Abstract: Based on a nonsmooth coherence condition, we construct and prove the convergence of a forward-backward splitting method that alternates between steps on a fine and a coarse grid. Our focus is on total variation regularised inverse imaging problems, specifically, their dual problems, for which we develop in detail the relevant coarse-grid problems. We demonstrate the performance of our method on total variation denoising and magnetic resonance imaging.

Abstract: Inverse Problems Purpose-Led Publishing logo. Paper Online optimisation for dynamic electrical impedance tomography Neil Dizon, Jyrki Jauhiainen and Tuomo Valkonen* Published 22 April 2025 • © 2025 IOP Publishing Ltd. All rights, including for text and data mining, AI training, and similar technologies, are reserved. Inverse Problems, Volume 41, Number 5Citation Neil Dizon et al 2025 Inverse Problems 41 055005DOI 10.1088/1361-6420/adcb66 Article metrics 104 Total downloads Submit Submit to this Journal Permissions Get permission to re-use this article Share this article Abstract Online optimisation studies the convergence of optimisation methods as the data embedded in the problem changes. Based on this idea, we propose a primal dual online method for nonlinear time-discrete inverse problems. We analyse the method through regret theory and demonstrate its performance in real-time monitoring of moving bodies in a fluid with electrical impedance tomography. To do so, we also prove the second-order differentiability of the complete electrode model solution operator on .

Abstract: We investigate differentiability and subdifferentiability properties of the solution mapping associated with variational inequalities (VI) of the second kind involving the discrete total-variation. Bouligand differentiability of the solution operator is established via a direct quotient analysis applied to a primal-dual reformulation of the VI. By exploiting the structure of the directional derivative and introducing a suitable subspace, we fully characterize the Bouligand subdifferential of the solution mapping.

Abstract: This study proposes a novel methodology to automatically parameterize atmospheric optical depths within the Radiative Transfer for TOVS (RTTOV) version 13 scheme using statistical thresholds across pressure levels and Least Absolute Shrinkage and Selection Operator (LASSO) regression to induce sparsity. Numerical experiments demonstrate that this approach significantly reduces computational costs while maintaining accuracy.

Abstract: We simulate the fow of a free-boundary biviscous material haracterized by a double-regime behavior determined by the yield stress. Developing a mathematical method to simulate this kind of fuid involves addressing hallenges such as precise formulation, complex interaction with boundaries, numerical stability, omputational efficiency, and experimental validation. In this paper, we propose a numerical scheme that combines the Semismooth-Newton method for solving the biviscous model with a Level-Set formulation describing the motion of the free boundary. We discuss the theoretical formulation of our method and the associated technical specifiations for its implementation. Finally, we conduct a series of numerical experiments to verify the reliability and performance of the proposed methodology.

Abstract: In this paper, we address the numerical solution of optimal control problems governed by elliptic partial diferential equations which contain Lq-quasinorms (q E (0, 1)) in the cost funntion. The nonconvexity and nonsmoothness of the quasinorm penalizers imply that the optimality conditions involves poinwise relations haracterizing optimal controls. Therefore, we discuss the application of the Sequential Quadratic Hamiltonian method and its comparison with state-of-the-art algorithms for solving this lass of problems.

Abstract: Online optimisation facilitates the solution of dynamic inverse problems, such as image stabilisation, fluid flow monitoring, and dynamic medical imaging. In this paper, we improve upon previous work on predictive online primal–dual methods on two fronts. Firstly, we provide a more concise analysis that symmetrises previously unsymmetric regret bounds, and relaxes previous restrictive conditions on the dual predictor. Secondly, based on the latter, we develop several improved dual predictors. We numerically demonstrate their efficacy in image stabilisation and dynamic positron emission tomography.

Abstract: We propose a new approach to solving bilevel optimization problems, intermediate between solving full-system optimality conditions with a Newton-type approach, and treating the inner problem as an implicit function. The overall idea is to solve the full-system optimality conditions, but to precondition them to alternate between taking steps of simple conventional methods for the inner problem, the adjoint equation, and the outer problem. While the inner objective has to be smooth, the outer objective may be nonsmooth subject to a prox-contractivity condition. We prove linear convergence of the approach for combinations of gradient descent and forward-backward splitting with exact and inexact solution of the adjoint equation. We demonstrate good performance on learning the regularization parameter for anisotropic total variation image denoising, and the convolution kernel for image deconvolution. © The Author(s) 2023.

Abstract: We introduce an efficient first-order primal-dual method for the solution of nonsmooth PDE-constrained optimization problems. We achieve this efficiency through not solving the PDE or its linearisation on each iteration of the optimization method. Instead, we run the method interwoven with a simple conventional linear system solver (Jacobi, Gauss–Seidel, conjugate gradients), always taking only one step of the linear system solver for each step of the optimization method. The control parameter is updated on each iteration as determined by the optimization method. We prove linear convergence under a second-order growth condition, and numerically demonstrate the performance on a variety of PDEs related to inverse problems involving boundary measurements.

Abstract: This paper focuses on the analysis of an optimal control problem governed by a nonsmooth quasilinear partial differential equation that models a stationary incompressible shear-thickening fluid. We start by studying the directional differentiability of the non-smooth term within the state equation as a prior step to demonstrate the directional differentiability of the solution operator. Thereafter, we establish a primal first order necessary optimality condition (Bouligand (B) stationarity), which is derived from the directional differentiability of the solution operator. By using a local regularization of the nonsmooth term and carrying out an asymptotic analysis thereafter, we rigourously derive a weak stationarity system for local minima. By combining the B-and weak stationarity conditions, and using the regularity of the Lagrange multiplier, we are able to obtain a strong stationarity system that includes a characterization of the Lagrange multiplier on the active and inactive sets. © 2024, American Institute of Mathematical Sciences. All rights reserved.

Abstract: We investigate the use of bilevel optimization for model learning in variational imaging problems. Bilevel learning is an alternative approach to traditional deep learning methods that leads to fully interpretable models. Our study encompasses the directional differentiability of the solution mapping, the derivation of optimality conditions, and the characterization of the Bouligand subdifferential of the solution operator.

Abstract: In this paper, we investigate the use of bilevel optimization for model learning in variational imaging problems. Bilevel learning is an alternative approach to deep learning methods, which leads to fully interpretable models. However, it requires a detailed analytical insight into the underlying mathematical model. We focus on the bilevel learning problem for total variation models with spatially- and patch-dependent parameters. Our study encompasses the directional differentiability of the solution mapping, the derivation of optimality conditions, and the characterization of the Bouligand subdifferential of the solution operator. We also propose a two-phase trust-region algorithm for solving the problem and present numerical tests using the CelebA dataset. © The Author(s) 2023.

Abstract: This paper presents a comprehensive study on Herschel–Bulkley flow, where the flow parameters are dependent on the density. The Herschel–Bulkley model is a generalized power-law model used to simulate viscoplastic fluids defined by a plasticity threshold. We consider the case where the plasticity threshold and the viscosity depend on the shear rate and fluid density. To analyze this model, we use a Huber regularization of the stress and propose an H(div)-conforming and discontinuous Galerkin (DG) numerical approximation for the coupled equations governing the flow. We discuss the stability and existence of discrete solutions and propose a semismooth Newton linearization for the numerical solution of the discretized system. Our numerical scheme is validated through several experiments that explore the behavior of Herschel–Bulkley flow under different conditions. The results demonstrate the robustness of our numerical method.

Abstract: This paper is devoted to the study of Bingham flow with variable density. We propose a local bi-viscosity regularization of the stress tensor based on a Huber smoothing step. Next, our computational approach is based on a second-order, divergence-conforming discretization of the Huber regularized Bingham constitutive equations, coupled with a discontinuous Galerkin scheme for the mass density. We take advantage of the properties of divergence-conforming and discontinuous Galerkin formulations to effectively incorporate upwind discretizations, thereby ensuring the stability of the formulation. The stability of the continuous problem and the fully discrete scheme are analyzed. Further, a semismooth Newton method is proposed for solving the obtained fully discretized system of equations at each time step. Finally, several numerical examples that illustrate the main features of the problem and the properties of the numerical scheme are presented.

Abstract: Optimization techniques have been widely used for image restoration tasks, as many imaging problems may be formulated as minimization ones with the recovered image as the target minimizer. Recently, novel optimization ideas also entered the scene in combination with machine learning approaches, to improve the reconstruction of images by optimally choosing different parameters/functions of interest in the models. This chapter provides a review of the latest developments concerning the latter, with special emphasis on bilevel optimization techniques and their use for learning local and nonlocal image restoration models in a supervised manner. Moreover, the use of related optimization ideas within the development of neural networks in imaging will be briefly discussed.

Abstract: We provide an overview of primal–dual algorithms for nonsmooth and nonconvex-concave saddle-point problems. This flows around a new analysis of such methods, using Bregman divergences to formulate simplified conditions for convergence. © Springer Nature Switzerland AG 2023.

Abstract: This paper aims to simulate viscoplastic flow in a shallow-water regime. We specifically use the Bingham model in which the material behaves as a solid if the stress is below a certain threshold, otherwise, it moves as a fluid. The main difficulty of this problem is the coupling of the shallow-water equations with the viscoplastic constitutive laws and the high computational effort needed in its solution. Although there have been many studies of this problem, most of these works use explicit methods with simplified empirical models. In our work, to accommodate non-uniform grids and complicated geometries, we use the discontinuous Galerkin method to solve shallow viscoplastic flows. This method is attractive due to its high parallelization, h- and p-adaptivity, and ability to capture shocks. Additionally, we treat the discontinuities in the interfaces between elements with numerical fluxes that ensure a stable solution of the nonlinear hyperbolic equations. To couple the Bingham model with the shallow-water equations, we regularize the problem with three alternatives. Finally, in order to show the effectiveness of our approach, we perform numerical examples for the usual benchmarks of the shallow-water equations.

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. © 2022 Informa UK Limited, trading as Taylor & Francis Group.

Abstract: We investigate a family of bilevel imaging learning problems where the lower-level instance corresponds to a convex variational model involving first- and second-order nonsmooth sparsity-based regularizers. By using geometric properties of the primal-dual reformulation of the lower-level problem and introducing suitable auxiliary variables, we are able to reformulate the original bilevel problems as mathematical programs with complementarity constraints (MPCC). For the latter, we prove tight constraint qualification conditions (MPCC-RCPLD and partial MPCC-LICQ) and derive Mordukhovich (M-) and strong (S-) stationarity conditions. The stationarity systems for the MPCC turn also into stationarity conditions for the original formulation. Second-order sufficient optimality conditions are derived as well, together with a local uniqueness result for stationary points. The proposed reformulation may be extended to problems in function spaces, leading to MPCC with constraints on the gradient of the state. The MPCC reformulation also leads to the efficient use of available large-scale nonlinear programming solvers, as shown in a companion paper, where different imaging applications are studied.

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.

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: This chapter provides a review of the latest developments concerning bilevel optimization techniques and their use for learning local and nonlocal image restoration models in a supervised manner. Moreover, the use of related optimization ideas within the development of neural networks in imaging is briefly discussed.

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: 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-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.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: 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: 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: 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-slow optimization methods applied to the rudin model, OSHE and Fatemi (ROF) Deneising Deneising and a chambolle-dual-dual-bock image renewal model.Due to the properties of the resulting numerical schemes, it is possible to handle these calculations Pixelwise, which allows parallel paradigm-based implementations that are useful in the context of high-resolution imaging. Translated with DeepL.com (free version)

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.

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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 formulate an optimal control problem for a Timoshenko beam model incorporating an L1 cost functional to promote sparsity, which is ideal for the optimal placement of control devices. We analyze the theoretical properties of the locking-free finite element discretization and propose an efficient numerical scheme to solve the non-smooth optimality system.

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

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 primary formulation, where the state system is a variational inequality of the second type.The necessary first-order optimality conditions are obtained by an approximation by a family of control problems with the state system regularized by huber-type smoothing, and a subsequent boundary analysis.The equivalence of the necessary boundary conditions is proved. The necessary first-order optimality conditions are obtained by an approximation by a family of control problems with the state system regularized by huber-type smoothing, and a subsequent boundary analysis.The equivalence of the optimality conditions with the C-stationarity system is demonstrated for the equivalent dual formulation of the problem.Numerical experiments are presented, demonstrating the feasibility of the huber-type smoothing approach. Translated with DeepL.com (free version)

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.

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.

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.

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.

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.

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

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

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

Abstract: We derive a-priori error estimates for the finite-element approximation of a distributed optimal control problem governed by the steady one-dimensional Burgers equation with pointwise box constraints on the control. Here the approximation of the state and the control is done by using piecewise linear functions. With this choice, a superlinear order of convergence for the control is obtained in the L2-norm; moreover, under a further assumption on the regularity structure of the optimal control this error estimate can be improved to h3/2, extending the results in Rösch (Optim. Methods Softw. 21(1): 121–134, 2006). The theoretical findings are tested experimentally by means of numerical examples.

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

Abstract: We study the qualitative properties of optimal regularisation parameters in variational models for image restoration. The parameters are solutions of bilevel optimisation problems with the image restoration problem as constraint. A general type of regulariser is considered, which encompasses total variation (TV), total generalised variation (TGV) and infimal-convolution total variation (ICTV). We prove that under certain conditions on the given data optimal parameters derived by bilevel optimisation problems exist. A crucial point in the existence proof turns out to be the boundedness of the optimal parameters away from 0 which we prove in this paper. The analysis is done on the original – in image restoration typically non-smooth variational problem – as well as on a smoothed approximation set in Hilbert space which is the one considered in numerical computations. For the smoothed bilevel problem we also prove that it $\gamma$ converges to the original problem as the smoothing vanishes. All analysis is done in function spaces rather than on the discretised learning problem.

Abstract: An accurate and fast solution scheme for parabolic bilinear optimization problems is presented. Parabolic models where the control plays the role of a reaction coefficient and the objective is to track a desired trajectory are formulated and investigated. Existence and uniqueness of optimal solution are proved. A space-time discretization is proposed and second-order accuracy for the optimal solution is discussed. The resulting optimality system is solved with a nonlinear multigrid strategy that uses a local semismooth Newton step as smoothing scheme. Results of numerical experiments validate the theoretical accuracy estimates and demonstrate the ability of the multigrid scheme to solve the given optimization problems with mesh-independent efficiency.

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

Abstract: We consider optimal control problems of quasilinear elliptic equations with gradient coefficients arising in variable viscosity fluid flow. The state equation is monotone and the controls are of distributed type. We prove that the control-to-state operator is twice Fréchet differentiable for this class of equations. A finite element approximation is studied and an estimate of optimal order h is obtained for the control. The result makes use of the distributed structure of the controls, together with a regularity estimate for elliptic equations with Hölder coefficients and a second order sufficient optimality condition. The paper ends with a numerical experiment, where the approximation order is computationally tested.

Abstract: We present a model for the dynamics of discrete deterministic systems, based on an extension of the Petri Net framework.Our model is based on the definition of a priority relation between conflicting transitions, which is compactly encoded by orienting the edges of a transition conflict graph.The benefit is that this allows the use of a successor oracle for the study of dynamical processes from a global point of view, independent of a particular initial state and the (complete) construction of the reachability graph. We provide a characterization, in terms of a local consistency condition, of those deterministic systems whose dynamical behavior can be encoded using our approach and consider the problem of recognizing when an orientation of the transition conflict graph is valid for this purpose.Finally, we address the problem of obtaining the information that allows to provide an appropriate priority relation that shapes the dynamical behavior of the studied system and dicuss some further implications and generalizations of the studied approach. Translated with DeepL.com (free version)

Abstract: We discuss numerical reduction methods for an optimal control problem of semi-infinite type with finitely many control parameters but infinitely many constraints. We invoke known a priori error estimates to reduce the number of constraints. In a first strategy, we apply uniformly refined meshes, whereas in a second more heuristic strategy we use adaptive mesh refinement and provide an a posteriori error estimate for the control based on perturbation arguments.

Abstract: 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.

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.

Abstract: We study minor related row family inequalities for the set covering polyhedron of circular matrices. We address the issue of generating these inequalities via the Chvátal-Gomory procedure and establish a general upper bound for their Chvátal-rank.

Abstract: We propose a nonsmooth continuum mechanical model for discontinuous shear thickening flow. The model obeys a formulation as energy minimization problem and its solution satisfies a Stokes type system with a nonsmooth constitute relation. Solutions have a free boundary at which the behavior of the fluid changes. We present Sobolev as well as H¨older regularity results and study the limit of the model as the viscosity in the shear thickened volume tends to infinity. A mixed problem formulation is discretized using finite elements and a semismooth Newton method is proposed for the solution of the resulting discrete system. Numerical problems for steady and unsteady shear thickening flows are presented and used to study the solution algorithm, properties of the flow and the free boundary. These numerical problems are motivated by recently reported experimental studies of dispersions with high particle-to-fluid volume fractions, which often show a sudden increase of viscosity at certain strain rates.

Abstract: We propose the use of the Kantorovich--Rubinstein norm from optimal transport in imaging problems. In particular, we discuss a variational regularization model endowed with a Kantorovich--Rubinstein discrepancy term and total variation regularization in the context of image denoising and cartoon-texture decomposition. We point out connections of this approach to several other recently proposed methods such as total generalized variation and norms capturing oscillating patterns. We also show that the respective optimization problem can be turned into a convex-concave saddle point problem with simple constraints and hence can be solved by standard tools. Numerical examples exhibit interesting features and favorable performance for denoising and cartoon-texture decomposition.

Abstract: We state and analyse one-shot methods in function space for the optimal control of nonlinear partial differential equations (PDEs) that can be formulated in terms of a fixed-point operator. A general convergence theorem is proved by generalizing the previously obtained results in finite dimensions. As application examples we consider two nonlinear elliptic model problems: the stationary solid fuel ignition model and the stationary viscous Burgers equation. For these problems we present a more detailed convergence analysis of the method. The resulting algorithms are computationally implemented in combination with an adaptive mesh refinement strategy, which leads to an improvement in the performance of the one-shot approach.

Abstract: We consider the bilevel optimisation approach proposed in [5] for learning the optimal parameters in a Total Variation (TV) denoising model featuring for multiple noise distributions. In applications, the use of databases (dictionaries) allows an accurate estimation of the parameters, but reflects in high computational costs due to the size of the databases and to the nonsmooth nature of the PDE constraints. To overcome this computational barrier we propose an optimisation algorithm that, by sampling dynamically from the set of constraints and using a quasi-Newton method, solves the problem accurately and in an efficient way.

Abstract: To model the dynamics of discrete deterministic systems, we extend the Petri nets framework by a priority relation between conflicting transitions, which is encoded by orienting the edges of a transition conflict graph. The aim of this paper is to gain some insight into the structure of this conflict graph and to characterize a class of suitable orientations by an analysis in the context of hypergraph theory.

Abstract: We propose a nonsmooth PDE-constrained optimization approach for the determination of the correct noise model in total variation (TV) image denoising. An optimization problem for the determination of the weights corresponding to different types of noise distributions is stated and existence of an optimal solution is proved. A tailored regularization approach for the approximation of the optimal parameter values is proposed thereafter and its consistency studied. Additionally, the differentiability of the solution operator is proved and an optimality system characterizing the optimal solutions of each regularized problem is derived. The optimal parameter values are numerically computed by using a quasi-Newton method, together with semismooth Newton type algorithms for the solution of the TV-subproblems.

Abstract: We extend the applicability of Newton’s method for k-Fréchet differentiable operators in a Banach space setting by using a more flexible way of computing upper bounds on the inverses of the operators involved. In particular, we improve and extend the recent works by Ezquerro et al. (2012, 2013) [13,15]. Moreover, we illustrate our study with some numerical examples involving Hammerstein integral equations.

Abstract: We present a tighter local convergence result for Newton’s method under generalized conditions of Kantorovich type than the one given by Dennis and Schnabel (Numerical methods for unconstrained optimization and nonlinear equations. SIAM, Philadelphia, 1996) and by Ezquerro et al. (J Comput Appl Math 236:2246–2258, 2012) by using more precise majorizing functions and sequences. These improvements are obtained under the same computational cost as in the earlier studies. Numerical examples are also provided to show that the new convergence radii are larger and the new error bounds are tighter than the older ones.

Abstract: We propose the use of the Kantorovich-Rubinstein norm from optimal transport in imaging problems. In particular, we discuss a variational regularisation model endowed with a Kantorovich-Rubinstein discrepancy term and total variation regularization in the context of image denoising and cartoon-texture decomposition. We also show that the respective optimization problem can be turned into a convex-concave saddle point problem with simple constraints and hence, can be solved by standard tools.

Abstract: We propose a nonsmooth PDE-constrained optimization approach for the determination of the correct noise model in total variation (TV) image denoising. An optimization problem for the determination of the weights corresponding to different types of noise distributions is stated and existence of an optimal solution is proved. A tailored regularization approach for the approximation of the optimal parameter values is proposed thereafter and its consistency studied.

Abstract: We study the numerical simulation of thermally convective viscoplastic fluids using semismooth second-order methods. The problem is formulated as a system of partial differential equations and variational inequalities, and we demonstrate the efficiency of the proposed algorithm in handling the non-smoothness of the flow.

Abstract: This paper addresses the optimization of mixed variational inequalities that model the flow of viscoplastic materials. We prove existence of solutions, derive optimality conditions, and propose a numerical framework based on semismooth Newton methods for large-scale simulations.

Abstract: We propose a combined backward differentiation formula (BDF) and semismooth Newton approach for the numerical solution of time-dependent Bingham fluid flow. The methodology allows for robust time-stepping while efficiently identifying the rigid and fluid regions at each step.

Abstract: We investigate optimality conditions for optimization problems constrained by a class of variational inequalities of the second kind. Based on a nonsmooth primal–dual reformulation of the governing inequality, the differentiability of the solution map is studied. Directional differentiability is proved both for finite-dimensional problems and for problems in function spaces, under suitable assumptions on the active set. A characterization of Bouligand and strong stationary points is obtained thereafter. Finally, based on the obtained first-order information, a trust-region algorithm is proposed for the solution of the optimization problems.

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: This paper focuses on the numerical solution of a variational inequality of the second kind, which arises as a model for the laminar flow of a Herschel-Bulkley fluid in the cross-section of a pipe. To tackle this problem, we develop a nonsmooth proximal bundle algorithm that bypasses the need for regularization techniques. We begin by formulating and analyzing an associated nonsmooth and convex optimization problem that characterizes the solution of the variational inequality. Following a discretize-then-optimize approach, we employ a first-order finite element discretization for the objective functional. The core of our method lies in the nonsmooth bundle algorithm, which leverages a Moreau-Yosida approximation combined with a quasi-Newton BFGS update. This approach approximates the function and gradient values through a finite inner bundle algorithm. We build and analyze the proposed algorithm, examining its convergence properties in the context of the flow model. Additionally, we demonstrate its efficiency through both theoretical analysis and numerical experiments.

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 is established.

Abstract: This paper develops a duality-based semismooth Newton framework for solving variational inequalities of the second kind. We demonstrate that the dual formulation leads to slantly differentiable operators, allowing for the application of fast Newton-type solvers with guaranteed local convergence.

Abstract: We propose a model order reduction strategy for the optimal control of evolution equations based on balanced truncation. The method allows for a significant reduction in the dimension of the state space while maintaining high fidelity in the control approximation.

Abstract: This work focuses on the optimization of Bingham fluid flow in cylindrical pipes. We derive optimality conditions for the flow model and implement a numerical algorithm based on second-order information to solve the resulting variational inequalities.

Abstract: We investigate nonlinear optimal control problems subject to vector-valued affine control constraints. We establish existence results and provide a characterization of the optimal control through Lagrange multipliers and semismoothness analysis.

Abstract: This paper investigates the primal-dual active set method for solving optimal control problems of the Stokes equations subject to control constraints. We prove the equivalence of the active set strategy with a semismooth Newton method and demonstrate its fast convergence behavior.

Abstract: We propose and analyze a semi-smooth Newton method for the numerical solution of optimal control problems governed by the stationary Navier-Stokes equations with pointwise state constraints. A regularization approach is used to handle the constraints, and the convergence of the method is verified through numerical experiments.

Abstract: We study the boundary optimal control of the Navier-Stokes equations with control constraints. A semismooth Newton method is applied to the optimality system, and we provide a detailed analysis of the differentiability properties of the boundary control-to-state mapping.

Abstract: This paper analyzes the numerical solution of distributed optimal control of the Navier-Stokes equations under bilateral pointwise control constraints. A primal-dual active set strategy is applied, showing global and local convergence properties.

Abstract: A comparison of three different but related numerical methods for control constrained optimal control of the Burgers equation is carried out. Principal ideas and detailed numerical examples are presented.

Abstract: A Sequential Quadratic Programming (SQP) method for the optimal control of the Burgers equation subject to constraints is investigated. The performance of the method is evaluated through numerical simulations.