96 research outputs found
Duality theory and optimal transport for sand piles growing in a silos
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
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 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 form, such as the divisible sandpile problem and
signum-Gordon. Addition of an 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 based
problems.Comment: 21 pages, 15 figure
Convergence of the Abelian sandpile
The Abelian sandpile growth model is a diffusion process for configurations
of chips placed on vertices of the integer lattice , 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 chips placed at a single vertex, the rescaled
stable configuration seems to converge to a particular fractal pattern as . 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 . 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
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
sites, where is the volume of the unit ball in , we show that
the inradius of the set of occupied sites is at least , while the
outradius is at most for any . 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 particles, we show that the inradius is at least , and the
outradius is at most . 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
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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