Showing papers in "Siam Review in 2001"

Atomic Decomposition by Basis Pursuit

Abstract: The time-frequency and time-scale communities have recently developed a large number of overcomplete waveform dictionaries---stationary wavelets, wavelet packets, cosine packets, chirplets, and warplets, to name a few. Decomposition into overcomplete systems is not unique, and several methods for decomposition have been proposed, including the method of frames (MOF), matching pursuit (MP), and, for special dictionaries, the best orthogonal basis (BOB). Basis pursuit (BP) is a principle for decomposing a signal into an "optimal"' superposition of dictionary elements, where optimal means having the smallest l1 norm of coefficients among all such decompositions. We give examples exhibiting several advantages over MOF, MP, and BOB, including better sparsity and superresolution. BP has interesting relations to ideas in areas as diverse as ill-posed problems, abstract harmonic analysis, total variation denoising, and multiscale edge denoising. BP in highly overcomplete dictionaries leads to large-scale optimization problems. With signals of length 8192 and a wavelet packet dictionary, one gets an equivalent linear program of size 8192 by 212,992. Such problems can be attacked successfully only because of recent advances in linear and quadratic programming by interior-point methods. We obtain reasonable success with a primal-dual logarithmic barrier method and conjugate-gradient solver.

An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations

Abstract: A practical and accessible introduction to numerical methods for stochastic differential equations is given. The reader is assumed to be familiar with Euler's method for deterministic differential equations and to have at least an intuitive feel for the concept of a random variable; however, no knowledge of advanced probability theory or stochastic processes is assumed. The article is built around $10$ MATLAB programs, and the topics covered include stochastic integration, the Euler--Maruyama method, Milstein's method, strong and weak convergence, linear stability, and the stochastic chain rule.

Strong Stability-Preserving High-Order Time Discretization Methods

Abstract: In this paper we review and further develop a class of strong stability-preserving (SSP) high-order time discretizations for semidiscrete method of lines approximations of partial differential equations. Previously termed TVD (total variation diminishing) time discretizations, these high-order time discretization methods preserve the strong stability properties of first-order Euler time stepping and have proved very useful, especially in solving hyperbolic partial differential equations. The new developments in this paper include the construction of optimal explicit SSP linear Runge--Kutta methods, their application to the strong stability of coercive approximations, a systematic study of explicit SSP multistep methods for nonlinear problems, and the study of the SSP property of implicit Runge--Kutta and multistep methods.

The Quadratic Eigenvalue Problem

Abstract: We survey the quadratic eigenvalue problem, treating its many applications, its mathematical properties, and a variety of numerical solution techniques. Emphasis is given to exploiting both the structure of the matrices in the problem (dense, sparse, real, complex, Hermitian, skew-Hermitian) and the spectral properties of the problem. We classify numerical methods and catalogue available software.

Nonuniform Sampling and Reconstruction in Shift-Invariant Spaces

Abstract: This article discusses modern techniques for nonuniform sampling and reconstruction of functions in shift-invariant spaces. It is a survey as well as a research paper and provides a unified framework for uniform and nonuniform sampling and reconstruction in shift-invariant subspaces by bringing together wavelet theory, frame theory, reproducing kernel Hilbert spaces, approximation theory, amalgam spaces, and sampling. Inspired by applications taken from communication, astronomy, and medicine, the following aspects will be emphasized: (a) The sampling problem is well defined within the setting of shift-invariant spaces. (b) The general theory works in arbitrary dimension and for a broad class of generators. (c) The reconstruction of a function from any sufficiently dense nonuniform sampling set is obtained by efficient iterative algorithms. These algorithms converge geometrically and are robust in the presence of noise. (d) To model the natural decay conditions of real signals and images, the sampling theory is developed in weighted L p-spaces.

Markowitz Revisited: Mean-Variance Models in Financial Portfolio Analysis

Abstract: Mean-variance portfolio analysis provided the first quantitative treatment of the tradeoff between profit and risk. We describe in detail the interplay between objective and constraints in a number of single-period variants, including semivariance models. Particular emphasis is laid on avoiding the penalization of overperformance. The results are then used as building blocks in the development and theoretical analysis of multiperiod models based on scenario trees. A key property is the possibility of removing surplus money in future decisions, yielding approximate downside risk minimization.

A Mathematical Tutorial on Synthetic Aperture Radar

Abstract: This paper presents the foundations of conventional strip-mode synthetic aperture radar (SAR) from a mathematical point of view. In particular, the paper shows how a simple antenna model can be used together with a linearized scattering approximation to predict the received signal. The conventional matched-filter processing is explained and analyzed to exhibit the resolution of the SAR system.

Synchronization of Elliptic Bursters

Abstract: Periodic bursting behavior in neurons is a recurrent transition between a quiescent state and repetitive spiking. When the transition to repetitive spiking occurs via a subcritical Andronov--Hopf bifurcation and the transition to the quiescent state occurs via fold limit cycle bifurcation, the burster is said to be of elliptic type (also known as a "subHopf/fold cycle" burster). Here we study the synchronization dynamics of weakly connected networks of such bursters. We find that the behavior of such networks is quite different from the behavior of weakly connected phase oscillators and resembles that of strongly connected relaxation oscillators. As a result, such weakly connected bursters need few (usually one) bursts to synchronize, and synchronization is possible for bursters having quite different quantitative features. We also find that interactions between bursters depend crucially on the spiking frequencies. Namely, the interactions are most effective when the presynaptic interspike frequency matches the frequency of postsynaptic oscillations. Finally, we use the FitzHugh--Rinzel, Morris--Lecar, and Hodgkin--Huxley models to illustrate our major results.

Statistical Regularization of Inverse Problems

Abstract: In experimental sciences we often need to solve inverse problems. That is, we want to obtain information about the internal structure of a physical system from indirect noisy observations. Often the problem is not whether a solution exists; on the contrary, there are too many solutions that fit the data to a chosen tolerance level. The goal is to use prior information to determine a physically meaningful solution. Here, we present some of the basic questions that arise. We describe methods that can be used to find inversion estimates as well as ways to assess their performance.

From Finite Covariance Windows to Modeling Filters: A Convex Optimization Approach

Abstract: The trigonometric moment problem is a classical moment problem with numerous applications in mathematics, physics, and engineering. The rational covariance extension problem is a constrained version of this problem, with the constraints arising from the physical realizability of the corresponding solutions. Although the maximum entropy method gives one well-known solution, in several applications a wider class of solutions is desired. In a seminal paper, Georgiou derived an existence result for a broad class of models. In this paper, we review the history of this problem, going back to Carath{eodory, as well as applications to stochastic systems and signal processing. In particular, we present a convex optimization problem for solving the rational covariance extension problem with degree constraint. Given a partial covariance sequence and the desired zeros of the shaping filter, the poles are uniquely determined from the unique minimum of the corresponding optimization problem. In this way we obtain an algorithm for solving the covariance extension problem, as well as a constructive proof of Georgiou's existence result and his conjecture, a generalized version of which we have recently resolved using geometric methods. We also survey recent related results on constrained Nevanlinna--Pick interpolation in the context of a variational formulation of the general moment problem.

Optimality Models in Behavioral Biology

Abstract: The action of natural selection results in organisms that are good at surviving and reproducing. We show how this intuitive idea can be given a formal definition in terms of fitness and reproductive value. An optimal strategy maximizes fitness, and reproductive value provides a common currency for comparing different actions. We provide a broad review of models and methods that have been used in this area, stressing the conceptual issues and exposing the logic of evolutionary explanations.

Periodicity versus Chaos in One-Dimensional Dynamics

Abstract: We survey recent results in one-dimensional dynamics and, as an application, we derive rigorous basic dynamical facts for two standard models in population dynamics, the Ricker and the Hassell families. We also informally discuss the concept of chaos in the context of one-dimensional discrete time models. First we use the model case of the quadratic family for an informal exposition. We then review precise generic results before turning to the population models. Our focus is on typical asymptotic behavior, seen for most initial conditions and for large sets of maps. Parameter sets corresponding to different types of attractors are described. In particular it is shown that maps with strong chaotic properties appear with positive frequency in parameter space in our population models. Natural measures (asymptotic distributions) and their stability properties are considered.

A Randomized Fully Polynomial Time Approximation Scheme for the All-Terminal Network Reliability Problem

Abstract: The classic all-terminal network reliability problem posits a graph, each of whose edges fails independently with some given probability. The goal is to determine the probability that the network becomes disconnected due to edge failures. This problem has obvious applications in the design of communication networks. Since the problem is ${\sharp {\cal P}}$-complete and thus believed hard to solve exactly, a great deal of research has been devoted to estimating the failure probability. In this paper, we give a fully polynomial randomized approximation scheme that, given any n-vertex graph with specified failure probabilities, computes in time polynomial in n and $1/\epsilon$ an estimate for the failure probability that is accurate to within a relative error of $1\pm\epsilon$ with high probability. We also give a deterministic polynomial approximation scheme for the case of small failure probabilities. Some extensions to evaluating probabilities of $k$-connectivity, strong connectivity in directed Eulerian graphs and $r$-way disconnection, and to evaluating the Tutte polynomial are also described. This version of the paper corrects several errata that appeared in the previous journal publication [D. R. Karger, SIAM J. Comput., 29 (1999), pp. 492--514].

A Matrix Version of the Fast Multipole Method

Abstract: We present a matrix interpretation of the three-dimensional fast multipole method (FMM). The FMM is for efficient computation of gravitational/electrostatic potentials and fields. It has found various applications and inspired the design of many efficient algorithms. The one-dimensional FMM is well interpreted in terms of matrix computations. The three-dimensional matrix version reveals the underlying matrix structures and computational techniques used in FMM. It also provides a unified view of algorithm variants as well as existing and emerging implementations of the FMM.

The Geometry of the Scroll Compressor

Abstract: The scroll compressor is an ingenious machine used for compressing air or refrigerant; it was originally invented in 1905 by Leon Creux. The classical design consists of two nested identical scrolls given by circle involutes, one of which is rotated through $\text{180}^{\circ}$ with respect to the other. By specifying not a parametrization of the curve, but instead the radius of curvature as a function of tangent direction and using the intrinsic equation of a planar curve, the design can be changed in a way that allows all relevant geometrical quantities to be calculated in closed analytical form.

The Lindstedt--Poincaré Technique as an Algorithm for Computing Periodic Orbits

Abstract: The Lindstedt--Poincare technique in perturbation theory is used to calculate periodic orbits of perturbed differential equations. It uses a nearby periodic orbit of the unperturbed differential equation as the first approximation. We derive a numerical algorithm based upon this technique for computing periodic orbits of dynamical systems. The algorithm, unlike the Lindstedt--Poincare technique, does not require the dynamical system to be a small perturbation of a solvable differential equation. This makes it more broadly applicable. The algorithm is quadratically convergent. It works with equal facility, as examples show, irrespective of whether the periodic orbit is attracting, or repelling, or a saddle. One of the examples presents what is possibly the most accurate computation of Hill's orbit of lunation since its justly celebrated discovery in 1878.

Matrix Exponentials---Another Approach

Abstract: The exponential of a matrix and the spectral decomposition of a matrix can be computed knowing nothing more than the eigenvalues of the matrix and the Cayley--Hamilton theorem. The arrangement of the ideas in this paper is simple enough to be taught to beginning students of ODEs.

A Fast Newton Algorithm for Entropy Maximization in Phase Determination

Abstract: A long-standing issue in the Bayesian statistical approach to the phase problem in X-ray crystallography is to solve an entropy maximization subproblem efficiently in every iteration of phase estimation. The entropy maximization problem is a semi-infinite convex program and can be solved in a finite dual space by using a standard Newton method. However, the Newton method is too expensive for this application since it requires O(n3) floating point operations per iteration, where n corresponds to the number of phases to be estimated. Other less expensive methods have been used, but they cannot guarantee fast convergence. In this paper, we present a fast Newton algorithm for the entropy maximization problem, which uses the Sherman--Morrison--Woodbury formula and the fast Fourier transform to compute the Newton step and requires only O(n log n) floating point operations per iteration, yet has the same convergence rate as the standard Newton. We describe the algorithm and discuss related computational issues. Numerical results on simple test cases will also be presented to demonstrate the behavior of the algorithm.

Schwarz--Christoffel Mapping of the Annulus

Abstract: A new derivation of the Schwarz--Christoffel (S--C) transformation for doubly connected domains is given. The work is based on constructing the pre-Schwarzian by reflection of singularities. A derivation of the simply connected S--C map without appeal to Liouville's theorem is a byproduct of this work.

An Application of Optimal Control to the Economics of Recycling

Abstract: A city's landfill is an exhaustible resource; recycling is a backstop method of waste disposal. We formulate an optimal control model that maximizes the total utility of a representative consumer by choosing the appropriate levels of the two disposal techniques. The model is simple enough that some general conclusions can be drawn, and yet specific solutions will require some computations that show a few of the possibilities in optimal control problems. Among the primary results is that once recycling begins, it will increase. It is also likely that the level of recycling will assume the values at the endpoints of its domain as well as the more standard values in the interior.

The 2000 International Mathematical Olympiad

Abstract: Report on the International Mathematical Olympiad of 2000 held in South Korea. The International Mathematical Olympiad is an international competition which started in 1959, and since, the competition has attracted the world's most eminent mathematicians and scientists.