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Conjugate Points Revisited and Neumann–Neumann Problems
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| Pengarang | : | |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 51-1, MARCH (No. 1) |
| Halaman | : | 193-212 |
| Abstrak | : | The theory of conjugate points in the calculus of variations is reconsidered with a perspective emphasizing the connection to finite-dimensional optimization. The object of central importance is the spectrum of the second-variation operator, analogous to the eigenvalues of the Hessian matrix in finite dimensions. With a few basic properties of this spectrum, one can gain a new perspective on the classic result that “stability requires the lack of conjugate points.” Furthermore, we show how the spectral perspective allows the extension of the conjugate point approach to variants of the classic problems in the literature, such as problems with Neumann–Neumann boundary conditions. |
Nonsmooth Coordination and Geometric Optimization via Distributed Dynamical Systems
| Pengarang | : | Es, Elizabeth A. van,Barnhart, Tara |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 51-1, MARCH (No. 1) |
| Halaman | : | 163-189 |
| Abstrak | : | Emerging applications for networked and cooperative robots motivate the study of motion coordination for groups of agents. For example, it is envisioned that groups of agents will perform a variety of useful tasks including surveillance, exploration, and environmental monitoring. This paper deals with basic interactions among mobile agents such as “move away from the closest other agent” or “move toward the furthest vertex of your own Voronoi polygon.” These simple interactions amount to distributed dynamical systems because their implementation requires only minimal information about neighboring agents. We characterize the close relationship between these distributed dynamical systems and the disk-covering and sphere-packing cost functions from geometric optimization. Our main results are as follows: (i) we characterize the smoothness properties of these geometric cost functions, (ii) we show that the interaction laws are variations of the nonsmooth gradient of the cost functions, and (iii) we establish various asymptotic convergence properties of the laws. The technical approach relies on concepts from computational geometry, nonsmooth analysis, and nonsmooth stability theory. |
Optimization and Performance Modeling of Stencil Computations on Modern Microprocessors
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 51-1, MARCH (No. 1) |
| Halaman | : | 129-159 |
| Abstrak | : | Stencil-based kernels constitute the core of many important scientific applications on block-structured grids. Unfortunately, these codes achieve a low fraction of peak performance, due primarily to the disparity between processor and main memory speeds. In this paper, we explore the impact of trends in memory subsystems on a variety of stencil optimization techniques and develop performance models to analytically guide our optimizations. Our work targets cache reuse methodologies across single and multiple stencil sweeps, examining cache-aware algorithms as well as cache-oblivious techniques on the Intel Itanium2, AMD Opteron, and IBM Power5. Additionally, we consider stencil computations on the heterogeneous multicore design of the Cell processor, a machine with an explicitly managed memory hierarchy. Overall our work represents one of the most extensive analyses of stencil optimizations and performance modeling to date. Results demonstrate that recent trends in memory system organization have reduced the efficacy of traditional cache-blocking optimizations. We also show that a cache-aware implementation is significantly faster than a cache-oblivious approach, while the explicitly managed memory on Cell enables the highest overall efficiency: Cell attains 88% of algorithmic peak while the best competing cache-based processor achieves only 54% of algorithmic peak performance. |
High Order Weighted Essentially Nonoscillatory Schemes for Convection Dominated Problems
| Pengarang | : | Chi-Wang Shu |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 51-1, MARCH (No. 1) |
| Halaman | : | 82-126 |
| Abstrak | : | High order accurate weighted essentially nonoscillatory (WENO) schemes are relatively new but have gained rapid popularity in numerical solutions of hyperbolic partial differential equations (PDEs) and other convection dominated problems. The main advantage of such schemes is their capability to achieve arbitrarily high order formal accuracy in smooth regions while maintaining stable, nonoscillatory, and sharp discontinuity transitions. The schemes are thus especially suitable for problems containing both strong discontinuities and complex smooth solution features. WENO schemes are robust and do not require the user to tune parameters. At the heart of the WENO schemes is actually an approximation procedure not directly related to PDEs, hence the WENO procedure can also be used in many non-PDE applications. In this paper we review the history and basic formulation of WENO schemes, outline the main ideas in using WENO schemes to solve various hyperbolic PDEs and other convection dominated problems, and present a collection of applications in areas including computational fluid dynamics, computational astronomy and astrophysics, semiconductor device simulation, traffic flow models, computational biology, and some non-PDE applications. Finally, we mention a few topics concerning WENO schemes that are currently under investigation. |
From Sparse Solutions of Systems of Equations to Sparse Modeling of Signals and Images
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 51-1, MARCH (No. 1) |
| Halaman | : | 34-81 |
| Abstrak | : | A full-rank matrix with generates an underdetermined system of linear equations having infinitely many solutions. Suppose we seek the sparsest solution, i.e., the one with the fewest nonzero entries. Can it ever be unique? If so, when? As optimization of sparsity is combinatorial in nature, are there efficient methods for finding the sparsest solution? These questions have been answered positively and constructively in recent years, exposing a wide variety of surprising phenomena, in particular the existence of easily verifiable conditions under which optimally sparse solutions can be found by concrete, effective computational methods. Such theoretical results inspire a bold perspective on some important practical problems in signal and image processing. Several well-known signal and image processing problems can be cast as demanding solutions of undetermined systems of equations. Such problems have previously seemed, to many, intractable, but there is considerable evidence that these problems often have sparse solutions. Hence, advances in finding sparse solutions to underdetermined systems have energized research on such signal and image processing problems—to striking effect. In this paper we review the theoretical results on sparse solutions of linear systems, empirical results on sparse modeling of signals and images, and recent applications in inverse problems and compression in image processing. This work lies at the intersection of signal processing and applied mathematics, and arose initially from the wavelets and harmonic analysis research communities. The aim of this paper is to introduce a few key notions and applications connected to sparsity, targeting newcomers interested in either the mathematical aspects of this area or its applications. |
Cloaking Devices, Electromagnetic Wormholes, and Transformation Optics
| Pengarang | : | Allan Greenleaf |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 51-1, MARCH (No. 1) |
| Halaman | : | 3-33 |
| Abstrak | : | We describe recent theoretical and experimental progress on making objects invisible to detection by electromagnetic waves. Ideas for devices that would once have seemed fanciful may now be at least approximately implemented physically using a new class of artificially structured materials called metamaterials. Maxwell's equations have transformation laws that allow for the design of electromagnetic material parameters that steer light around a hidden region, returning it to its original path on the far side. Not only would observers be unaware of the contents of the hidden region, they would not even be aware that something was being hidden. An object contained in the hidden region, which would have no shadow, is said to be cloaked. Proposals for, and even experimental implementations of, such cloaking devices have received the most attention, but other designs having striking effects on wave propagation are possible. All of these designs are initially based on the transformation laws of the equations that govern wave propagation but, due to the singular parameters that give rise to the desired effects, care needs to be taken in formulating and analyzing physically meaningful solutions. We recount the recent history of the subject and discuss some of the mathematical and physical issues involved. |
Unstable Solutions of Nonautonomous Linear Differential Equations
| Pengarang | : | Masters, Heidi |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 50-3, SEPTEMBER (No. 3) |
| Halaman | : | 570-584 |
| Abstrak | : | The fact that the eigenvalues of the family of matrices do not determine the stability of nonautonomous differential equations is well known. This point is often illustrated using examples in which the matrices have constant eigenvalues with negative real part, but the solutions of the corresponding differential equation grow in time. Here we provide an intuitive, geometric explanation of the idea that underlies these examples. The discussion is accompanied by a number of animations and easily modifiable Mathematica programs. We conclude with a discussion of possible extensions of the ideas that may provide suitable topics for undergraduate research. |
On Spiking Models for Synaptic Activity and Impulsive Differential Equations
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 50-3, SEPTEMBER (No. 3) |
| Halaman | : | 553-569 |
| Abstrak | : | We illustrate the problems that can arise in writing differential equations that include Dirac delta functions to model equations with state-dependent impulsive forcing. Specifically, difficulties arise in the interpretation of the products of distributions with discontinuous functions. We suggest several methods to resolve these ambiguities, such as using limiting sequences and asymptotic analysis, with applications of the results given for discrete maps. These suggestions are applied to a popular model describing synaptic connections in the brain. |
A Method to Compute Statistics of Large, Noise-Induced Perturbations of Nonlinear Schrödinger Solitons
| Pengarang | : | Campbell, Todd,Melville, Wayne ,Verma, Geeta |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 50-3, SEPTEMBER (No. 3) |
| Halaman | : | 523-549 |
| Abstrak | : | We describe in detail the application of importance sampling to numerical simulations of large noise-induced perturbations in soliton-based optical transmission systems governed by the nonlinear Schrödinger equation. The method allows one to concentrate the samples in Monte Carlo simulations around those noise realizations that are most likely to produce the large pulse deformations connected with errors, and it yields computational speedups of several orders of magnitude over standard Monte Carlo simulations. We demonstrate the method by using it to calculate the probability density functions associated with pulse amplitude, frequency, timing, and phase fluctuations in a prototypical soliton-based communication system. |
The Bad Truth about Laplace's Transform
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 50-3, SEPTEMBER (No. 3) |
| Halaman | : | 504-520 |
| Abstrak | : | Inverting the Laplace transform is a paradigm for exponentially ill-posed problems. For a class of operators, including the Laplace transform, we give forward and inverse formulae that have fast implementations using the fast Fourier transform. These formulae lead easily to regularized inverses whose effects on noise and filtered data can be precisely described. Our results give cogent reasons for the general sense of dread most mathematicians feel about inverting the Laplace transform. |