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POD a-posteriori error based inexact SQP method for bilinear elliptic optimal control problems

Martin Kahlbacher, Stefan Volkwein (2012)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

An optimal control problem governed by a bilinear elliptic equation is considered. This problem is solved by the sequential quadratic programming (SQP) method in an infinite-dimensional framework. In each level of this iterative method the solution of linear-quadratic subproblem is computed by a Galerkin projection using proper orthogonal decomposition (POD). Thus, an approximate (inexact) solution of the subproblem is determined. Based on a POD a-posteriori error estimator developed by Tröltzsch...

POD a-posteriori error based inexact SQP method for bilinear elliptic optimal control problems∗

Martin Kahlbacher, Stefan Volkwein (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

An optimal control problem governed by a bilinear elliptic equation is considered. This problem is solved by the sequential quadratic programming (SQP) method in an infinite-dimensional framework. In each level of this iterative method the solution of linear-quadratic subproblem is computed by a Galerkin projection using proper orthogonal decomposition (POD). Thus, an approximate (inexact) solution of the subproblem is determined. Based on a POD...

Poincaré theorem and nonlinear PDE's

Maria E. Pliś (1998)

Annales Polonici Mathematici

A family of formal solutions of some type of nonlinear partial differential equations is found. Terms of such solutions are Laplace transforms of some Laplace distributions. The series of these distributions are locally finite.

Poincaré-invariant structures in the solution manifold of a nonlinear wave equation.

Irving E. Segal (1986)

Revista Matemática Iberoamericana

The solution manifold M of the equation ⎯φ + gφ3 = 0 in Minkowski space is studied from the standpoint of the establishment of differential-geometric structures therein. It is shown that there is an almost Kähler structure globally defined on M that is Poincaré invariant. In the vanishing curvature case g = 0 the structure obtained coincides with the complex Hilbert structure in the solution manifold of the real wave equation. The proofs are based on the transfer of the equation to an ambient universal...

Point fixe d'une application non contractante.

Pierre Gilles Lemarié-Rieusset (2006)

Revista Matemática Iberoamericana

We study a multilinear fixed-point equation in a closed ball of a Banach space where the application is 1-Lipschitzian: existence, uniqueness, approximations, regularity.

Pointwise and locally uniform convergence of holomorphic and harmonic functions

Libuše Štěpničková (1999)

Commentationes Mathematicae Universitatis Carolinae

We shall characterize the sets of locally uniform convergence of pointwise convergent sequences. Results obtained for sequences of holomorphic functions by Hartogs and Rosenthal in 1928 will be generalized for many other sheaves of functions. In particular, our Hartogs-Rosenthal type theorem holds for the sheaf of solutions to the second order elliptic PDE's as well as it has applications to the theory of harmonic spaces.

Pointwise convergence to the initial data for nonlocal dyadic diffusions

Marcelo Actis, Hugo Aimar (2016)

Czechoslovak Mathematical Journal

We solve the initial value problem for the diffusion induced by dyadic fractional derivative s in ℝ + . First we obtain the spectral analysis of the dyadic fractional derivative operator in terms of the Haar system, which unveils a structure for the underlying “heat kernel”. We show that this kernel admits an integrable and decreasing majorant that involves the dyadic distance. This allows us to provide an estimate of the maximal operator of the diffusion by the Hardy-Littlewood dyadic maximal operator....

Pointwise decay for solutions of the 2D Neumann exterior problem for the wave equation

Paolo Secchi (2004)

Bollettino dell'Unione Matematica Italiana

We consider the exterior problem in the plane for the wave equation with a Neumann boundary condition. We are interested to the asymptotic behavior for large times for the solution, and in particular to the dependence on the norms of the initial data in the estimate for the pointwise decay rate. In the paper we prove such an estimate, by a combination of the estimate of the local energy decay and decay estimates for the free space solution.

Pointwise estimates and rigidity results for entire solutions of nonlinear elliptic pde’s

Alberto Farina, Enrico Valdinoci (2013)

ESAIM: Control, Optimisation and Calculus of Variations

We prove pointwise gradient bounds for entire solutions of pde’s of the form      ℒu(x) = ψ(x, u(x), ∇u(x)), where ℒ is an elliptic operator (possibly singular or degenerate). Thus, we obtain some Liouville type rigidity results. Some classical results of J. Serrin are also recovered as particular cases of our approach.

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