5,554 research outputs found
Robust preconditioners for PDE-constrained optimization with limited observations
Regularization robust preconditioners for PDE-constrained optimization
problems have been successfully developed. These methods, however, typically
assume that observation data is available throughout the entire domain of the
state equation. For many inverse problems, this is an unrealistic assumption.
In this paper we propose and analyze preconditioners for PDE-constrained
optimization problems with limited observation data, e.g. observations are only
available at the boundary of the solution domain. Our methods are robust with
respect to both the regularization parameter and the mesh size. That is, the
condition number of the preconditioned optimality system is uniformly bounded,
independently of the size of these two parameters. We first consider a
prototypical elliptic control problem and thereafter more general
PDE-constrained optimization problems. Our theoretical findings are illuminated
by several numerical results
Primal dual mixed finite element methods for indefinite advection--diffusion equations
We consider primal-dual mixed finite element methods for the
advection--diffusion equation. For the primal variable we use standard
continuous finite element space and for the flux we use the Raviart-Thomas
space. We prove optimal a priori error estimates in the energy- and the
-norms for the primal variable in the low Peclet regime. In the high
Peclet regime we also prove optimal error estimates for the primal variable in
the norm for smooth solutions. Numerically we observe that the method
eliminates the spurious oscillations close to interior layers that pollute the
solution of the standard Galerkin method when the local Peclet number is high.
This method, however, does produce spurious solutions when outflow boundary
layer presents. In the last section we propose two simple strategies to remove
such numerical artefacts caused by the outflow boundary layer and validate them
numerically.Comment: 25 pages, 6 figures, 5 table
Orbitally stable standing waves of a mixed dispersion nonlinear Schr\"odinger equation
We study the mixed dispersion fourth order nonlinear Schr\"odinger equation
\begin{equation*} %\tag{\protect{4NLS}}\label{4nls} i \partial_t \psi -\gamma
\Delta^2 \psi +\beta \Delta \psi +|\psi|^{2\sigma} \psi =0\ \text{in}\ \R
\times\R^N, \end{equation*} where and . We
focus on standing wave solutions, namely solutions of the form , for some . This ansatz yields the
fourth-order elliptic equation \begin{equation*}
%\tag{\protect{*}}\label{4nlsstar} \gamma \Delta^2 u -\beta \Delta u +\alpha u
=|u|^{2\sigma} u. \end{equation*} We consider two associated constrained
minimization problems: one with a constraint on the -norm and the other on
the -norm. Under suitable conditions, we establish existence of
minimizers and we investigate their qualitative properties, namely their sign,
symmetry and decay at infinity as well as their uniqueness, nondegeneracy and
orbital stability.Comment: 37 pages. To appear in SIAM J. Math. Ana
Shock Formation in Small-Data Solutions to Quasilinear Wave Equations: An Overview
In his 2007 monograph, D. Christodoulou proved a remarkable result giving a
detailed description of shock formation, for small -initial conditions
( sufficiently large), in solutions to the relativistic Euler equations in
three space dimensions. His work provided a significant advancement over a
large body of prior work concerning the long-time behavior of solutions to
higher-dimensional quasilinear wave equations, initiated by F. John in the mid
1970's and continued by S. Klainerman, T. Sideris, L. H\"ormander, H. Lindblad,
S. Alinhac, and others. Our goal in this paper is to give an overview of his
result, outline its main new ideas, and place it in the context of the above
mentioned earlier work. We also introduce the recent work of J. Speck, which
extends Christodoulou's result to show that for two important classes of
quasilinear wave equations in three space dimensions, small-data shock
formation occurs precisely when the quadratic nonlinear terms fail the classic
null condition
Inverse problems for linear hyperbolic equations using mixed formulations
We introduce in this document a direct method allowing to solve numerically
inverse type problems for linear hyperbolic equations. We first consider the
reconstruction of the full solution of the wave equation posed in - a bounded subset of - from a partial
distributed observation. We employ a least-squares technique and minimize the
-norm of the distance from the observation to any solution. Taking the
hyperbolic equation as the main constraint of the problem, the optimality
conditions are reduced to a mixed formulation involving both the state to
reconstruct and a Lagrange multiplier. Under usual geometric optic conditions,
we show the well-posedness of this mixed formulation (in particular the inf-sup
condition) and then introduce a numerical approximation based on space-time
finite elements discretization. We prove the strong convergence of the
approximation and then discussed several examples for and . The
problem of the reconstruction of both the state and the source term is also
addressed
Preconditioners for state constrained optimal control problems\ud with Moreau-Yosida penalty function tube
Optimal control problems with partial differential equations play an important role in many applications. The inclusion of bound constraints for the state poses a significant challenge for optimization methods. Our focus here is on the incorporation of the constraints via the Moreau-Yosida regularization technique. This method has been studied recently and has proven to be advantageous compared to other approaches. In this paper we develop preconditioners for the efficient solution of the Newton steps associated with the fast solution of the Moreau-Yosida regularized problem. Numerical results illustrate the competitiveness of this approach. \ud
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