1,973 research outputs found
Optimal control of semilinear elliptic equations in measure spaces
Optimal control problems in measure spaces governed by semilinear elliptic equations are considered. First order optimality conditions are derived and structural properties of their solutions, in particular sparsity, are discussed. Necessary and sufficient second order optimality conditions are obtained as well. On the basis of the sufficient conditions, stability of the solutions is analyzed. Highly nonlinear terms can be incorporated by utilizing an L∞(Ω) regularity result for solutions of the first order necessary optimality conditions.This author’s research was supported by Spanish Ministerio de Economía y Competitividad under
project MTM2011-22711
Superconvergence for Neumann boundary control problems governed by semilinear elliptic equations
This paper is concerned with the discretization error analysis of semilinear
Neumann boundary control problems in polygonal domains with pointwise
inequality constraints on the control. The approximations of the control are
piecewise constant functions. The state and adjoint state are discretized by
piecewise linear finite elements. In a postprocessing step approximations of
locally optimal controls of the continuous optimal control problem are
constructed by the projection of the respective discrete adjoint state.
Although the quality of the approximations is in general affected by corner
singularities a convergence order of is proven for domains
with interior angles smaller than using quasi-uniform meshes. For
larger interior angles mesh grading techniques are used to get the same order
of convergence
Mild solutions of semilinear elliptic equations in Hilbert spaces
This paper extends the theory of regular solutions ( in a suitable
sense) for a class of semilinear elliptic equations in Hilbert spaces. The
notion of regularity is based on the concept of -derivative, which is
introduced and discussed. A result of existence and uniqueness of solutions is
stated and proved under the assumption that the transition semigroup associated
to the linear part of the equation has a smoothing property, that is, it maps
continuous functions into -differentiable ones. The validity of this
smoothing assumption is fully discussed for the case of the Ornstein-Uhlenbeck
transition semigroup and for the case of invertible diffusion coefficient
covering cases not previously addressed by the literature. It is shown that the
results apply to Hamilton-Jacobi-Bellman (HJB) equations associated to infinite
horizon optimal stochastic control problems in infinite dimension and that, in
particular, they cover examples of optimal boundary control of the heat
equation that were not treatable with the approaches developed in the
literature up to now
Discontinuous Galerkin Methods for Mass Transfer through Semi-Permeable Membranes
A discontinuous Galerkin (dG) method for the numerical solution of
initial/boundary value multi-compartment partial differential equation (PDE)
models, interconnected with interface conditions, is presented and analysed.
The study of interface problems is motivated by models of mass transfer of
solutes through semi-permeable membranes. More specifically, a model problem
consisting of a system of semilinear parabolic advection-diffusion-reaction
partial differential equations in each compartment, equipped with respective
initial and boundary conditions, is considered. Nonlinear interface conditions
modelling selective permeability, congestion and partial reflection are applied
to the compartment interfaces. An interior penalty dG method is presented for
this problem and it is analysed in the space-discrete setting. The a priori
analysis shows that the method yields optimal a priori bounds, provided the
exact solution is sufficiently smooth. Numerical experiments indicate agreement
with the theoretical bounds and highlight the stability of the numerical method
in the advection-dominated regime
Calculus via regularizations in Banach spaces and Kolmogorov-type path-dependent equations
The paper reminds the basic ideas of stochastic calculus via regularizations
in Banach spaces and its applications to the study of strict solutions of
Kolmogorov path dependent equations associated with "windows" of diffusion
processes. One makes the link between the Banach space approach and the so
called functional stochastic calculus. When no strict solutions are available
one describes the notion of strong-viscosity solution which alternative (in
infinite dimension) to the classical notion of viscosity solution.Comment: arXiv admin note: text overlap with arXiv:1401.503
Adaptive Pseudo-Transient-Continuation-Galerkin Methods for Semilinear Elliptic Partial Differential Equations
In this paper we investigate the application of pseudo-transient-continuation
(PTC) schemes for the numerical solution of semilinear elliptic partial
differential equations, with possible singular perturbations. We will outline a
residual reduction analysis within the framework of general Hilbert spaces,
and, subsequently, employ the PTC-methodology in the context of finite element
discretizations of semilinear boundary value problems. Our approach combines
both a prediction-type PTC-method (for infinite dimensional problems) and an
adaptive finite element discretization (based on a robust a posteriori residual
analysis), thereby leading to a fully adaptive PTC-Galerkin scheme. Numerical
experiments underline the robustness and reliability of the proposed approach
for different examples.Comment: arXiv admin note: text overlap with arXiv:1408.522
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