This issue of SIAM Review presents two papers in the Education section. The first paper, “Deforming $\|\cdot\|_1$ into $\|\cdot\|_\infty$ via Polyhedral Norms: A Pedestrian Approach,” is contributed by Manlio Gaudioso and Jean-Baptiste Hiriart-Urruty. The paper starts from the well-known properties of the $\ell_p$-norms in the Euclidean space $\mathbb{R}^n$. For $x\in\mathbb{R}^n$ and $p\geq 1$, the norm is given by $\|x\|_p=\big(\sum_{i=1}^n |x_i|^p\big)^\frac{1}{p}$ with the limiting case $p\to\infty$ providing the norm $\|x\|_\infty=\max_{1\leq i\leq n} |x_i|.$ The unit balls determined by the norms for $p=1$ and $p=\infty$ are polyhedral, while for $p\in (1,\infty)$ the unit ball has a smooth surface. Furthermore, the unit balls become larger when $p$ grows. One can represent the polyhedral norms (for $p=1$ or $p=\infty$) as maxima of linear forms, and this observation motivates the following notion. For an integer $k$ with value between 1 and $n$, we define the following real-valued function: \[ N_k(x)=\max\|x_i_1|+|x_i_2|+\dots+|x_i_k|: 1łeq i_1 <\dots <i_k łeq n\. \] It is not difficult to check that $N_k$ is a norm, $N_1$ is the infinity norm, $N_n$ is the $\ell_1$-norm, and $N_k$ provides a value between them. In this way, the $\ell_1$ ball is deformed into the $\ell_\infty$ ball using only polyhedral convex sets. The authors discuss the properties and some representations of these norms. They provide the distance between the unit balls when $k$ changes, and discuss the relations of $N_k$ to support functions and gauges. Special cases for $n=3$ or 4 are examined and illustrated. Conjugate duality is invoked to present the dual norm to $N_k$ and the respective dual ball. The authors point out the relevance of these norms to some modern optimization problems, in which a sparse solution is desired. The polyhedral representation of the norms also facilitates numerical methods. The paper is written in a clear and engaging manner and is accessible to advanced undergraduate students who are familiar with basic notions of convex analysis. The paper “Analyzing Pattern Formation in the Gray--Scott Model: An XPPAUT Tutorial” is presented by Demi L. Gandy and Martin R. Nelson. It discusses a mathematical model of autocatalytic chemical processes that may lead to spontaneous formation of spatial and spatio-temporal patterns. The authors focus on a reaction-diffusion model, introduced by Gray and Scott, in which a pair of two chemicals, one reacting and one diffusing, can give rise to formation of spots, stripes, spirals, and other patterns. The model consists of two partial differential equations (PDEs) describing the process, in which two generic chemicals U and V react to produce a product P. It assumes that U is continuously supplied and the inert product P is continuously removed. The model uses the concentrations of U and V, the rate of replenishment of U, and the rate at which V decays to produce P. It also uses two diffusion constants, representing the rate at which each chemical moves spatially. The paper explains the main steps of the analysis of this model for the purpose of identifying pattern formation and illustrates this using an example. The first step is to understand the dynamics of the corresponding homogeneous system, which is obtained by neglecting the diffusion terms. Once the spatial dependence is removed the PDE system reduces to a pair of ordinary differential equations. The steady states are identified and their stability is characterized. The key step is to understand how changes of the parameters impact the steady states. At this point, software named XPPAUT may be used to compute bifurcation diagrams in order to gain more detailed information about periodic structures. Further, the spatial terms are reintroduced and one has to examine how these terms give rise to spatially inhomogeneous solutions. The authors describe the capabilities of XPPAUT and give instructions for its use. They also use MATLAB to visualize some patterns and make all code used in the paper freely available online at https://github.com/martinrnelson/GrayScott. The targeted audience is students who have completed an introductory course on dynamical systems and bifurcation theory, have some familiarity with PDEs such as the heat equation, and have knowledge of basic numerical method techniques.
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Sep 2, 2024
Deepsikha Mahanta + 2
Open Access