**Abstract**: A wave field is a spatial expanse of the ocean surface in which waves of a single meteorological event are generated and propagate until their energy dissipates and disperses to the point where they are not detectable any further. Since water as a transport medium allows the superposition of waves of different sizes traveling in different directions, the typical sea surface is composed of several coexisting and overlapping fields. Historically, the methods for describing waves were based on bulk parameters (e.g., Hs, Tz), but the modern methods based on the wave spectrum, provide us with all the information necessary to distinguish individual wave fields out of the composed set. Weather prediction centers archive nowadays point spectra time series with global coverage and spanning long periods of time. The statistical characterization of such data shows that long-term spectral patterns are few and well defined at each location, and they can be associated with a specific meteorological forcing (e.g., distant swells, trade-winds, local jets). The objective of this work is to consolidate the local point information as to obtain spatially coherent wave fields, discerning them from each other to determine their characteristics. Individual wave fields can be regarded as a new source of information, useful for a wide range of applications such as data assimilation, sediment transport, biomass productivity, among others. For climate-related purposes time and space variability can be analyzed, identifying trends, anomalies, and tele-connections. These climate aspects are explored here through an illustrative example focused on the southern trade-winds' field.

## A new mechanochemical model for apical constriction: Coupling calcium signalling and viscoelasticity

**Abstract**: Embryonic epithelial cells exhibit strong coupling of mechanical responses to chemical signals and most notably to calcium. Recent experiments have shown that the disruption of calcium signals during neurulation strongly correlates with the appearance of neural tube defects. We, thus, develop a multi-dimensional mechanochemical model and use it to reproduce important experimental findings that describe anterior neural plate morphogenetic behaviour during neural tube closure. The governing equations consist of an advection-diffusion-reaction system for calcium concentration which is coupled to a force balance equation for the tissue. The tissue is modelled as a linear viscoelastic material that includes a calcium-dependent contraction stress. We implement a random distribution of calcium sparks that is compatible with experimental findings.

**Abstract**: A very severe storm in the Antarctic belt is analysed that sent a very large swell throughout the South-Pacific Ocean. The reasons for the storm were a deep depression passing over an anomalous warm sea area, with consequent increased intensity, more active wind input, gustiness, with also dynamical generation. Wind and wave model results are verified with scatterometer and altimeter data. We follow the swell evolution during the five days required to reach the Galapagos Islands and a buoy off the Peruvian coast. The first forerunners peaked at 0.032 Hz at these locations, well represented in the model thanks to a purposely extended frequency range used in the WAM model. A nonlinear combined analysis is carried out to estimate the overall maximum single wave heights that may have impinged on the Galapagos coasts. Single wave heights up to 6 m have been estimated. Once generated, the swell conditions at Galapagos and the buoy are perfectly anticipated. Including generation, useful forecasts extend till at least eight days before the event. The lack of any local communication is discussed. An analysis using ERA5 winds, but a respectively higher resolution long-term wave hindcast, shows that a similar, actually stronger, event happened in 2006. A simple, but sound method, based on physical principles and elementary geometry, is proposed to estimate, firsthand and after any time, the maximum height of a once generated swell. The results for the 2015 storm are correct within 5% of the model values.

**Abstract**: We address the problem of optimal scale-dependent parameter learning in total variation image denoising. Such problems are formulated as bilevel optimization instances with total variation denoising problems as lower-level constraints. For the bilevel problem, we are able to derive M-stationarity conditions, after characterizing the corresponding Mordukhovich generalized normal cone and verifying suitable constraint qualification conditions. We also derive B-stationarity conditions, after investigating the Lipschitz continuity and directional differentiability of the lower-level solution operator. A characterization of the Bouligand subdifferential of the solution mapping, by means of a properly defined linear system, is provided as well. Based on this characterization, we propose a two-phase nonsmooth trust-region algorithm for the numerical solution of the bilevel problem and test it computationally for two particular experimental settings.

**Abstract**: The feasibility of a Linear Inversion Method applied to Synthetic Aperture Radar cross spectra to retrieve wave directional spectra is investigated. The proposed method is computationally efficient and does not require any a priori information from wave or atmospheric models, as is usually the case with other approaches. It is also comparatively less sensitive to low signal-to-noise ratio image spectra, which in general hinders the disambiguation of the retrieved directional wave spectrum. Several test cases were computed employing a simulator of complex SAR image spectra considering a broad range of values of steepness for the Envisat and Sentinel 1 satellite configurations.

**Abstract**: In embryogenesis, epithelial cells acting as individual entities or as coordinated aggregates in a tissue, exhibit strong coupling between mechanical responses to internally or externally applied stresses and chemical signalling. One of the most important chemical signals in this process is calcium. This mechanochemical coupling and intercellular communication drive the coordination of morphogenetic movements which are characterised by drastic changes in the concentration of calcium in the tissue. In this paper we extend the recent mechanochemical model in Kaouri et al. (J. Math. Biol. 78, 2059–2092, 2019), for an epithelial continuum in one dimension, to a more realistic multi-dimensional case. The resulting parametrised governing equations consist of an advection-diffusion-reaction system for calcium signalling coupled with active-stress linear viscoelasticity and equipped with pure Neumann boundary conditions. We implement a finite element method in perturbed saddle-point form for the simulation of this complex multiphysics problem.

**Abstract**: Safe execution of marine operations (MOs) depends on accurate prediction of vessel dynamic responses, which are necessary to help a superintendent make on-board decisions. However, it can be challenging to conduct complex numerical simulations prior to execute a MO, especially at offshore sites characterized by complex wave conditions, i.e. multimodal directional (2D) wave spectra. In this context, this article introduces a methodology for assessing vessel dynamic responses using characteristics of actual 2D wave spectra and regression models from machine learning. First, a state-of-the-art algorithm is used for 2D spectra partitioning and spectral characterization (climate). This allows identifying wave systems that can occur at a study location, and their integral parameters. For that, a data set for the North Atlantic Ocean, spanning 36 years is used. Then, the main wave parameters of each wave system including significant wave height, peak period and peak direction, are used as features to train and test regression models from machine learning tool kits. Accurate predictions of dynamic responses were obtained with a “boosted trees” regression model considering all six long-term wave systems (18 features) detected at the site. These findings are valuable as new tools to help on-board personnel make safe and quick decisions.

**Abstract**: Electrical impedance tomography is an imaging modality for extracting information on the interior structure of a physical body from boundary measurements of current and voltage. This work studies a new robust way of modeling the contact electrodes used for driving current patterns into the examined object and measuring the resulting voltages. The idea is to not define the electrodes as strict geometric objects on the measurement boundary but only to assume approximate knowledge about their whereabouts and let a boundary admittivity function determine the actual locations of the current inputs. Such an approach enables reconstructing the boundary admittivity, i.e., the locations and strengths of the contacts, at the same time and with analogous methods as the interior admittivity. The functionality of the new model is verified by two-dimensional numerical experiments based on water tank data.

**Abstract**: In electrical impedance tomography (EIT), we aim to solve the conductivity within a target body through electrical measurements made on the surface of the target. This inverse conductivity problem is severely ill-posed, especially in real applications with only partial boundary data available. Thus regularization has to be introduced. Conventionally regularization promoting smooth features is used, however, the Mumford–Shah (M–S) regularizer familiar for image segmentation is more appropriate for targets consisting of several distinct objects or materials. It is, however, numerically challenging. We show theoretically through gamma-convergence that a modification of the Ambrosio–Tortorelli approximation of the M–S regularizer is applicable to EIT, in particular the complete electrode model of boundary measurements. With numerical and experimental studies, we confirm that this functional works in practice and produces higher quality results than typical regularizations employed in EIT when the conductivity of the target consists of distinct smoothly-varying regions.

**Abstract**: In this paper, we propose a dual-mixed formulation for stationary viscoplastic flows with yield, such as the Bingham or the Herschel-Bulkley flow. The approach is based on a Huber regularization of the viscosity term and a two-fold saddle point nonlinear operator equation for the resulting weak formulation. We provide the uniqueness of solutions for the continuous formulation and propose a discrete scheme based on Arnold-Falk-Winther finite elements. The discretization scheme yields a system of slantly differentiable nonlinear equations, for which a semismooth Newton algorithm is proposed and implemented. Local superlinear convergence of the method is also proved. Finally, we perform several numerical experiments in two and three dimensions to investigate the behavior and efficiency of the method.

**Abstract**: We investigate the choice of finite-dimensional parameter spaces within a bilevel optimization framework for selecting scale-dependent weights in total variation image denoising with non-uniform noise. Due to the pointwise box constraints on the parameter function, we prove existence of Lagrange multipliers in low regularity spaces, and derive a first-order optimality system. To cope with the difficulties related to the lack of regularity, a Moreau-Yosida regularization is introduced and convergence of the regularized optimal parameters towards the optimal weight for the original problem is verified. For each regularized bilevel problem a second-order quasi-Newton algorithm is proposed, together with a semismooth Newton scheme for solving the lower-level problem. Finally, several numerical tests are carried out to compare the different parameter space choices and draw some conclusions.

**Abstract**: We consider the exact penalization of the incompressibility condition div(u)=0 for the velocity field of a bi-viscous fluid in terms of the L1–norm. This penalization procedure results in a nonsmooth optimization problem for which we propose an algorithm using generalized second-order information. Our method solves the resulting nonsmooth problem by considering the steepest descent direction and extra generalized second-order information associated to the nonsmooth term. This method has the advantage that the divergence-free property is enforced by the descent direction proposed by the method without the need of build-in divergence-free approximation schemes. The inexact penalization approach, given by the L2-norm, is also considered in our discussion and comparison.

**Abstract**: Planning and execution of marine operations requires proper estimation of vessel dynamic responses and their corresponding operational limits, including considerations of fatigue damage. Current guidelines for marine operations are based on a design wave height without considering the wave energy distribution in frequency and direction. This can be critical for ships operating in open seas where multimodal wave spectra may occur frequently. This study provides criteria for heading selection, with the aim of reducing fatigue damage of vessels under action of directional (2D) bimodal and multimodal wave spectra. In addition, some consequences of using analytical 2D JONSWAP spectra are also addressed. Based on a hydrodynamic model of a vessel, stresses at the midships section are computed using a spectral method. For bimodal wave spectra and considering that all dynamic responses are acceptable, fatigue damage can be reduced in about 50% when the vessel is heading to the least energetic wave component of 2D wave spectra. Moreover, fatigue damage obtained from actual 2D bimodal spectra can be well represented by its corresponding JONSWAP counterpart computed from the spectral parameters of the largest wave component. These findings can be used for vessel heading selection during planning and execution of marine operations.

**Abstract**: The Synthetic Aperture Radar (SAR) carried on-board satellites yields invaluable data of global wave spectra since the early 1990s, with several satellites in orbit at present and more launches scheduled in the near future. However, the retrieval of wave information from SAR images constitutes a complex set of procedures. In this context, we have presented here a methodology to simulate SAR image spectra of ocean swell waves. SAR simulators are important tools for the implementation and evaluation of wave spectra retrieval schemes. The one proposed here is based on the Hasselmann Transform whose Modulation Transfer Functions (MTF’s) account for the main physical processes involved in the imaging of ocean waves. A detailed description of its structure is provided. Through several test cases, we highlight some particularities of the relationship between SAR image spectra and wave spectra. We have evaluated the impact of the parameters settings and input information on the retrieval process, pinpointing possible shortcomings. The results indicated that useful information about the processes involved in the imaging of ocean swells can be derived from the SAR simulator.

**Abstract**: The discovery of the theory of compressed sensingbrought the realisation that many inverse problems can be solvedeven when measurements are "incomplete". This is particularlyinteresting in magnetic resonance imaging (MRI), where longacquisition times can limit its use. In this work, we considerthe problem of learning a sparse sampling pattern that can beused to optimally balance acquisition time versus quality of thereconstructed image. We use a supervised learning approach,making the assumption that our training data is representativeenough of new data acquisitions. We demonstrate that this isindeed the case, even if the training data consists of just 7training pairs of measurements and ground-truth images; with atraining set of brain images of size 192 by 192, for instance, oneof the learned patterns samples only 35% of k-space, howeverresults in reconstructions with mean SSIM 0.914 on a test setof similar images. The proposed framework is general enoughto learn arbitrary sampling patterns, including common patternssuch as Cartesian, spiral and radial sampling.

**Abstract**: Online optimisation revolves around new data being introduced into a problem while it is still being solved; think of deep learning as more training samples become available. We adapt the idea to dynamic inverse problems such as video processing with optical flow. We introduce a corresponding predictive online primal-dual proximal splitting method. The video frames now exactly correspond to the algorithm iterations. A user-prescribed predictor describes the evolution of the primal variable. To prove convergence we need a predictor for the dual variable based on (proximal) gradient flow. This affects the model that the method asymptotically minimises. We show that for inverse problems the effect is, essentially, to construct a new dynamic regulariser based on infimal convolution of the static regularisers with the temporal coupling. We finish by demonstrating excellent real-time performance of our method in computational image stabilisation and convergence in terms of regularisation theory.

**Abstract**: We propose a general trust-region method for the minimization of nonsmooth and nonconvex, locally Lipschitz continuous functions that can be applied, e.g., to optimization problems constrained by elliptic variational inequalities. The convergence of the considered algorithm to C-stationary points is verified in an abstract setting and under suitable assumptions on the involved model functions. For a special instance of a variational inequality constrained problem, we are able to properly characterize the Bouligand subdifferential of the reduced cost function, and, based on this characterization result, we construct a computable trust-region model which satisfies all hypotheses of our general convergence analysis. The article concludes with numerical experiments that illustrate the main properties of the proposed algorithm.

**Abstract**: In this work, a multi-constraint graph partitioning problem is introduced. The input is an undirected graph with costs on the edges and multiple weights on the nodes. The problem calls for a partition of the node set into a fixed number of clusters, such that each cluster satisfies a collection of node weight constraints, and the total cost of the edges whose end nodes are in the same cluster is minimized. It arises as a sub-problem of an integrated vehicle and pollster problem from a real-world application. Two integer programming formulations are provided, and several families of valid inequalities associated with the respective polyhedra are proved. An exact algorithm based on Branch & Bound and cutting planes is proposed, and it is tested on real-world instances.

**Abstract**: In this work, a multi-constraint graph partitioning problem is introduced. The input is an undirected graph with costs on the edges and multiple weights on the nodes. The problem calls for a partition of the node set into a fixed number of clusters, such that each cluster satisfies a collection of node weight constraints, and the total cost of the edges whose end nodes are in the same cluster is minimized. It arises as a sub-problem of an integrated vehicle and pollster problem from a real-world application. Two integer programming formulations are provided, and several families of valid inequalities associated with the respective polyhedra are proved. An exact algorithm based on Branch & Bound and cutting planes is proposed, and it is tested on real-world instances.

**Abstract**: In this work, we derive an a priori error estimate of order $h^2|log(h)|$ for the finite element approximation of a sparse optimal control problem governed by an elliptic equation, which is controlled in a finite dimensional space. Furthermore, box-constrains on the control are considered and finitely many pointwise state-constrains are imposed on specific points in the domain. With this choice for the control space, the achieved order of approximation for the optimal control is optimal, in the sense that the order of the error for the optimal control is of the same order of the approximation for the state equation.

**Abstract**: While moderate wind and wave conditions prevail in the eastern equatorial Pacific, modeling waves in this area remains challenging due to the presence of multiple wave systems converging from different parts of the ocean. This area is covered by swells originated far away including the storm belts of both hemispheres, coexisting with local generation due to the regular action of both the southern trade winds and the wind jets from Central America. In this context, our ability to predict waves in the area depends on the overall quality (i.e., at Pacific scale) of the meteorological input, and also on the skills of the wave model itself. Clearly any error at the remote generation areas translates into larger errors the further waves go, especially if attention is focused on coastal areas. A relevant aspect is that the traditional integral parameters do not offer the possibility to properly assess the errors associated with the different parts of the spectrum (e.g., wind sea and swell). To gain insight in this direction, we make use of partitioning techniques, which enables us to neatly cross-assign and evaluate three spectral components. Not surprisingly, the performance for the swell part is lower than that of the corresponding wind sea. This is further explored with a couple of tests modifying both the wind input and the wave model physics. We find that although at first sight the initial scheme (i.e., ST4) seems to provide the better estimate, the spectral analysis reveals a substantial underestimation of wind sea, compensated with a substantial overestimation of swell. This suggests a problem with too high winds and wave generation in the storm belts together with a likely lack of dissipation or dispersion of swell. In turn, local waves are generally underestimated due to a corresponding underestimation of the local winds. This insight emphasizes the need and advantages of evaluation methods able to look at the different sectors of the wave spectrum.

**Abstract**: This paper is devoted to the numerical solution of the non-isothermal instationary Bingham flow with temperature dependent parameters by semismooth Newton methods. We discuss the main theoretical aspects regarding this problem. Mainly, we discuss the existence of solutions for the problem, and focus on a multiplier formulation which leads us to a coupled system of PDEs involving a Navier–Stokes type equation and a parabolic energy PDE. Further, we propose a Huber regularization for this coupled system of partial differential equations, and we briefly discuss the well posedness of the regularized problem. A detailed finite element discretization, based on the so called (cross-grid ) - elements, is proposed for the space variable, involving weighted stiffness and mass matrices. After discretization in space, a second order BDF method is used as a time advancing technique, leading, in each time iteration, to a nonsmooth system of equations, which is suitable to be solved by a semismooth Newton (SSN) algorithm. Therefore, we propose and discuss the main properties of a SSN algorithm, including the convergence properties. The paper finishes with two computational experiments that exhibit the main properties of the numerical approach

**Abstract**: Covariate Extreme Value Analysis Using Wave Spectral PartitioningJESÚSPORTILLA-YANDÚNResearch Center of Mathematical Modelling (MODEMAT), and Department of MechanicalEngineering, Escuela Politécnica Nacional, Quito, EcuadorEDWINJÁCOMEDepartment of Mechanical Engineering, Escuela Politécnica Nacional, Quito, Ecuador(Manuscript received 5 December 2019, in final form 16 March 2020)ABSTRACTAn important requirement in extreme value analysis (EVA) is for the working variable to be identicallydistributed. However, this is typically not the case in wind waves, because energy components with differentorigins belong to separate data populations, with different statistical properties. Although this information isavailable in the wave spectrum, the working variable in EVA is typically the total significant wave heightHs,aparameter that does not contain information of the spectral energy distribution, and therefore does not fulfillthis requirement. To gain insight in this aspect, we develop here a covariate EVA application based onspectral partitioning. We observe that in general the totalHsis inappropriate for EVA, leading to potentialover- or underestimation of the projected extremes. This is illustrated with three representative cases undersignificantly different wave climate conditions. It is shown that the covariate analysis provides a meaningfulunderstanding of the individual behavior of the wave components, in regard to the consequences for pro-jecting extreme values.

**Abstract**: We examine the possibility of making useful climate extrapolations in inner basins. Stressing the role of the local geographic features, for a practical example we focus our attention on the Red Sea. We observe that in spite of being an enclosed and relatively small Sea, its climate conditions are heavily affected by those of the larger neighboring regions, in particular the Mediterranean and the Arabian Seas. Using existing high-resolution information of the recent decades, we use both reasoned extrapolation and knowledge of, past and future, longer term general climatic information to frame what is presently possible to assess for the Red Sea. Specifically, the northern part, influenced by the Mediterranean Sea, shows a clear decreasing trend of both the meteorological and wave conditions in the recent decades.

**Abstract**: We develop block structure adapted primal-dual algorithms for non-convex non-smoothoptimisation problems whose objectives can be written as compositionsG(x)+F(K(x))of non-smooth block-separable convex functionsGandFwith a non-linear Lipschitz-differentiable op-eratorK. Our methods are renements of the non-linear primal-dual proximal splitting methodfor such problems without the block structure, which itself is based on the primal-dual proximalsplitting method of Chambolle and Pock for convex problems. We propose individual step lengthparameters and acceleration rules for each of the primal and dual blocks of the problem. This allowsthem to convergence faster by adapting to the structure of the problem. For the squared distanceof the iterates to a critical point, we show localO(1/N),O(1/N2)and linear rates under varyingconditions and choices of the step lengths parameters. Finally, we demonstrate the performanceof the methods on practical inverse problems: diffusion tensor imaging and electrical impedancetomography

**Abstract**: In this paper we propose a bilevel optimization approach for the placement of observations in variational data assimilation problems. Within the framework of supervised learning, we consider a bilevel problem where the lower level task is the variational reconstruction of the initial condition of a semilinear system, and the upper level problem solves the optimal placement with help of a sparsity inducing norm. Due to the pointwise nature of the observations, an optimality system with regular Borel measures on the right-hand side is obtained as necessary optimality condition for the lower level problem. The latter is then considered as constraint for the upper level instance, yielding an optimization problem constrained by a multi-state system with measures. We demonstrate existence of Lagrange multipliers and derive a necessary optimality system characterizing the optimal solution of the bilevel problem. The numerical solution is carried out also on two levels. The lower level problem is solved using a standard BFGS method, while the upper level one is solved by means of a projected BFGS algorithm based on the estimation of $\epsilon$-active sets. A penalty function is also considered for enhancing sparsity of the location weights. Finally some numerical experiments are presented to illustrate the main features of our approach.

**Abstract**: Given the growing availability of directional spectra of ocean waves, we explore two different statistical approaches to mine large spectra databases: Spectral Partitions Statistics (SPS) and Self-Organizing Maps (SOM). The first method is not new in the literature, while the second one is for the first time here applied to directional wave spectra. The main goal is to improve the characterization of the directional wave climate at a site, providing a more complete and consistent description than that obtained from traditional statistical methods based on integral spectral parameters (e.g., $H_s$, $T_m$, $\theta_m$). Indeed, while the use of integral parameters allows a direct application of standard techniques for statistical analysis, important information related to the physics of the processes may be overlooked (e.g., the presence of multiple wave systems, for instance locally and remotely generated). The two proposed methods do not exclude integral parameters analysis, but they further allow accounting for different events (e.g., with different genesis) independently. Although SPS and SOM are equally valid for both numerical model and observational data, we illustrate their potential using a 37-year long (1979–2015) model dataset of directional wave spectra at a study site in the western Mediterranean Sea. We show that standard integral parameters fail to show the complex and even multimodal conditions at this site, that are otherwise revealed by the directional spectra statistical analysis. Although the processing pathways and the resulting indicators of both SPS and SOM are substantially different, we observe that their results are mutually consistent, and provide a better insight into the physical processes at work.

**Abstract**: A new approach for assessing the quality of Synthetic Aperture Radar (SAR) wave spectra is presented here. The algorithm addresses two specific issues, related to the 180° directional ambiguity in the propagation direction inherent to SAR measurements, and the removal of noise. In spite of several and progressive advances in the mapping and retrieval of SAR wave spectra, these issues are persistent in the existing official databases and hinder the use of these data for practical uses. This new algorithm is based on a recently developed database of long-term global wave spectral characteristics, which allows estimating the occurrence probability of every individual wave system found in the observed spectra, and that of its ambiguous pair. In addition, assuring the spatial consistency of the wave systems along track, helps obtaining more robust results. In this sense, wave spectra within a track are processed in several steps, so that wave systems with more likelihood of being correct are processed first, and consequently used to evaluate the more challenging components. Accordingly, a specific quality label is assigned to every wave system depending on the certainty achieved. An Envisat track over a challenging area with multiple crossing swells is used to illustrate the performance of the algorithm.

**Abstract**: We propose a second-order total generalized variation (TGV) regularization for the reconstruction of the initial condition in variational data assimilation problems. After showing the equivalence between TGV regularization and a Bayesian MAP estimator, we focus on the detailed study of the inviscid Burgers' data assimilation problem. Due to the difficult structure of the governing hyperbolic conservation law, we consider a discretize–then–optimize approach and rigorously derive a first-order optimality condition for the problem. For the numerical solution, we propose a globalized reduced Newton-type method together with a polynomial line-search strategy, and prove convergence of the algorithm to stationary points. The paper finishes with some numerical experiments where, among others, the performance of TGV–regularization compared to TV–regularization is tested.

**Abstract**: We propose a local regularization of elliptic optimal control problems which involves the nonconvex $L^q$ quasi-norm penalization in the cost function. The proposed Huber type regularization allows us to formulate the PDE constrained optimization instance as a DC programming problem (difference of convex functions) that is useful to obtain necessary optimality conditions and tackle its numerical solution by applying the well known DC algorithm used in nonconvex optimization problems. By this procedure we approximate the original problem in terms of a consistent family of parameterized nonsmooth problems for which there are efficient numerical methods available. Finally, we present numerical experiments to illustrate our theory with different configurations associated to the parameters of the problem.

**Abstract**: The primal-dual hybrid gradient method, modified (PDHGM, also known as the Chambolle{Pock method), has proved very successful for convex optimization problems involving linear operators arising in image processing and inverse problems. In this paper, we analyze an extension to nonconvex problems that arise if the operator is nonlinear. Based on the idea of testing, we derive new step-length parameter conditions for the convergence in infinite-dimensional Hilbert spaces and provide acceleration rules for suitably (locally and/or partially) monotone problems. Importantly, we prove linear convergence rates as well as global convergence in certain cases. We demonstrate the efficacy of these step-length rules for PDE-constrained optimization problems.

**Abstract**: We study and develop (stochastic) primal-dual block-coordinate descent methods for convex problems based on the method due to Chambolle and Pock. Our methods have known convergence rates for the iterates and the ergodic gap of $O(1/N^2)$ if each block is strongly convex, $O(1/N)$ if no convexity is present, and more generally a mixed rate $O(1/N2) + O(1/N)$ for strongly convex blocks if only some blocks are strongly convex. Additional novelties of our methods include blockwise-adapted step lengths and acceleration as well as the ability to update both the primal and dual variables randomly in blocks under a very light compatibility condition. In other words, these variants of our methods are doubly-stochastic. We test the proposed methods on various image processing problems, where we employ pixelwise-adapted acceleration.

**Abstract**: Employing the ideas of non-linear preconditioning and testing of the classical proximal point method, we formalise common arguments in convergence rate and convergence proofs of optimisation methods to the verification of a simple iteration-wise inequality. When applied to fixed point operators, the latter can be seen as a generalisation of firm non-expansivity or the $\alpha$-averaged property. The main purpose of this work is to provide the abstract background theory for our companion paper “Block-proximal methods with spatially adapted acceleration”. In the present account we demonstrate the effectiveness of the general approach on several classical algorithms, as well as their stochastic variants. Besides, of course, the proximal point method, these method include the gradient descent, forward–backward splitting, Douglas–Rachford splitting, Newton’s method, as well as several methods for saddle-point problems, such as the Alternating Directions Method of Multipliers, and the Chambolle–Pock method.

**Abstract**: Variational inequalities are an important mathematical tool for modelling free boundary problems that arise in different application areas. Due to the intricate nonsmooth structure of the resulting models, their analysis and optimization is a difficult task that has drawn the attention of researchers for several decades. In this paper we focus on a class of variational inequalities, called of the second kind, with a twofold purpose. First, we aim at giving a glance at some of the most prominent applications of these types of variational inequalities in mechanics, and the related analytical and numerical difficulties. Second, we consider optimal control problems constrained by these variational inequalities and provide a thorough discussion on the existence of Lagrange multipliers and the different types of optimality systems that can be derived for the characterization of local minima. The article ends with a discussion of the main challenges and future perspectives of this important problem class.

**Abstract**: Fractional differential equations are becoming increasingly popular as a modeling tool to describe a wide range of non-classical phenomena with spatial heterogeneities throughout the applied science and engineering. A recently developed nonlocal vector calculus is exploited to provide a variational analysis for a general class of nonlocal operators which include fractional Laplacians on bounded domains in $\mathbb{R}^n$. We develop the Galerkin method to prove existence and uniqueness of weak solutions to nonlocal parabolic problems. Moreover, we study the existence of orthonormal basis of eigenvectors associated to these nonlocal operators.

**Abstract**: In this work a balanced $k$-way partitioning problem with weight constraints is defined to model the sports team realignment. Sports teams must be partitioned into a fixed number of groups according to some regulations, where the total distance of the road trips that all teams must travel to play a double round robin tournament in each group is minimized. Two integer programming formulations for this problem are introduced, and the validity of three families of inequalities associated to the polytope of these formulations is proved. The performance of a tabu search procedure and a branch and cut algorithm, which uses the valid inequalities as cuts, is evaluated over simulated and real-world instances. In particular, an optimal solution for the realignment of the Ecuadorian football league is reported and the methodology can be suitable adapted for the realignment of other sports leagues.

**Abstract**: A global atlas of ocean wave spectra is developed and presented. The development is based on a new technique for deriving wave spectral statistics, which is applied to the extensive ERA-Interim database from European Centre of Medium-Range Weather Forecasts. Spectral statistics is based on the idea of long-term wave systems, which are unique and distinct at every geographical point. The identification of those wave systems allows their separation from the overall spectrum using the partition technique. Their further characterization is made using standard integrated parameters, which turn out much more meaningful when applied to the individual components than to the total spectrum. The parameters developed include the density distribution of spectral partitions, which is the main descriptor; the identified wave systems; the individual distribution of the characteristic frequencies, directions, wave height, wave age, seasonal variability of wind and waves; return periods derived from extreme value analysis; and crossing-sea probabilities. This information is made available in web format for public use at http://www.modemat.epn.edu.ec/#/nereo. It is found that wave spectral statistics offers the possibility to synthesize data while providing a direct and comprehensive view of the local and regional wave conditions.

**Abstract**: In this paper we propose a multigrid optimization algorithm (MG/OPT) for the numerical solution of a class of quasilinear variational inequalities of the second kind. This approach is enabled by the fact that the solution of the variational inequality is given by the minimizer of a nonsmooth energy functional, involving the pp-Laplace operator. We propose a Huber regularization of the functional and a finite element discretization for the problem. Further, we analyze the regularity of the discretized energy functional, and we are able to prove that its Jacobian is slantly differentiable. This regularity property is useful to analyze the convergence of the MG/OPT algorithm. In fact, we demonstrate that the algorithm is globally convergent by using a mean value theorem for semismooth functions. Finally, we apply the MG/OPT algorithm to the numerical simulation of the viscoplastic flow of Bingham, Casson and Herschel–Bulkley fluids in a pipe. Several experiments are carried out to show the efficiency of the proposed algorithm when solving this kind of fluid mechanics problems.

**Abstract**: In this paper we are interested in comparing the performance of some of the most relevant first order non-smooth optimization methods applied to the Rudin, Osher and Fatemi (ROF) Image Denoising Model and a Primal-Dual Chambolle-Pock Image Denoising Model. Because of the properties of the resulting numerical schemes it is possible to handle these computations pixelwise, allowing implementations based on parallel paradigms which are helpful in the context of high resolution imaging.

**Abstract**: Based on the needs of convergence proofs of preconditioned proximal point methods, we introduce notions of partial strong submonotonicity and partial (metric) subregularity of set-valued maps. We study relationships between these two concepts, neither of which is generally weaker or stronger than the other one. For our algorithmic purposes, the novel submonotonicity turns out to be easier to employ than more conventional error bounds obtained from subregularity. Using strong submonotonicity, we demonstrate the linear convergence of the Primal-Dual Proximal splitting method to some strictly complementary solutions of example problems from image processing and data science. This is without the conventional assumption that all the objective functions of the involved saddle point problem are strongly convex.

**Abstract**: In this paper, we deal with a new class of non-local operators that we term integro-differential systems of mixed type. We study the behaviour of solutions of this system when the diffusion term involves higher order fractional powers of the Laplacian. Moreover, we prove that the solution of the system decays faster than a power with an exponent given by the smallest index of the fractional power of the Laplacian.

**Abstract**: We consider the problem of image denoising in the presence of noise whose statistical properties are a combination of two different distributions. We focus on noise distributions frequently considered in applications, such as salt & pepper and Gaussian, and Gaussian and Poisson noise mixtures. We derive a variational image denoising model that features a total variation regularization term and a data discrepancy encoding the mixed noise as an infimal convolution of discrepancy terms of the single-noise distributions. We give a statistical derivation of this model by joint maximum a posteriori (MAP) estimation. Classical single-noise models are recovered asymptotically as the weighting parameters go to infinity. The numerical solution of the model is computed using second order Newton-type methods. Numerical results show the decomposition of the noise into its constituting components. The paper is furnished with several numerical experiments, and comparisons with other methods dealing with the mixed noise case are shown.

**Abstract**: We consider a bilevel optimization approach in function space for the choice of spatially dependent regularization parameters in TV image denoising models. First- and second-order optimality conditions for the bilevel problem are studied when the spatially-dependent parameter belongs to the Sobolev space ${{H}^{1}}\left(\Omega \right)$ . A combined Schwarz domain decomposition-semismooth Newton method is proposed for the solution of the full optimality system and local superlinear convergence of the semismooth Newton method is verified. Exhaustive numerical computations are finally carried out to show the suitability of the approach.

**Abstract**: We review some recent learning approaches in variational imaging, based on bilevel optimisation, and emphasize the importance of their treatment in function space. The paper covers both analytical and numerical techniques. Analytically, we include results on the existence and structure of minimisers, as well as optimality conditions for their characterisation. Based on this information, Newton type methods are studied for the solution of the problems at hand, combining them with sampling techniques in case of large databases. The computational verification of the developed techniques is extensively documented, covering instances with different type of regularisers, several noise models, spatially dependent weights and large image databases.

**Abstract**: In this paper we are interested in the well-posedness of fully nonlinear Cauchy problems in which the time derivative is of Caputo type. We address this question in the framework of viscositysolutions, obtain-ing the existence via Perron’s method, and comparison for bounded sub and supersolutions by a suitable regularization through inf and sup convolution in time. As an application, we prove the steady-state large time behavior in the case of proper nonlinearities and provide a rate of convergence by using the Mittag–Leffler operator.

**Abstract**: We present a second order algorithm, based on orthantwise directions, for solving optimization problems involving the sparsity enhancing $\ell_1$-norm. The main idea of our method consists in modifying the descent orthantwise directions by using second order information both of the regular term and (in weak sense) of the $\ell_1$-norm. The weak second order information behind the $\ell_1$-term is incorporated via a partial Huber regularization. One of the main features of our algorithm consists in a faster identification of the active set. We also prove that a reduced version of our method is equivalent to a semismooth Newton algorithm applied to the optimality condition, under a specific choice of the algorithm parameters. We present several computational experiments to show the efficiency of our approach compared to other state-of-the-art algorithms.

**Abstract**: This paper is concerned with the numerical solution of a class of variational inequalities of the second kind, involving the p-Laplacian operator. This kind of problems arise, for instance, in the mathematical modelling of non-Newtonian fluids. We study these problems by using a regularization approach, based on a Huber smoothing process. Well posedness of the regularized problems is proved, and convergence of the regularized solutions to the solution of the original problem is verified. We propose a preconditioned descent method for the numerical solution of these problems and analyze the convergence of this method in function spaces. The existence of admissible descent directions is established by variational methods and admissible steps are obtained by a backtracking algorithm which approximates the objective functional by polynomial models. Finally, several numerical experiments are carried out to show the efficiency of the methodology here introduced.

**Abstract**: The fact that ocean surface waves are an integrated effect of meteorological activity has the interesting consequence that the memory of the wave systems is larger than that of the wind and storms that generated them. At each single point the related information is stored as its wave spectrum, a matrix containing the energy distribution of wave systems with different origins in space and time. We describe the concept of spectral partitioning and the technique used to obtain spectral statistics, whose outcome we associate with the physical reality. Using long series of spectral data we derive information of the, possibly very far, generation zones climatologically connected at a confluent point. Working on the eastern equatorial Pacific we focus on the prominent effects of El Niño events, for which interactions of mesoscale phenomena are revealed from the analysis of the local situation.

**Abstract**: An optimal control problem of static plasticity with linear kinematic hardening and von Mises yield condition is studied. The problem is treated in its primal formulation, where the state system is a variational inequality of the second kind. First-order necessary optimality conditions are obtained by means of an approximation by a family of control problems with state system regularized by Huber-type smoothing, and a subsequent limit analysis. The equivalence of the optimality conditions with the C-stationarity system for the equivalent dual formulation of the problem is proved. Numerical experiments are presented, which demonstrate the viability of the Huber-type smoothing approach.