104,075 research outputs found
Pooling or sampling: Collective dynamics for electrical flow estimation
The computation of electrical flows is a crucial primitive for many recently proposed optimization algorithms on weighted networks. While typically implemented as a centralized subroutine, the ability to perform this task in a fully decentralized way is implicit in a number of biological systems. Thus, a natural question is whether this task can provably be accomplished in an efficient way by a network of agents executing a simple protocol. We provide a positive answer, proposing two distributed approaches to electrical flow computation on a weighted network: a deterministic process mimicking Jacobi's iterative method for solving linear systems, and a randomized token diffusion process, based on revisiting a classical random walk process on a graph with an absorbing node. We show that both processes converge to a solution of Kirchhoff's node potential equations, derive bounds on their convergence rates in terms of the weights of the network, and analyze their time and message complexity
Pooling or Sampling: Collective Dynamics for Electrical Flow Estimation
The computation of electrical flows is a crucial primitive for many recently
proposed optimization algorithms on weighted networks. While typically
implemented as a centralized subroutine, the ability to perform this task in a
fully decentralized way is implicit in a number of biological systems. Thus, a
natural question is whether this task can provably be accomplished in an
efficient way by a network of agents executing a simple protocol.
We provide a positive answer, proposing two distributed approaches to
electrical flow computation on a weighted network: a deterministic process
mimicking Jacobi's iterative method for solving linear systems, and a
randomized token diffusion process, based on revisiting a classical random walk
process on a graph with an absorbing node. We show that both processes converge
to a solution of Kirchhoff's node potential equations, derive bounds on their
convergence rates in terms of the weights of the network, and analyze their
time and message complexity
A random map implementation of implicit filters
Implicit particle filters for data assimilation generate high-probability
samples by representing each particle location as a separate function of a
common reference variable. This representation requires that a certain
underdetermined equation be solved for each particle and at each time an
observation becomes available. We present a new implementation of implicit
filters in which we find the solution of the equation via a random map. As
examples, we assimilate data for a stochastically driven Lorenz system with
sparse observations and for a stochastic Kuramoto-Sivashinski equation with
observations that are sparse in both space and time
Reflected scheme for doubly reflected BSDEs with jumps and RCLL obstacles
We introduce a discrete time reflected scheme to solve doubly reflected
Backward Stochastic Differential Equations with jumps (in short DRBSDEs),
driven by a Brownian motion and an independent compensated Poisson process. As
in Dumitrescu-Labart (2014), we approximate the Brownian motion and the Poisson
process by two random walks, but contrary to this paper, we discretize directly
the DRBSDE, without using a penalization step. This gives us a fully
implementable scheme, which only depends on one parameter of approximation: the
number of time steps (contrary to the scheme proposed in Dumitrescu-Labart
(2014), which also depends on the penalization parameter). We prove the
convergence of the scheme, and give some numerical examples.Comment: arXiv admin note: text overlap with arXiv:1406.361
Stochastic fiber dynamics in a spatially semi-discrete setting
We investigate a spatially discrete surrogate model for the dynamics of a
slender, elastic, inextensible fiber in turbulent flows. Deduced from a
continuous space-time beam model for which no solution theory is available, it
consists of a high-dimensional second order stochastic differential equation in
time with a nonlinear algebraic constraint and an associated Lagrange
multiplier term. We establish a suitable framework for the rigorous formulation
and analysis of the semi-discrete model and prove existence and uniqueness of a
global strong solution. The proof is based on an explicit representation of the
Lagrange multiplier and on the observation that the obtained explicit drift
term in the equation satisfies a one-sided linear growth condition on the
constraint manifold. The theoretical analysis is complemented by numerical
studies concerning the time discretization of our model. The performance of
implicit Euler-type methods can be improved when using the explicit
representation of the Lagrange multiplier to compute refined initial estimates
for the Newton method applied in each time step.Comment: 20 pages; typos removed, references adde
Convergence of numerical methods for stochastic differential equations in mathematical finance
Many stochastic differential equations that occur in financial modelling do
not satisfy the standard assumptions made in convergence proofs of numerical
schemes that are given in textbooks, i.e., their coefficients and the
corresponding derivatives appearing in the proofs are not uniformly bounded and
hence, in particular, not globally Lipschitz. Specific examples are the Heston
and Cox-Ingersoll-Ross models with square root coefficients and the Ait-Sahalia
model with rational coefficient functions. Simple examples show that, for
example, the Euler-Maruyama scheme may not converge either in the strong or
weak sense when the standard assumptions do not hold. Nevertheless, new
convergence results have been obtained recently for many such models in
financial mathematics. These are reviewed here. Although weak convergence is of
traditional importance in financial mathematics with its emphasis on
expectations of functionals of the solutions, strong convergence plays a
crucial role in Multi Level Monte Carlo methods, so it and also pathwise
convergence will be considered along with methods which preserve the positivity
of the solutions.Comment: Review Pape
A Milstein scheme for SPDEs
This article studies an infinite dimensional analog of Milstein's scheme for
finite dimensional stochastic ordinary differential equations (SODEs). The
Milstein scheme is known to be impressively efficient for SODEs which fulfill a
certain commutativity type condition. This article introduces the infinite
dimensional analog of this commutativity type condition and observes that a
certain class of semilinear stochastic partial differential equation (SPDEs)
with multiplicative trace class noise naturally fulfills the resulting infinite
dimensional commutativity condition. In particular, a suitable infinite
dimensional analog of Milstein's algorithm can be simulated efficiently for
such SPDEs and requires less computational operations and random variables than
previously considered algorithms for simulating such SPDEs. The analysis is
supported by numerical results for a stochastic heat equation and stochastic
reaction diffusion equations showing signifficant computational savings.Comment: The article is slightly revised and shortened. In particular, some
numerical simulations are remove
Stable and fast semi-implicit integration of the stochastic Landau-Lifshitz equation
We propose new semi-implicit numerical methods for the integration of the
stochastic Landau-Lifshitz equation with built-in angular momentum
conservation. The performance of the proposed integrators is tested on the 1D
Heisenberg chain. For this system, our schemes show better stability properties
and allow us to use considerably larger time steps than standard explicit
methods. At the same time, these semi-implicit schemes are also of comparable
accuracy to and computationally much cheaper than the standard midpoint
implicit method. The results are of key importance for atomistic spin dynamics
simulations and the study of spin dynamics beyond the macro spin approximation.Comment: 24 pages, 5 figure
- âŠ