difficulty of identifying the range over which power-law behavior holds. Commonly used methods for analyzing power-law data, such as least-squares fitting, can produce substantially inaccurate estimates of parameters for power-law distributions, and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all. Here we present a principled statistical framework for discerning and quantifying power-law behavior in empirical data. Our approach combines maximum-likelihood fitting methods with goodness-of-fit tests based on the Kolmogorov–Smirnov (KS) statistic and likelihood ratios. We evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches. We also apply the proposed methods to twenty-four real-world data sets from a range of different disciplines, each of which has been conjectured to follow a power-law distribution. In some cases we find these conjectures to be consistent with the data, while in others the power law is ruled out. |
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Lin & Segel's Standing Gradient Problem Revisited: A Lesson in Mathematical Modeling and Asymptotics SIAM Rev. 53, pp. 775-796 (22 pages) Online Publication Date: November 07, 2011
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We revisit a physiological standing gradient problem of Lin and Segel from their landmark text on mathematical modeling [C. C. Lin and L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences, SIAM, Philadelphia, 1988] with a view to giving it an up-to-date perspective. In particular, via an alternative nondimensionalization, we show that the problem can be analyzed using the tools of singular perturbation theory and matched asymptotic expansions. In the spirit of the aforementioned authors, the development is didactic in style. Solving the problem requires many of the necessary skills of continuous modern mathematical modeling: formulation from a physical description of the process, scaling and asymptotic simplification, and solution using advanced perturbation (boundary layer) techniques. |
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Stabilization in a State-Dependent Model of Turning Processes SIAM J. Appl. Math. 72, pp. 1-24 (24 pages) Online Publication Date: January 03, 2012
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We consider a two-degree-of-freedom model for turning processes which involves a system of differential equations with state-dependent delay. Depending on process parameters (e.g., spindle speed, depth of cut) the cutting tool can exhibit unwanted vibrations, resulting in a nonsmooth surface of the workpiece. In this paper we propose a feedback law to stabilize the turning process for a large range of system parameters. The feedback law introduces a generic nonhyperbolic stationary point into the model, which generates the main technical challenge of this work. We establish the stability equivalence between the differential equations with state-dependent delay and a corresponding nonlinear system with the delay fixed at its stationary value. Then we show the stability of that nonlinear system with constant delay by computing its normal form. Finally, we obtain conditions on system parameters which guarantee the stability of the state-dependent delay model at the nonhyperbolic stationary point. |
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Tensor Decompositions and Applications SIAM Rev. 51, pp. 455-500 (46 pages) Online Publication Date: August 05, 2009
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This survey provides an overview of higher-order tensor decompositions, their applications, and available software. A tensor is a multidimensional or $N$-way array. Decompositions of higher-order tensors (i.e., $N$-way arrays with $N \geq 3$) have applications in psycho-metrics, chemometrics, signal processing, numerical linear algebra, computer vision, numerical analysis, data mining, neuroscience, graph analysis, and elsewhere. Two particular tensor decompositions can be considered to be higher-order extensions of the matrix singular value decomposition: CANDECOMP/PARAFAC (CP) decomposes a tensor as a sum of rank-one tensors, and the Tucker decomposition is a higher-order form of principal component analysis. There are many other tensor decompositions, including INDSCAL, PARAFAC2, CANDELINC, DEDICOM, and PARATUCK2 as well as nonnegative variants of all of the above. The N-way Toolbox, Tensor Toolbox, and Multilinear Engine are examples of software packages for working with tensors. |
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Lyapunov Functions and Global Stability for Age-Structured HIV Infection Model SIAM J. Appl. Math. 72, pp. 25-38 (14 pages) Online Publication Date: January 03, 2012
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We study the basic age-structured population model describing the HIV infection process, which is defined by PDEs. The model allows the production rate of viral particles and the death rate of productively infected cells to vary and depend on the infection age. By using the direct Lyapunov method and constructing suitable Lyapunov functions, dynamical properties of the age-structured model without (or with) drug treatment are established. The results show that the global asymptotic stability of the infection-free steady state and the infected steady state depends only on the basic reproductive number determined by the burst size. Further, we establish mathematically that the typical ODE and DDE (delay differential equation) models of HIV infection are equivalent to two special cases of the above PDE models. |
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Controlled Drug Delivery in Cancer Immunotherapy: Stability, Optimization, and Monte Carlo Analysis SIAM J. Appl. Math. 71, pp. 2229-2245 (17 pages) Online Publication Date: December 20, 2011
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A discussion on controlled drug delivery in cancer immunotherapy is presented in this paper. A fifth-order model is adopted to describe the dynamics of the tumor–immune interaction. Natural equilibrium points of this system are sought, and their stability is analyzed. An optimal control problem is stated and solved numerically. Both continuous and discrete controls are treated, and their implications on the therapy protocol are discussed. The robustness of the optimal therapies is assessed a posteriori with a Monte Carlo analysis. This shows that the control policy is effective even when the initial patient conditions are affected by uncertainties. |
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SIAM J. Control Optim. 50, pp. 1-22 (22 pages) Online Publication Date: January 03, 2012
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This paper is concerned with a Lyapunov inequality characterization of the eigenstructure assignment–based low gain feedback laws. With this characterization and our earlier characterizations of other low gain feedback design approaches, all existing low gain feedback designs are unified under this Lyapunov inequality framework, which in turn implies that all of these low gain feedback laws are both $L_{\infty}$ and $L_{2}$ low gain feedback. This Lyapunov inequality characterization also leads to a quadratic Lyapunov function for the closed-loop system, which is expected to play an important role in solving other control problems. This characterization also motivates a new Riccati inequality–based low gain feedback design, which not only possesses the appealing features of the existing low gain designs but also is computationally easy to carry out. |
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Sparse Optimization with Least-Squares Constraints SIAM J. Optim. 21, pp. 1201-1229 (29 pages) Online Publication Date: October 04, 2011
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The use of convex optimization for the recovery of sparse signals from incomplete or compressed data is now common practice. Motivated by the success of basis pursuit in recovering sparse vectors, new formulations have been proposed that take advantage of different types of sparsity. In this paper we propose an efficient algorithm for solving a general class of sparsifying formulations. For several common types of sparsity we provide applications, along with details on how to apply the algorithm, and experimental results. |
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The Mathematics of Atmospheric Dispersion Modeling SIAM Rev. 53, pp. 349-372 (24 pages) Online Publication Date: May 05, 2011
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The Gaussian plume model is a standard approach for studying the transport of airborne contaminants due to turbulent diffusion and advection by the wind. This paper reviews the assumptions underlying the model, its derivation from the advection-diffusion equation, and the key properties of the plume solution. The results are then applied to solving an inverse problem in which emission source rates are determined from a given set of ground-level contaminant measurements. This source identification problem can be formulated as an overdetermined linear system of equations that is most easily solved using the method of least squares. Various generalizations of this problem are discussed, and we illustrate our results with an application to the study of zinc emissions from a smelting operation. |
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Simulating Size-constrained Galton–Watson Trees SIAM J. Comput. 41, pp. 1-11 (11 pages) Online Publication Date: January 03, 2012
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We discuss various methods for generating random Galton–Watson trees conditional on their sizes being equal to $n$. A linear expected time algorithm is proposed. |
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Optimal Structural Policies for Ambiguity and Risk Averse Inventory and Pricing Models SIAM J. Control Optim. 50, pp. 133-146 (14 pages) Online Publication Date: January 05, 2012
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This paper discusses multiperiod stochastic joint inventory and pricing models when the decision maker is risk and ambiguity averse. We study infinite horizon models with discounted and long run average optimization criteria. The main result of this paper is establishing the optimality of stationary $(s,S,p)$ policies for the infinite horizon inventory and pricing models. |
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The Split Bregman Method for L1-Regularized Problems SIAM J. Imaging Sci. 2, pp. 323-343 (21 pages) Online Publication Date: April 01, 2009
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The class of L1-regularized optimization problems has received much attention recently because of the introduction of “compressed sensing,” which allows images and signals to be reconstructed from small amounts of data. Despite this recent attention, many L1-regularized problems still remain difficult to solve, or require techniques that are very problem-specific. In this paper, we show that Bregman iteration can be used to solve a wide variety of constrained optimization problems. Using this technique, we propose a “split Bregman” method, which can solve a very broad class of L1-regularized problems. We apply this technique to the Rudin–Osher–Fatemi functional for image denoising and to a compressed sensing problem that arises in magnetic resonance imaging. |
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Heterogeneous Multiscale Simulations of Suspension Flow Multiscale Model. Simul. 9, pp. 1301-1326 (26 pages) Online Publication Date: October 20, 2011
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The macroscopically emergent rheology of suspensions is dictated by details of fluid-particle and particle-particle interactions. For systems where the typical spatial scale on the particle level is much smaller than that of macroscopic properties, the scales can be split. We present a heterogeneous multiscale method (HMM) approach to modeling suspension flow in which at the macroscale the suspension is treated as a non-Newtonian fluid. The local shear-rate and particle volume fraction are input to a simulation of fully resolved suspension microdynamics. With the help of these simulations, the apparent viscosity and shear-induced diffusivities can be computed for a given shear-rate and volume fraction, and are then used to complete the information needed in the constitutional relations on the macroscopic level. On both levels, the lattice-Boltzmann method (LBM) is applied to model the fluid phase and coupled to a Lagrangian model for the advection-diffusion of the solid phase. Down and upward mapping of viscosity and diffusivity related quantities will be discussed, as well as information exchanged between the phases on both scales. Temporal scale splitting between viscous and diffusive dynamics has also been exploited to accelerate the macroscopic equilibration dynamics. Additionionally, Galileian and rotational symmetries allow us to make very efficient use of a database where the results of previous simulations can be stored, again reducing the computational effort by factors of several orders of magnitude. The HMM suspension model is applied to the simulation of a 2-dimensional flow through a straight channel of macroscopic width. The equilibration dynamics of flow and volume fraction profiles and equilibrium profiles of volume fraction, diffusivity, velocity, shear-rate, and viscosity are discussed. We show that the proposed HMM model not only reproduces experimental findings for low Reynolds numbers but also predicts additional dependencies introduced by shear-thickening effects not covered by existing macroscopic suspension flow models. |
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Fast Analytical Methods for Macroscopic Electrostatic Models in Biomolecular Simulations SIAM Rev. 53, pp. 683-720 (38 pages) Online Publication Date: November 07, 2011
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We review recent developments of fast analytical methods for macroscopic electrostatic calculations in biological applications, including the Poisson–Boltzmann (PB) and the generalized Born models for electrostatic solvation energy. The focus is on analytical approaches for hybrid solvation models, especially the image charge method for a spherical cavity, and also the generalized Born theory as an approximation to the PB model. This review places much emphasis on the mathematical details behind these methods. |
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A Distance For Multistage Stochastic Optimization Models SIAM J. Optim. 22, pp. 1-23 (23 pages) Online Publication Date: January 05, 2012
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We describe multistage stochastic programs in a purely in-distribution setting, i.e., without any reference to a concrete probability space. The concept is based on the notion of nested distributions, which encompass in one mathematical object the scenario values as well as the information structure under which decisions have to be made. The nested distance between these distributions is introduced and turns out to be a generalization of the Wasserstein distance for stochastic two-stage problems. We give characterizations of this distance and show its usefulness in examples. The main result states that the difference of the optimal values of two multistage stochastic programs, which are Lipschitz and differ only in the nested distribution of the stochastic parameters, can be bounded by the nested distance of these distributions. This theorem generalizes the well-known Kantorovich–Rubinstein theorem, which is applicable only in two-stage situations, to multistage. Moreover, a dual characterization for the nested distance is established. The setup is applicable both for general stochastic processes and for finite scenario trees. In particular, the nested distance between general processes and scenario trees is well defined and becomes the important tool for judging the quality of the scenario tree generation. Minimizing—at least heuristically—this distance is what good scenario tree generation is all about. |
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Convergence Speed in Distributed Consensus and Averaging SIAM Rev. 53, pp. 747-772 (26 pages) Online Publication Date: November 07, 2011
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We study the convergence speed of distributed iterative algorithms for the consensus and averaging problems, with emphasis on the latter. We first consider the case of a fixed communication topology. We show that a simple adaptation of a consensus algorithm leads to an averaging algorithm. We prove lower bounds on the worst-case convergence time for various classes of linear, time-invariant, distributed consensus methods, and provide an algorithm that essentially matches those lower bounds. We then consider the case of a time-varying topology, and provide a polynomial-time averaging algorithm. |
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Stable Computation of the CS Decomposition: Simultaneous Bidiagonalization SIAM. J. Matrix Anal. & Appl. 33, pp. 1-21 (21 pages) Online Publication Date: January 05, 2012
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Since its discovery in 1977, the CS decomposition (CSD) has resisted computation, even though it is a sibling of the well-understood eigenvalue and singular value decompositions. Several algorithms have been developed for the reduced 2-by-1 form of the decomposition, but none have been extended to the complete 2-by-2 form of the decomposition in Stewart's original paper. In this article, we present an algorithm for simultaneously bidiagonalizing the four blocks of a unitary matrix partitioned into a 2-by-2 block structure. This serves as the first, direct phase of a two-stage algorithm for the CSD, much as Golub–Kahan–Reinsch bidiagonalization serves as the first stage in computing the singular value decomposition. Backward stability is proved. |
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