14,664 research outputs found
External optimal control of fractional parabolic PDEs
In this paper we introduce a new notion of optimal control, or source
identification in inverse, problems with fractional parabolic PDEs as
constraints. This new notion allows a source/control placement outside the
domain where the PDE is fulfilled. We tackle the Dirichlet, the Neumann and the
Robin cases. For the fractional elliptic PDEs this has been recently
investigated by the authors in \cite{HAntil_RKhatri_MWarma_2018a}. The need for
these novel optimal control concepts stems from the fact that the classical PDE
models only allow placing the source/control either on the boundary or in the
interior where the PDE is satisfied. However, the nonlocal behavior of the
fractional operator now allows placing the control in the exterior. We
introduce the notions of weak and very-weak solutions to the parabolic
Dirichlet problem. We present an approach on how to approximate the parabolic
Dirichlet solutions by the parabolic Robin solutions (with convergence rates).
A complete analysis for the Dirichlet and Robin optimal control problems has
been discussed. The numerical examples confirm our theoretical findings and
further illustrate the potential benefits of nonlocal models over the local
ones.Comment: arXiv admin note: text overlap with arXiv:1811.0451
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
Gradient-based estimation of Manning's friction coefficient from noisy data
We study the numerical recovery of Manning's roughness coefficient for the
diffusive wave approximation of the shallow water equation. We describe a
conjugate gradient method for the numerical inversion. Numerical results for
one-dimensional model are presented to illustrate the feasibility of the
approach. Also we provide a proof of the differentiability of the weak form
with respect to the coefficient as well as the continuity and boundedness of
the linearized operator under reasonable assumptions using the maximal
parabolic regularity theory.Comment: 19 pages, 3 figure
Singularly perturbed forward-backward stochastic differential equations: application to the optimal control of bilinear systems
We study linear-quadratic stochastic optimal control problems with bilinear
state dependence for which the underlying stochastic differential equation
(SDE) consists of slow and fast degrees of freedom. We show that, in the same
way in which the underlying dynamics can be well approximated by a reduced
order effective dynamics in the time scale limit (using classical
homogenziation results), the associated optimal expected cost converges in the
time scale limit to an effective optimal cost. This entails that we can well
approximate the stochastic optimal control for the whole system by the reduced
order stochastic optimal control, which is clearly easier to solve because of
lower dimensionality. The approach uses an equivalent formulation of the
Hamilton-Jacobi-Bellman (HJB) equation, in terms of forward-backward SDEs
(FBSDEs). We exploit the efficient solvability of FBSDEs via a least squares
Monte Carlo algorithm and show its applicability by a suitable numerical
example
- …