44,031 research outputs found
Branching diffusion representation of semilinear PDEs and Monte Carlo approximation
We provide a representation result of parabolic semi-linear PD-Es, with
polynomial nonlinearity, by branching diffusion processes. We extend the
classical representation for KPP equations, introduced by Skorokhod (1964),
Watanabe (1965) and McKean (1975), by allowing for polynomial nonlinearity in
the pair , where is the solution of the PDE with space gradient
. Similar to the previous literature, our result requires a non-explosion
condition which restrict to "small maturity" or "small nonlinearity" of the
PDE. Our main ingredient is the automatic differentiation technique as in Henry
Labordere, Tan and Touzi (2015), based on the Malliavin integration by parts,
which allows to account for the nonlinearities in the gradient. As a
consequence, the particles of our branching diffusion are marked by the nature
of the nonlinearity. This new representation has very important numerical
implications as it is suitable for Monte Carlo simulation. Indeed, this
provides the first numerical method for high dimensional nonlinear PDEs with
error estimate induced by the dimension-free Central limit theorem. The
complexity is also easily seen to be of the order of the squared dimension. The
final section of this paper illustrates the efficiency of the algorithm by some
high dimensional numerical experiments
Information Acquisition with Sensing Robots: Algorithms and Error Bounds
Utilizing the capabilities of configurable sensing systems requires
addressing difficult information gathering problems. Near-optimal approaches
exist for sensing systems without internal states. However, when it comes to
optimizing the trajectories of mobile sensors the solutions are often greedy
and rarely provide performance guarantees. Notably, under linear Gaussian
assumptions, the problem becomes deterministic and can be solved off-line.
Approaches based on submodularity have been applied by ignoring the sensor
dynamics and greedily selecting informative locations in the environment. This
paper presents a non-greedy algorithm with suboptimality guarantees, which does
not rely on submodularity and takes the sensor dynamics into account. Our
method performs provably better than the widely used greedy one. Coupled with
linearization and model predictive control, it can be used to generate adaptive
policies for mobile sensors with non-linear sensing models. Applications in gas
concentration mapping and target tracking are presented.Comment: 9 pages (two-column); 2 figures; Manuscript submitted to the 2014
IEEE International Conference on Robotics and Automatio
Exact Simulation of Non-stationary Reflected Brownian Motion
This paper develops the first method for the exact simulation of reflected
Brownian motion (RBM) with non-stationary drift and infinitesimal variance. The
running time of generating exact samples of non-stationary RBM at any time
is uniformly bounded by where is the
average drift of the process. The method can be used as a guide for planning
simulations of complex queueing systems with non-stationary arrival rates
and/or service time
Steady-state simulation of reflected Brownian motion and related stochastic networks
This paper develops the first class of algorithms that enable unbiased
estimation of steady-state expectations for multidimensional reflected Brownian
motion. In order to explain our ideas, we first consider the case of compound
Poisson (possibly Markov modulated) input. In this case, we analyze the
complexity of our procedure as the dimension of the network increases and show
that, under certain assumptions, the algorithm has polynomial-expected
termination time. Our methodology includes procedures that are of interest
beyond steady-state simulation and reflected processes. For instance, we use
wavelets to construct a piecewise linear function that can be guaranteed to be
within distance (deterministic) in the uniform norm to Brownian
motion in any compact time interval.Comment: Published at http://dx.doi.org/10.1214/14-AAP1072 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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