Abstract. We consider the aggregation equationWe assume that K is rotationally invariant, nonnegative, decaying at infinity, with at worst a Lipschitz point at the origin. We prove existence, uniqueness, and continuation of solutions. Finite time blow-up (in the L ∞ norm) of solutions is proved when the kernel has precisely a Lipschitz point at the origin.
Eigenvalue problems are fundamental to mathematics and science. We present a simple algorithm for determining eigenvalues and eigenfunctions of the Laplace-Beltrami operator on rather general curved surfaces. Our algorithm, which is based on the Closest Point Method, relies on an embedding of the surface in a higher-dimensional space, where standard Cartesian finite difference and interpolation schemes can be easily applied. We show that there is a one-to-one correspondence between a problem defined in the embedding space and the original surface problem. For open surfaces, we present a simple way to impose Dirichlet and Neumann boundary conditions while maintaining second-order accuracy. Convergence studies and a series of examples demonstrate the effectiveness and generality of our approach. or, more generally, the elliptic operatorThe Laplace-Beltrami eigenvalue problem has played a prominent role in recent years in data analysis. For example, in [1], eigenvalues of the Laplace-Beltrami operator were used to extract "fingerprints" which characterize surfaces and solid objects. In [2, 3], Laplace-Beltrami eigenvalues and eigenfunctions were used for dimensionality reduction and data representation. Other application areas include smoothing of surfaces [4] and the segmentation and registration of shape [5].Analytical solutions to the Laplace-Beltrami eigenvalue problem are rarely available, so it is crucial to be able to numerically approximate them in an accurate and efficient manner. Partial differential equations on surfaces, including eigenvalue problems, have traditionally been approximated using either (a) discretizations based on a parameterization of the surface [6], (b) finite element discretizations on a triangulation of the surface [7], or (c) embedding techniques which solve some embedding PDE in a small region near the surface [8] (see also the related works [9,10,11,12,13,14,15]).Parameterization methods (a) are often effective for simple surfaces [6], but for more complicated geometries have the deficiency of introducing distortions and singularities into the method through the parameterization [16]. Approaches based on the finite element method can be deceptively difficult to implement; as described in [7], "even though this method seems to be very simple, it is quite tricky to implement". Embedding methods (c) have gained a considerable following because they permit PDEs on surfaces to be solving using standard finite differences.This paper proposes a simple and effective embedding method for the Laplace-Beltrami eigenvalue problem based on the Closest Point Method. The Closest Point Method is a recent embedding method that has been used to compute the numerical solution to a variety of partial differential equations [17,18,19,20], including in-surface heat flow, reaction-diffusion equations, and higher-order motions involving biharmonic and "surface diffusion" terms. Unlike traditional embedding methods, which are built around level set representatives of the surface, the Closest Point Method i...
We investigate low-energy deformations of a thin elastic sheet subject to a displacement boundary condition consistent with a conical deformation. Under the assumption that the displacement near the sheet's center is of order h| log h|, where h 1 is the thickness of the sheet, we establish matching upper and lower bounds of order h 2 | log h| for the minimum elastic energy per unit thickness, with a prefactor determined by the geometry of the associated conical deformation. These results are established first for a 2D model problem and then extended to 3D elasticity.
Abstract. Methane emissions from oil/gas fields originate from a large number of relatively small and densely clustered point sources. A small fraction of high-mode emitters can make a large contribution to the total methane emission. Here we conduct observation system simulation experiments (OSSEs) to examine the potential of recently launched or planned satellites to detect and locate these high-mode emitters through measurements of atmospheric methane columns. We simulate atmospheric methane over a generic oil/gas field (20–500 production sites of different size categories in a 50×50 km2 domain) for a 1-week period using the WRF-STILT meteorological model with 1.3×1.3 km2 horizontal resolution. The simulations consider many random realizations for the occurrence and distribution of high-mode emitters in the field by sampling bimodal probability density functions (PDFs) of emissions from individual sites. The atmospheric methane fields for each realization are observed virtually with different satellite and surface observing configurations. Column methane enhancements observed from satellites are small relative to instrument precision, even for high-mode emitters, so an inverse analysis is necessary. We compare L1 and L2 regularizations and show that L1 regularization effectively provides sparse solutions for a bimodally distributed variable and enables the retrieval of high-mode emitters. We find that the recently launched TROPOMI instrument (low Earth orbit, 7×7 km2 nadir pixels, daily return time) and the planned GeoCARB instrument (geostationary orbit, 2.7×3.0 km2 pixels, 2 times or 4 times per day return times) are successful (> 80 % detection rate, < 20 % false alarm rate) at locating high-emitting sources for fields of 20–50 emitters within the 50×50 km2 domain as long as skies are clear. They are unsuccessful for denser fields. GeoCARB does not benefit significantly from more frequent observations (4 times per day vs. 2 times per day) because of a temporal error correlation in the inversion, unless under partly cloudy conditions where more frequent observation increases the probability of clear sky. It becomes marginally successful when allowing a 5 km error tolerance for localization. A next-generation geostationary satellite instrument with 1.3×1.3 km2 pixels, hourly return time, and 1 ppb precision can successfully detect and locate the high-mode emitters for a dense field with up to 500 sites in the 50×50 km2 domain. The capabilities of TROPOMI and GeoCARB can be usefully augmented with a surface air observation network of 5–20 sites, and in turn the satellite instruments increase the detection capability that can be achieved from the surface sites alone.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.