49M25 Discrete approximations
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- English (3)
Keywords
- Markov chains (1)
- Trajectory planning, optimal control problem, collision avoidance, (1)
- Wasserstein gradient flow (1)
- adaptive mesh refinement (1)
- entropy/entropy-dissipation formulation (1)
- finite-volume scheme (1)
- goal oriented error estimation (1)
- gradient structures (1)
- graph search algorithm, initialization, robotics (1)
- optimal control (1)
A coupling of discrete and continuous optimization to solve kinodynamic motion planning problems
(2014)
This paper studies the relationship between the material derivative method, the shape derivative method, the min-max formulation of Correa and Seeger, and the Lagrange method introduced by Cea. A theorem is formulated
which allows a rigorous proof of the shape differentiability without the usage of material derivative;
the domain expression is automatically obtained and the boundary expression is easy to derive.
Furthermore, the theorem is applied to a cost function which depends on a quasi-linear transmission
problem. Using a Gagliardo penalization the existence of optimal shapes is established.
We study the approximation of Wasserstein gradient structures
by their finite-dimensional analog. We show that simple
finite-volume discretizations of the linear Fokker-Planck
equation exhibit the recently established entropic gradient-flow
structure for reversible Markov chains. Then we reprove the
convergence of the discrete scheme in the limit of vanishing
mesh size using only the involved gradient-flow structures.
In particular, we make no use of the linearity of the equations
nor of the fact that the Fokker-Planck equation is of second order.
The paper proposes goal-oriented error estimation and mesh refinement
for optimal control problems with elliptic PDE constraints using the value
of the reduced cost functional as quantity of interest. Error representation,
hierarchical error estimators, and greedy-style error indicators are derived and
compared to their counterparts when using the all-at-once cost functional as
quantity of interest. Finally, the efficiency of the error estimator and generated
meshes are demonstrated on numerical examples.