We study the qualitative properties of optimal regularisation parameters in variational models for image restoration. The parameters are solutions of bilevel optimisation problems with the image restoration problem as constraint. A general type of regulariser is considered, which encompasses total variation (TV), total generalised variation (TGV) and infimal-convolution total variation (ICTV). We prove that under certain conditions on the given data optimal parameters derived by bilevel optimisation problems exist. A crucial point in the existence proof turns out to be the boundedness of the optimal parameters away from 0 which we prove in this paper. The analysis is done on the original – in image restoration typically non-smooth variational problem – as well as on a smoothed approximation set in Hilbert space which is the one considered in numerical computations. For the smoothed bilevel problem we also prove that it Γ converges to the original problem as the smoothing vanishes. All analysis is done in function spaces rather than on the discretised learning problem.
We derive a-priori error estimates for the finite-element approximation of a distributed optimal control problem governed by the steady one-dimensional Burgers equation with pointwise box constraints on the control. Here the approximation of the state and the control is done by using piecewise linear functions. With this choice, an $L^2$ superlinear order of convergence for the control is obtained; moreover, under a further assumption on the regularity structure of the optimal control this error estimate can be improved to $h^{3/2}$. The theoretical findings are tested experimentally by means of numerical examples.
We study the qualitative properties of optimal regularisation parameters in variational models for image restoration. The parameters are solutions of bilevel optimisation problems with the image restoration problem as constraint. A general type of regulariser is considered, which encompasses total variation (TV), total generalised variation (TGV) and infimal-convolution total variation (ICTV). We prove that under certain conditions on the given data optimal parameters derived by bilevel optimisation problems exist. A crucial point in the existence proof turns out to be the boundedness of the optimal parameters away from 0 which we prove in this paper. The analysis is done on the original – in image restoration typically non-smooth variational problem – as well as on a smoothed approximation set in Hilbert space which is the one considered in numerical computations. For the smoothed bilevel problem we also prove that it Γ converges to the original problem as the smoothing vanishes. All analysis is done in function spaces rather than on the discretised learning problem.
The Ecuadorian proposal to keep 846 million barrels of crude oil in the Yasuní National Park underground—for the purposes of avoiding CO2 emissions and to protect both the biological diversity and the indigenous peoples in isolation who inhabit this area of the Amazon—is evaluated from a “multi-criteria” analysis. The main purpose of the paper is to compare this policy option with other alternatives across different values. An analytical framework is used that recognises the inherent complexity of a problem of this nature, in which the financial values are indeed relevant for policy, but other values are also relevant: the economic (in a broad sense), social, environmental, cultural and political. The results confirm that from a financial standpoint, extracting the oil is preferable, but by incorporating the non-monetary values into the multi-criteria decision process, one can plausibly defend the Yasuní-ITT Initiative as the most desirable policy option. Indeed, the social and environmental benefits (or “criteria”) signalling an economic transition towards a model based on renewable sources of energy, along with the protection of critical environmental and social capital, make up for the financial gap.
This book introduces, in an accessible way, the basic elements of Numerical PDE-Constrained Optimization, from the derivation of optimality conditions to the design of solution algorithms. Numerical optimization methods in function-spaces and their application to PDE-constrained problems are carefully presented. The developed results are illustrated with several examples, including linear and nonlinear ones. In addition, MATLAB codes, for representative problems, are included. Furthermore, recent results in the emerging field of nonsmooth numerical PDE constrained optimization are also covered. The book provides an overview on the derivation of optimality conditions and on some solution algorithms for problems involving bound constraints, state-constraints, sparse cost functionals and variational inequality constraints.
We investigate optimality conditions for optimization problems constrained by a class of variational inequalities of the second kind. Based on a nonsmooth primal-dual reformulation of the governing inequality, the differentiability of the solution map is studied. Directional differentiability is proved both for finite-dimensional and function space problems, under suitable assumptions on the active set. A characterization of B-stationary optimal solutions is obtained thereafter. Finally, based on the obtained first-order information, a trust-region algorithm is proposed for the solution of the optimization problems.
In this paper we deal with quasilinear singular parabolic equations with L∞ coefficients, whose prototypes are the p-Laplacian (View the MathML source) equations. In this range of the parameters, we are in the so called fast diffusion case. Extending a recent result (Ragnedda et al. 2013), we are able to prove Harnack estimates at large, i.e. starting from the value attained in a point by the solution, we are able to give explicit and sharp pointwise estimates, from below by using the Barenblatt solutions. In the last section we briefly show how these results can be adapted to equations of porous medium type in the fast diffusion range i.e. View the MathML source.
An accurate and fast solution scheme for parabolic bilinear optimization problems is presented. Parabolic models where the control plays the role of a reaction coefficient and the objective is to track a desired trajectory are formulated and investigated. Existence and uniqueness of optimal solution are proved. A space-time discretization is proposed and second-order accuracy for the optimal solution is discussed. The resulting optimality system is solved with a nonlinear multigrid strategy that uses a local semismooth Newton step as smoothing scheme. Results of numerical experiments validate the theoretical accuracy estimates and demonstrate the ability of the multigrid scheme to solve the given optimization problems with mesh-independent efficiency.
We discuss numerical reduction methods for an optimal control problem of semi-infinite type with finitely many control parameters but infinitely many constraints. We invoke known a priori error estimates to reduce the number of constraints. In a first strategy, we apply uniformly refined meshes, whereas in a second more heuristic strategy we use adaptive mesh refinement and provide an a posteriori error estimate for the control based on perturbation arguments.
We consider optimal control problems of quasilinear elliptic equations with gradient coefficients arising in variable viscosity fluid flow. The state equation is monotone and the controls are of distributed type. We prove that the control-to-state operator is twice Fréchet differentiable for this class of equations. A finite element approximation is studied and an estimate of optimal order h is obtained for the control. The result makes use of the distributed structure of the controls, together with a regularity estimate for elliptic equations with Hölder coefficients and a second order sufficient optimality condition. The paper ends with a numerical experiment, where the approximation order is computationally tested.
We present a model for the dynamics of discrete deterministic systems, based on an extension of the Petri net framework. Our model relies on the definition of a priority relation between conflicting transitions, which is encoded in a compact manner by orienting the edges of a transition conflict graph. The benefit is that this allows the use of a successor oracle for the study of dynamic processes from a global point of view, independent from a particular initial state and the (complete) construction of the reachability graph. We provide a characterization, in terms of a local consistency condition, of those deterministic systems whose dynamic behavior can be encoded using our approach and consider the problem of recognizing when an orientation of the transition conflict graph is valid for this purpose . Finally, we address the problem of gaining the information that allows to provide an appropriate priority relation gouverning the dynamic behavior of the studied system and dicuss some further implications and generalizations of the studied approach.
We derive a-priori error estimates for the finite-element approximation of a distributed optimal control problem governed by the steady one-dimensional Burgers equation with pointwise box constraints on the control. Here the approximation of the state and the control is done by using piecewise linear functions. With this choice, a superlinear order of convergence for the control is obtained in the L2L2 -norm; moreover, under a further assumption on the regularity structure of the optimal control this error estimate can be improved to h3/2h3/2 , extending the results in Rösch (Optim. Methods Softw. 21(1): 121–134, 2006). The theoretical findings are tested experimentally by means of numerical examples.
We study the time asymptotic propagation of solutions to the reaction–diffusion cooperative systems with fractional diffusion. We prove that the propagation speed is exponential in time, and we find the precise exponent of propagation. This exponent depends on the smallest index of the fractional laplacians and on the principal eigenvalue of the matrix DF(0) where F is the reaction term. We also note that this speed does not depend on the space direction.