96 research outputs found

    Duality theory and optimal transport for sand piles growing in a silos

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    We prove existence and uniqueness of solutions for a system of PDEs which describes the growth of a sandpile in a silos with flat bottom under the action of a vertical, measure source. The tools we use are a discrete approximation of the source and the duality theory for optimal transport (or Monge-Kantorovich) problems

    PDEs with Compressed Solutions

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    Sparsity plays a central role in recent developments in signal processing, linear algebra, statistics, optimization, and other fields. In these developments, sparsity is promoted through the addition of an L1L^1 norm (or related quantity) as a constraint or penalty in a variational principle. We apply this approach to partial differential equations that come from a variational quantity, either by minimization (to obtain an elliptic PDE) or by gradient flow (to obtain a parabolic PDE). Also, we show that some PDEs can be rewritten in an L1L^1 form, such as the divisible sandpile problem and signum-Gordon. Addition of an L1L^1 term in the variational principle leads to a modified PDE where a subgradient term appears. It is known that modified PDEs of this form will often have solutions with compact support, which corresponds to the discrete solution being sparse. We show that this is advantageous numerically through the use of efficient algorithms for solving L1L^1 based problems.Comment: 21 pages, 15 figure

    Convergence of the Abelian sandpile

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    The Abelian sandpile growth model is a diffusion process for configurations of chips placed on vertices of the integer lattice Zd\mathbb{Z}^d, in which sites with at least 2d chips {\em topple}, distributing 1 chip to each of their neighbors in the lattice, until no more topplings are possible. From an initial configuration consisting of nn chips placed at a single vertex, the rescaled stable configuration seems to converge to a particular fractal pattern as nn\to \infty. However, little has been proved about the appearance of the stable configurations. We use PDE techniques to prove that the rescaled stable configurations do indeed converge to a unique limit as nn \to \infty. We characterize the limit as the Laplacian of the solution to an elliptic obstacle problem.Comment: 12 pages, 2 figures, acroread recommended for figure displa

    Strong Spherical Asymptotics for Rotor-Router Aggregation and the Divisible Sandpile

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    The rotor-router model is a deterministic analogue of random walk. It can be used to define a deterministic growth model analogous to internal DLA. We prove that the asymptotic shape of this model is a Euclidean ball, in a sense which is stronger than our earlier work. For the shape consisting of n=ωdrdn=\omega_d r^d sites, where ωd\omega_d is the volume of the unit ball in Rd\R^d, we show that the inradius of the set of occupied sites is at least rO(logr)r-O(\log r), while the outradius is at most r+O(rα)r+O(r^\alpha) for any α>11/d\alpha > 1-1/d. For a related model, the divisible sandpile, we show that the domain of occupied sites is a Euclidean ball with error in the radius a constant independent of the total mass. For the classical abelian sandpile model in two dimensions, with n=πr2n=\pi r^2 particles, we show that the inradius is at least r/3r/\sqrt{3}, and the outradius is at most (r+o(r))/2(r+o(r))/\sqrt{2}. This improves on bounds of Le Borgne and Rossin. Similar bounds apply in higher dimensions.Comment: [v3] Added Theorem 4.1, which generalizes Theorem 1.4 for the abelian sandpile. [v4] Added references and improved exposition in sections 2 and 4. [v5] Final version, to appear in Potential Analysi

    Apollonian structure in the Abelian sandpile

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    The Abelian sandpile process evolves configurations of chips on the integer lattice by toppling any vertex with at least 4 chips, distributing one of its chips to each of its 4 neighbors. When begun from a large stack of chips, the terminal state of the sandpile has a curious fractal structure which has remained unexplained. Using a characterization of the quadratic growths attainable by integer-superharmonic functions, we prove that the sandpile PDE recently shown to characterize the scaling limit of the sandpile admits certain fractal solutions, giving a precise mathematical perspective on the fractal nature of the sandpile.National Science Foundation (U.S.) (grants DMS-1004696, DMS-1004595 and DMS- 1243606
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