A study on vessel fatigue damage as a criterion for heading selection by application of 2D actual bimodal and JONSWAP wave spectra

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.

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A simulator of Synthetic Aperture Radar (SAR) image spectra: the applications on oceanswell waves

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.

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Learning the Sampling Pattern for MRI

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.

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Predictive Online Optimisation with Applications to Optical Flow

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.

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A Nonsmooth Trust-Region Method for Locally Lipschitz Functions with Application to Optimization Problems Constrained by Variational Inequalities

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.

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An exact approach for the multi-constraint graph partitioning problem

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.

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An exact approach for the multi-constraint graph partitioning problem

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.

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Error estimates for the FEM approximation of optimal sparse control of elliptic equations with pointwise state constraints and finite-dimensional control space

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.

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Modeling multiple wave systems in the eastern equatorial Pacific

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.

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A BDF2-semismooth Newton algorithm for the numerical solution of the Bingham flow with temperature dependent parameters

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

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Covariate Extreme Value Analysis Using Wave Spectral Partitioning

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.

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Can we extrapolate climate in an inner basin? The case of the Red Sea

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.

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Primal-dual block-proximal splitting for a class of non-convex problems

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

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A bilevel learning approach for optimal observation placement in variational data assimilation

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.

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On the statistical analysis of ocean wave directional spectra

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.

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A Hybrid Physical-Statistical Algorithm for SAR Wave Spectra Quality Assessment

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.

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Total generalized variation regularization in data assimilation for Burgers' equation

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.

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A difference-of-convex functions approach for sparse PDE optimal control problems with nonconvex costs

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.

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Acceleration and global convergence of a first-order primal-dual method for nonconvex problems

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.

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Block-proximal methods with spatially adapted acceleration

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.

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Testing and Non-linear Preconditioning of the Proximal Point Method

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.

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On the optimal control of some nonsmooth distributed parameter systems arising in mechanics

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.

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Existence and uniqueness of weak solutions for nonlocal parabolic problems via the Galerkin method

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.

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An exact approach for the balanced k-way partitioning problem with weight constraints and its application to sports team realignment

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.

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The Global Signature of Ocean Wave Spectra

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.

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A multigrid optimization algorithm for the numerical solution of quasilinear variational inequalities involving the p-Laplacian

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.

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First Order Methods for High Resolution Image Denoising

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.

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Preconditioned Proximal Point Methods and Notions of Partial Subregularity

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.

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Integro-differential systems of mixed type involving higher order fractional Laplacian, Integral Transforms and Special Functions

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.

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Infimal Convolution of Data Discrepancies for Mixed Noise Removal

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.

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Learning optimal spatially-dependent regularization parameters in total variation image denoising

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.

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The jump set under geometric regularisation. Part 2: Higher-order approaches

Abstract:

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Bilevel approaches for learning of variational imaging models

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.

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Existence and uniqueness for parabolic problems with Caputo time derivative

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.

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Second-order orthant-based methods with enriched Hessian information for sparse $\ell_1$-optimization

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.

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A preconditioned descent algorithm for variational inequalities of the second kind involving the p-Laplacian operator

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.

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Climate patterns derived from ocean wave spectra

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.

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Optimal Control of Static Elastoplasticity in Primal Formulation

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.

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Propagation Speed for Fractional Cooperative Systems with Slowly Decaying Initial Conditions

Abstract: The aim of this paper is to study the time asymptotic propagation for mild solutions to the fractional reaction diffusion cooperative systems when at least one entry of the initial condition decays slower than a power. We state that the solution spreads at least exponentially fast with an exponent depending on the diffusion term and on the smallest index of fractional Laplacians.

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Asymptotic behaviour of solutions to one-dimensional reaction diffusion cooperative systems involving infinitesimal generators

Abstract: The aim of this paper is to study the large-time behaviour of mild solutions to the one-dimensional cooperative systems with anomalous diffusion when at least one entry of the initial condition decays slower than a power. We prove that the solution moves at least exponentially fast as time goes to infinity. Moreover, the exponent of propagation depends on the decay of the initial condition and of the reaction term.

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Balanced Partition of a Graph for Football Team Realignment in Ecuador

Abstract: In the second category of the Ecuadorian football league, a set of football teams must be grouped into $k$ geographical zones 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 zone is minimized. This problem can be modeled as a $k$-clique partitioning problem with constraints on the sizes and weights of the cliques. An integer programming formulation and a heuristic approach were developed to provide a solution to the problem which has been implemented in the 2015 edition of the aforementioned football championship.

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Bilevel Parameter Learning for Higher-Order Total Variation Regularisation Models

Abstract: We consider a bilevel optimisation approach for parameter learning in higher-order total variation image reconstruction models. Apart from the least squares cost functional, naturally used in bilevel learning, we propose and analyse an alternative cost based on a Huber-regularised TV seminorm. Differentiability properties of the solution operator are verified and a first-order optimality system is derived. Based on the adjoint information, a combined quasi-Newton/semismooth Newton algorithm is proposed for the numerical solution of the bilevel problems. Numerical experiments are carried out to show the suitability of our approach and the improved performance of the new cost functional. Thanks to the bilevel optimisation framework, also a detailed comparison between $TGV^2$ and $ICTV$ is carried out, showing the advantages and shortcomings of both regularisers, depending on the structure of the processed images and their noise level.

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On the specification of background errors for wave data assimilation systems

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.

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Minor related row family inequalities for the set covering polyhedron of circulant matrices

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.

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Limiting Aspects of Nonconvex ${TV}^{\phi}$ Models

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.

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Finite element error estimates for an optimal control problem governed by the Burgers equation

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.

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Optimal control of electrorheological fluids through the action of electric fields

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.

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The structure of optimal parameters for image restoration problems

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.

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Second-order approximation and fast multigrid solution of parabolic bilinear optimization problems

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.

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Wave spectra partitioning and long term statistical distribution

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.

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