2 research outputs found
Efficient approximation of the solution of certain nonlinear reaction--diffusion equation II: the case of large absorption
We study the positive stationary solutions of a standard finite-difference
discretization of the semilinear heat equation with nonlinear Neumann boundary
conditions. We prove that, if the absorption is large enough, compared with the
flux in the boundary, there exists a unique solution of such a discretization,
which approximates the unique positive stationary solution of the "continuous"
equation. Furthermore, we exhibit an algorithm computing an
-approximation of such a solution by means of a homotopy continuation
method. The cost of our algorithm is {\em linear} in the number of nodes
involved in the discretization and the logarithm of the number of digits of
approximation required
MPC-MPNet: Model-Predictive Motion Planning Networks for Fast, Near-Optimal Planning under Kinodynamic Constraints
Kinodynamic Motion Planning (KMP) is to find a robot motion subject to
concurrent kinematics and dynamics constraints. To date, quite a few methods
solve KMP problems and those that exist struggle to find near-optimal solutions
and exhibit high computational complexity as the planning space dimensionality
increases. To address these challenges, we present a scalable, imitation
learning-based, Model-Predictive Motion Planning Networks framework that
quickly finds near-optimal path solutions with worst-case theoretical
guarantees under kinodynamic constraints for practical underactuated systems.
Our framework introduces two algorithms built on a neural generator,
discriminator, and a parallelizable Model Predictive Controller (MPC). The
generator outputs various informed states towards the given target, and the
discriminator selects the best possible subset from them for the extension. The
MPC locally connects the selected informed states while satisfying the given
constraints leading to feasible, near-optimal solutions. We evaluate our
algorithms on a range of cluttered, kinodynamically constrained, and
underactuated planning problems with results indicating significant
improvements in computation times, path qualities, and success rates over
existing methods