**Abstract**: In this paper, a new methodology is proposed for the computation of Background Errors in wave data assimilation systems. Background errors define the spatial influence of an observation in the model domain. Since at present the directional wave spectrum is the fundamental variable of both state-ofthe- art numerical models and most modern instrumentation, this is at the core of the proposed methodology. The advantage of the spectral approach is that the wave spectrum contains detailed information of the different wave systems and physical processes at work (e.g., wind-sea or swells). These systems have different origins and may be driven by different mechanisms, having therefore different spatial structures, length scales, and sensitivity to local wind conditions. The presented method enables making consistent and specific corrections to each component of the spectrum, in time and space. The innovations presented here require an integral look at the data assimilation algorithm for which a suitable scheme is also proposed. Examples of computed background errors are presented for shelf and oceanic basins showing the spatial structures of the different wave systems active in these areas.

**Abstract**: Row family inequalities defined in [Argiroffo, G. and S. Bianchi, Row family inequalities for the set covering polyhedron, Electronic Notes in Discrete Mathematics 36 (2010), pp. 1169–1176] are revisited in the context of the set covering polyhedron of circulant matrices $Q^*(C_n^k)$. A subclass of these inequalities, together with boolean facets, provides a complete linear description of $Q^*(C_n^k)$. The relationship between row family inequalities and minor inequalities is further studied.

**Abstract**: Recently, nonconvex regularization models have been introduced in order to provide a better prior for gradient distributions in real images. They are based on using concave energies $\phi$ in the total variation--type functional ${TV}^\phi(u) := \int \phi(|\nabla u(x)|)\,d x$. In this paper, it is demonstrated that for typical choices of $\phi$, functionals of this type pose several difficulties when extended to the entire space of functions of bounded variation, ${BV}(\Omega)$. In particular, if $\phi(t)=t^q$ for $q \in (0, 1)$, and ${TV}^\phi$ is defined directly for piecewise constant functions and extended via weak* lower semicontinuous envelopes to ${BV}(\Omega)$, then it still holds that ${TV}^\phi(u)=\infty$ for $u$ not piecewise constant. If, on the other hand, ${TV}^\phi$ is defined analogously via continuously differentiable functions, then ${TV}^\phi \equiv 0$ (!). We study a way to remedy the models through additional multiscale regularization and area strict convergence, provided that the energy $\phi(t)=t^q$ is linearized for high values. The fact that such energies actually better match reality and improve reconstructions is demonstrated by statistics and numerical experiments.

**Abstract**: 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 L2-norm; moreover, under a further assumption on the regularity structure of the optimal control this error estimate can be improved to h3/2, extending the results in Rösch (Optim. Methods Softw. 21(1): 121–134, 2006). The theoretical findings are tested experimentally by means of numerical examples.

**Abstract**: This paper is concerned with an optimal control problem of steady-state electrorheological fluids based on an extended Bingham model. Our control parameters are given by finite real numbers representing applied direct voltages, which enter in the viscosity of the electrorheological fluid via an electrostatic potential. The corresponding optimization problem belongs to a class of nonlinear optimal control problems of variational inequalities with control in the coefficients. We analyze the associated variational inequality model and the optimal control problem. Thereafter, we introduce a family of Huber-regularized optimal control problems for the approximation of the original one and verify the convergence of the regularized solutions. Differentiability of the solution operator is proved and an optimality system for each regularized problem is established. In the last part of the paper, an algorithm for the numerical solution of the regularized problem is constructed and numerical experiments are carried out.

**Abstract**: 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 $\Gamma$ converges to the original problem as the smoothing vanishes. All analysis is done in function spaces rather than on the discretised learning problem.

**Abstract**: 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.

**Abstract**: A new method is presented for a physically based statistical description of wind wave climatology. The method applies spectral partitioning to identify individual wave systems (partitions) in time series of 2D-wave spectra, followed by computing the probability of occurrence of their (peak) position in frequency–direction space. This distribution can be considered as a spectral density function to which another round of partitioning is applied to obtain spectral domains, each representing a typical wave system or population in a statistical sense. This two-step partitioning procedure allows identifying aggregate wave systems without the need to discuss specific characteristics as wind sea and swell systems. We suggest that each of these aggregate wave systems (populations) is linked to a specific generation pattern opening the way to dedicated analyses. Each population (of partitions) can be subjected to further analyses to add dimension carrying information based on integrated wave parameters of each partition, such as significant wave height, wave age, mean wave period and direction, among others. The new method is illustrated by analysing model spectra from a numerical wave prediction model and measured spectra from a directional wave buoy located in the Southern North Sea. It is shown that these two sources of information yield consistent results. Examples are given of computing the statistical distribution of significant wave height, spectral energy distribution and the spatial variation of wind wave characteristics along a north–south transect in the North Sea. Wind or wave age information can be included as an extra attribute of the members of a population to label them as wind sea or swell systems. Finally, suggestions are given for further applications of this new method.

**Abstract**: 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.

**Abstract**: 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.

**Abstract**: 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.

**Abstract**: 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 problems and for problems in function spaces, under suitable assumptions on the active set. A characterization of Bouligand and strong stationary points is obtained thereafter. Finally, based on the obtained first-order information, a trust-region algorithm is proposed for the solution of the optimization problems.