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Augmented Phase Reduction of (Not So) Weakly Perturbed Coupled Oscillators
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| Pengarang | : | Dan Wilson |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 61 (No. 2) |
| Halaman | : | 277-315 |
| Abstrak | : | While phase reduction is a tremendously useful tool for understanding the dynamics of weakly perturbed limit cycle oscillators, its assumptions break down as perturbations become larger, limiting its practical utility in many applications. This fundamental limitation is often apparent when studying coupled populations of oscillators when the collective behavior approaches a limit cycle with transient behavior that decays slowly. Using the notion of isostables of periodic orbits, which define a coordinate system transverse to the limit cycle, we develop a strategy to augment the standard phase reduction in large populations of coupled oscillators to allow for the application of larger perturbations. The resulting augmented phase reduction yields a tremendous decrease in model complexity with a resulting dimensionality that does not depend on the number of oscillators in the population. Additionally, the augmented phase reduction replicates model behavior that standard phase reduction cannot. Finally, we propose and implement a direct method for the application of augmented phase reduction that would be applicable in systems of biological oscillators. Applying the augmented phase reduction to a model of coupled circadian oscillators yields novel insight about the possible mechanism behind the difference in reported jet-lag severity between eastward and westward travel. |
Network Modularity in the Presence of Covariates
| Pengarang | : | Beate Ehrhardt |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 61 (No. 2) |
| Halaman | : | 261-276 |
| Abstrak | : | We characterize the large-sample properties of network modularity in the presence of covariates, under a natural and flexible null model. This provides for the first time an objective measure of whether or not a particular value of modularity is meaningful. In particular, our results quantify the strength of the relation between observed community structure and the interactions in a network. Our technical contribution is to provide limit theorems for modularity when a community assignment is given by nodal features or covariates. These theorems hold for a broad class of network models over a range of sparsity regimes, as well as for weighted, multiedge, and power-law networks. This allows us to assign -values to observed community structure, which we validate using several benchmark examples from the literature. We conclude by applying this methodology to investigate a multiedge network of corporate email interactions. |
Dynamics over Signed Networks
| Pengarang | : | Guodong Shi |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 61 (No. 2) |
| Halaman | : | 229-257 |
| Abstrak | : | A signed network is a network in which each link is associated with a positive or negative sign. Models for nodes interacting over such signed networks arise from various biological, social, political, and economic systems. As modifications to the conventional DeGroot dynamics for positive links, two basic types of negative interactions along negative links, namely, the opposing rule and the repelling rule, have been proposed and studied in the literature. This paper reviews a few fundamental convergence results for such dynamics over deterministic or random signed networks under a unified algebraic-graphical method. We show that a systematic tool for studying node state evolution over signed networks can be obtained utilizing generalized Perron--Frobenius theory, graph theory, and elementary algebraic recursions. |
The Renormalization Group: A Perturbation Method for the Graduate Curriculum
| Pengarang | : | Eleftherios Kirkinis |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 54 (No. 2) |
| Halaman | : | 374-388 |
| Abstrak | : | In this paper the renormalization group (RG) method of Chen, Goldenfeld, and Oono [Phys. Rev. Lett., 73 (1994), pp. 1311–1315; Phys. Rev. E, 54 (1996), pp. 376–394] is presented in a pedagogical way to increase its visibility in applied mathematics and to argue favorably for its incorporation into the corresponding graduate curriculum. The method is illustrated by some linear and nonlinear singular perturbation problems. |
Modified Gershgorin Disks for Companion Matrices
| Pengarang | : | Aaron Melman |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 54 (No. 2) |
| Halaman | : | 355-373 |
| Abstrak | : | All the zeros of a polynomial are contained in the union of Gershgorin disks derived from its companion matrix, a consequence of Gershgorin's theorem. However, this theorem does not exploit the structure of the companion matrix. We will use this structure to obtain smaller zero inclusion regions, thereby providing some nonstandard results to accompany and illustrate this frequently covered topic in numerical and matrix analysis. |
Chemical Reactions as Gamma-Limit of Diffusion
| Pengarang | : | Mark A. Peletier |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 54 (No. 2) |
| Halaman | : | 327-352 |
| Abstrak | : | We study the limit of high activation energy of a special Fokker–Planck equation known as the Kramers–Smoluchowski equation (KS). This equation governs the time evolution of the probability density of a particle performing a Brownian motion under the influence of a chemical potential . We choose H having two wells corresponding to two chemical states A and B. We prove that after a suitable rescaling the solution to the KS converges, in the limit of high activation energy (), to the solution of a simpler system modeling the spatial diffusion of A and B combined with the reaction . With this result we give a rigorous proof of Kramers's formal derivation, and we show how chemical reactions and diffusion processes can be embedded in a common framework. This allows one to derive a chemical reaction as a singular limit of a diffusion process, thus establishing a connection between two worlds often regarded as separate. The proof rests on two main ingredients. One is the formulation of the two disparate equations as evolution equations for measures. The second is a variational formulation of both equations that allows us to use the tools of variational calculus and, specifically, -convergence. |
Synthesis, as Opposed to Separation, of Variables
| Pengarang | : | A. S. Fokas |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 54 (No. 2) |
| Halaman | : | 291-324 |
| Abstrak | : | Every applied mathematician has used separation of variables. For a given boundary value problem (BVP) in two dimensions, the starting point of this powerful method is the separation of the given PDE into two ODEs. If the spectral analysis of either of these ODEs yields an appropriate transform pair, i.e., a transform consistent with the given boundary conditions, then the given BVP can be reduced to a BVP for an ODE. For simple BVPs it is straightforward to choose an appropriate transform and hence the spectral analysis can be avoided. In spite of its enormous applicability, this method has certain limitations. In particular, it requires the given domain, PDE, and boundary conditions to be separable, and also may not be applicable if the BVP is non-self-adjoint. Furthermore, it expresses the solution as either an integral or a series, neither of which are uniformly convergent on the boundary of the domain (for nonvanishing boundary conditions), which renders such expressions unsuitable for numerical computations. This paper describes a recently introduced transform method that can be applied to certain nonseparable and non-self-adjoint problems. Furthermore, this method expresses the solution as an integral in the complex plane that is uniformly convergent on the boundary of the domain. The starting point of the method is to write the PDE as a one-parameter family of equations formulated in a divergence form, and this allows one to consider the variables together. In this sense, the method is based on the “synthesis” as opposed to the “separation” of variables. The new method has already been applied to a plethora of BVPs and furthermore has led to the development of certain novel numerical techniques. However, a large number of related analytical and numerical questions remain open. This paper illustrates the method by applying it to two particular non-self-adjoint BVPs: one for the linearized KdV equation formulated on the half-line, and the other for the Helmholtz equation in the exterior of the disc (the latter is non-self-adjoint due to the radiation condition). The former problem played a crucial role in the development of the new method, whereas the latter problem was instrumental in the full development of the classical transform method. Although the new method can now be presented using only classical techniques, it actually originated in the theory of certain nonlinear PDEs called integrable, whose crucial feature is the existence of a Lax pair formulation. It is shown here that Lax pairs provide the generalization of the divergence formulation from a separable linear to an integrable nonlinear PDE. |
Mixed-Mode Oscillations with Multiple Time Scales
| Pengarang | : | Mathieu Desroches |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 54 (No. 2) |
| Halaman | : | 211-288 |
| Abstrak | : | Mixed-mode oscillations (MMOs) are trajectories of a dynamical system in which there is an alternation between oscillations of distinct large and small amplitudes. MMOs have been observed and studied for over thirty years in chemical, physical, and biological systems. Few attempts have been made thus far to classify different patterns of MMOs, in contrast to the classification of the related phenomena of bursting oscillations. This paper gives a survey of different types of MMOs, concentrating its analysis on MMOs whose small-amplitude oscillations are produced by a local, multiple-time-scale “mechanism.” Recent work gives substantially improved insight into the mathematical properties of these mechanisms. In this survey, we unify diverse observations about MMOs and establish a systematic framework for studying their properties. Numerical methods for computing different types of invariant manifolds and their intersections are an important aspect of the analysis described in this paper. |
From Random Polygon to Ellipse: An Eigenanalysis
| Pengarang | : | Adam N. Elmachtoub |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 52 (No. 1) |
| Halaman | : | 151-170 |
| Abstrak | : | Suppose x and y are unit 2-norm n-vectors whose components sum to zero. Let be the polygon obtained by connecting in order. We say that is the normalized average of if it is obtained by connecting the midpoints of its edges and then normalizing the resulting vertex vectors and so that they have unit 2-norm. If this process is repeated starting with , then in the limit the vertices of the polygon iterates converge to an ellipse that is centered at the origin and whose semiaxes are tilted forty-five degrees from the coordinate axes. An eigenanalysis together with the singular value decomposition is used to explain this phenomenon. The problem and its solution is a metaphor for matrix-based research in computational science and engineering. |
Image Denoising Methods. A New Nonlocal Principle
| Pengarang | : | J. M. Morel |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 52 (No. 1) |
| Halaman | : | 113-147 |
| Abstrak | : | The search for efficient image denoising methods is still a valid challenge at the crossing of functional analysis and statistics. In spite of the sophistication of the recently proposed methods, most algorithms have not yet attained a desirable level of applicability. All show an outstanding performance when the image model corresponds to the algorithm assumptions but fail in general and create artifacts or remove fine structures in images. The main focus of this paper is, first, to define a general mathematical and experimental methodology to compare and classify classical image denoising algorithms and, second, to propose a nonlocal means (NL-means) algorithm addressing the preservation of structure in a digital image. The mathematical analysis is based on the analysis of the “method noise,” defined as the difference between a digital image and its denoised version. The NL-means algorithm is proven to be asymptotically optimal under a generic statistical image model. The denoising performance of all considered methods is compared in four ways; mathematical: asymptotic order of magnitude of the method noise under regularity assumptions; perceptual-mathematical: the algorithms artifacts and their explanation as a violation of the image model; quantitative experimental: by tables of distances of the denoised version to the original image. The fourth and perhaps most powerful evaluation method is, however, the visualization of the method noise on natural images. The more this method noise looks like a real white noise, the better the method. |