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Stability Criteria for Switched and Hybrid Systems
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| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 49 (No. 4) |
| Halaman | : | 545-592 |
| Abstrak | : | The study of the stability properties of switched and hybrid systems gives rise to a number of interesting and challenging mathematical problems. The objective of this paper is to outline some of these problems, to review progress made in solving them in a number of diverse communities, and to review some problems that remain open. An important contribution of our work is to bring together material from several areas of research and to present results in a unified manner. We begin our review by relating the stability problem for switched linear systems and a class of linear differential inclusions. Closely related to the concept of stability are the notions of exponential growth rates and converse Lyapunov theorems, both of which are discussed in detail. In particular, results on common quadratic Lyapunov functions and piecewise linear Lyapunov functions are presented, as they represent constructive methods for proving stability and also represent problems in which significant progress has been made. We also comment on the inherent difficulty in determining stability of switched systems in general, which is exemplified by NP-hardness and undecidability results. We then proceed by considering the stability of switched systems in which there are constraints on the switching rules, through both dwell-time requirements and state-dependent switching laws. Also in this case the theory of Lyapunov functions and the existence of converse theorems are reviewed. We briefly comment on the classical Lur'e problem and on the theory of stability radii, both of which contain many of the features of switched systems and are rich sources of practical results on the topic. Finally we present a list of questions and open problems which provide motivation for continued research in this area. |
As Flat As Possible
| Pengarang | : | Jon Jacobsen |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 49 (No. 3) |
| Halaman | : | 491-507 |
| Abstrak | : | How does one determine a surface which is as flat as possible, such as those created by soap film surfaces? What does it mean to be as flat as possible? In this paper we address this question from two distinct points of view, one local and one global in nature. Continuing with this theme, we put a temporal twist on the question and ask how to evolve a surface so as to flatten it as efficiently as possible. This elementary discussion provides a platform to introduce a wide range of advanced topics in partial differential equations and helps students build geometric and analytic understanding of solutions of certain elliptic and parabolic partial differential equations. |
Deciding the Nature of the Coarse Equation through Microscopic Simulations: The Baby-Bathwater Scheme
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 49 (No. 3) |
| Halaman | : | 469-487 |
| Abstrak | : | Recent developments in multiscale computation allow the solution of coarse equations for the expected macroscopic behavior of microscopically evolving particles without ever obtaining these coarse equations in closed form. The closure is obtained on demand through appropriately initialized bursts of microscopic simulation. The effective coupling of microscopic simulators with macroscopic behavior requires certain decisions about the nature of the unavailable coarse equation. Such decisions include (a) the highest spatial derivative active in the coarse equation, (b) whether the equation satisfies certain conservation laws, or (c) whether the coarse dynamics is Hamiltonian or dissipative. These decisions affect the number and type of boundary conditions as well as the algorithms employed. In the absence of an explicit formula for the temporal derivative, we propose, implement, and validate a simple scheme for deciding these and other similar questions about the coarse equation using only the microscopic simulator. Simulations under periodic boundary conditions are carried out for appropriately chosen families of random initial conditions; evaluating the sample variance of certain statistics over the simulation ensemble allows us to infer the highest order of spatial derivatives active in the coarse equation. In the same spirit we show how to determine whether a certain coarse conservation law exists or not, and we discuss plausibility tests for the existence of a coarse Hamiltonian or integrability. We believe that such schemes constitute an important part of the equation-free approach to multiscale computation. |
A Reduced-Order Kalman Filter for Data Assimilation in Physical Oceanography
| Pengarang | : | D. Rozier |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 49 (No. 3) |
| Halaman | : | 449-465 |
| Abstrak | : | A central task of physical oceanography is the prediction of ocean circulation at various time scales. Mathematical techniques are used in this domain not only for the modeling of ocean circulation but also for the enhancement of simulation through data assimilation. The ocean circulation model of concern here, namely, HYCOM, is briefly presented through its variables, equations, and specific vertical coordinate system. The main part of this paper focuses on the Kalman filter as a data assimilation method, and especially on how this mathematical technique, usually associated with a prohibitively high computing cost for operational sciences, is simplified in order to make it applicable to the simulation of realistic ocean circulation models. Some practical issues are presented, such as a brief explanation about ocean observation systems, together with examples of data assimilation results. |
A Direct Formulation for Sparse PCA Using Semidefinite Programming
| Pengarang | : | Laurent El Ghaoui |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 49 (No. 3) |
| Halaman | : | 434-448 |
| Abstrak | : | iven a covariance matrix, we consider the problem of maximizing the variance explained by a particular linear combination of the input variables while constraining the number of nonzero coefficients in this combination. This problem arises in the decomposition of a covariance matrix into sparse factors or sparse principal component analysis (PCA), and has wide applications ranging from biology to finance. We use a modification of the classical variational representation of the largest eigenvalue of a symmetric matrix, where cardinality is constrained, and derive a semidefinite programming–based relaxation for our problem. We also discuss Nesterov's smooth minimization technique applied to the semidefinite program arising in the semidefinite relaxation of the sparse PCA problem. The method has complexity , where n is the size of the underlying covariance matrix and is the desired absolute accuracy on the optimal value of the problem. |
Buchberger's Algorithm and the Two-Locus, Two-Allele Model
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 49 (No. 3) |
| Halaman | : | 421-433 |
| Abstrak | : | The present paper uses results from algebraic geometry to study the classical model describing the equilibrium frequencies of the four gametic types in the two-locus, two-allele genetic model under the assumption that the fitnesses are constant and that the cis- and trans- fitnesses are equal. It shows that there is a finite process for determining an upper bound on the maximum number of equilibrium frequencies for almost all choices of recombination and fitness parameters, and that this number is finite. It also shows that the solutions are locally continuous for almost all choices of parameters. The paper studies the equilibrium frequencies as functions of the recombination parameter, with attention to the cases when the recombination becomes infinite and when one of the equilibrium frequencies becomes infinite. |
A Survey of the S-Lemma
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 49 (No. 3) |
| Halaman | : | 371-418 |
| Abstrak | : | In this survey we review the many faces of the S-lemma, a result about the correctness of the S-procedure. The basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization, convex geometry, and linear algebra as well. These were all active research areas, but as there was little interaction between researchers in these different areas, their results remained mainly isolated. Here we give a unified analysis of the theory by providing three different proofs for the S-lemma and revealing hidden connections with various areas of mathematics. We prove some new duality results and present applications from control theory, error estimation, and computational geometry. |
Determining Sets for the Discrete Laplacian
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 49 (No. 2) |
| Halaman | : | 315-324 |
| Abstrak | : | We define a notion of determining sets for the discrete Laplacian in a domain Ω. A set D is called determining if harmonic functions are uniquely determined by providing their values on D, and if D has the same size as the boundary of Ω. It is shown that there exist determining sets that are fairly evenly distributed in Ω. A number of basic properties of determining sets are derived. |
Fundamental Solutions for Some Partial Differential Operators from Fluid Dynamics and Statistical Physics
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 49 (No. 2) |
| Halaman | : | 303-314 |
| Abstrak | : | We present a method to find fundamental solutions for a class of partial differential equations that often arise in fluid dynamics and in transport problems. The method is elementary in the sense that it uses only linear algebra and ODEs. |
Error Estimation for Reduced?Order Models of Dynamical Systems
| Pengarang | : | Chris Homescu |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 49 (No. 2) |
| Halaman | : | 277-299 |
| Abstrak | : | The use of reduced?order models to describe a dynamical system is pervasive in science and engineering. Often these models are used without an estimate of their error or range of validity. In this paper we consider dynamical systems and reduced models built using proper orthogonal decomposition. We show how to compute estimates and bounds for these errors by a combination of small sample statistical condition estimation and error estimation using the adjoint method. Most important, the proposed approach allows the assessment of regions of validity for reduced models, i.e., ranges of perturbations in the original system over which the reduced model is still appropriate. Numerical examples validate our approach: the error norm estimates approximate well the forward error, while the derived bounds are within an order of magnitude. |