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En este artículo presentamos el modelo matemático de tipo compartimental con individuos asintomáticos utilizado por el Centro de Modelización Matemática (MODEMAT) para estudiar la propagación del SARS-CoV-2 en Ecuador, así como el esquema variacional de tipo bayesiano desarrollado para estimar los diferentes parámetros del modelo, en presencia de incertidumbre de los datos observados. Esta estimación permite realizar actualizaciones periódicas del número efectivo de reproducción, así como proyecciones a corto plazo, mediante métodos de ensamble de la incidencia de la epidemia.

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 non-smooth trust-region algorithm for the numerical solution of the bilevel problem and test it computationally for two particular experimental settings., 33 pages, 9 figures, 4 tables

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.

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

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 Frechet 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 Holder coefficients and a second order sufficient optimality condition. The paper ends with a numerical experiment, where the approximation order is computationally tested.

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.

We propose a nonsmooth trust-region method for solving optimization problems with locally Lipschitz continuous functions, with application to problems constrained by variational inequalities of the second kind. Under suitable assumptions on the model functions, convergence of the general algorithm to a C-stationary point is verified. For variational inequality constrained problems, we are able to properly characterize the Bouligand subdifferential of the reduced cost function and, based on that, we propose a computable trust-region model which fulfills the convergence hypotheses of the general algorithm. The article concludes with the experimental study of the main properties of the proposed method based on two different numerical instances.

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.

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 that are frequently considered in applications, in particular mixtures of salt & pepper and Gaussian noise, and Gaussian and Poisson noise. We derive a variational image denoising model that features a total variation regularisation term and a data discrepancy that features 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, and discuss in particular its interpretation as the MAP of a so-called infinity convolution of two noise distributions. Moreover, 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 existing methods dealing with the mixed noise case are shown.

Optimal control problems of mixed variational inequalities of the second kind arising in flow of Bingham viscoplastic materials are considered. Two type of active-inactive set regularizing functions for the control problems are proposed and approximation properties and optimality conditions are investigated. A detailed first order optimality system for the control problem is obtained as limit of the regularized optimality conditions. For the solution of each regularized system a globalized semismooth Newton algorithm is constructed and its computational performance is investigated.

Optimal control problems governed by a class of elliptic variational inequalities of the second kind are investigated. Applications include the optimal control of viscoplastic fluid flow and of simplified friction. Based on a Tikhonov regularization of the dual problem, a family of primal-dual regularized control problems is introduced, and convergence of the regularized solutions towards a solution of the original control problem is verified. For each regularized problem an optimality condition is derived, and an optimality system for the original control problem is obtained as a limit of the regularized ones. Thanks to the structure of the proposed regularization, complementarity relations between the variables involved are derived. Since the regularized optimality systems involve Newton differentiable functions, a semismooth Newton algorithm is proposed and its numerical performance investigated.

The numerical solution of the Dirichlet boundary optimal control problem of the Navier–Stokes equations in presence of pointwise state constraints is investigated. Two different regularization techniques are considered. First, a Moreau–Yosida regularization of the problem is studied. Optimality conditions are derived and the convergence of the regularized solutions towards the original one is proved. A source representation of the control combined with a Lavrentiev type regularization strategy is also presented. The analysis concerning optimality conditions and convergence of the regularized solutions is carried out. In the last part of the paper numerical experiments are presented. For the numerical solution of each regularized problem a semi-smooth Newton method is applied.

This paper is devoted to the numerical solution of stationary laminar Bingham fluids by path-following methods. By using duality theory, a system that characterizes the solution of the original problem is derived. Since this system is ill-posed, a family of regularized problems is obtained and the convergence of the regularized solutions to the original one is proved. For the update of the regularization parameter, a path-following method is investigated. Based on the differentiability properties of the path, a model of the value functional and a correspondent algorithm are constructed. For the solution of the systems obtained in each path-following iteration a semismooth Newton method is proposed. Numerical experiments are performed in order to investigate the behavior and efficiency of the method, and a comparison with a penalty-Newton-Uzawa-conjugate gradient method, proposed in [Dean et al. , J. Non-Newtonian Fluid Mech. 142 (2007) 36–62], is carried out.

In this chapter we present some basic elements of the analysis of partial differential equations, and of their numerical discretization by finite differences. Our aim is to introduce some notions that enable the reader to follow the material developed in the subsequent chapters. Both the analysis and the numerical solution of partial differential equations (PDEs) are research areas by themselves, with a large amount of related literature. We refer, for instance, to the books [9,19] for the analysis of PDEs and to, e.g., [23, 52] for their numerical approximation.

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.

In this chapter we present optimality conditions of first and second order for optimization problems with PDE constraints. The theory is illustrated with several examples.

In this chapter we present and analyze some infinite dimensional optimization methods for the solution of problem (.1). Once the infinite dimensional algorithm is posed, the discretization of the partial differential equations is carried out. Such an approach is known as optimize-then-discretize in contrast to the discretize-then—optimize one, where the equations and the cost functional are first discretized and the problem is then solved using large-scale optimization tools.

In this chapter we present optimality conditions and solution methods for PDE-constrained optimization problems with so-called box constraints.

We consider the bilevel optimisation approach proposed by De Los Reyes, Sch\"onlieb (2013) 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.

In an appropriate function space setting, semismooth Newton methods are proposed for iteratively computing the solution of a rather general class of variational inequalities (VIs) of the second kind. The Newton scheme is based on the Fenchel dual of the original VI problem which is regularized if necessary. In the latter case, consistency of the regularization with respect to the original problem is studied. The application of the general framework to specific model problems including Bingham flows, simplified friction, or total variation regularization in mathematical imaging is described in detail. Finally, numerical experiments are presented in order to verify the theoretical results.

In this paper, we present a balanced truncation based strategy for the numerical solution of optimal control problems governed by nonlinear evolution partial differential equations. The idea consists in utilizing a balanced truncation model reduction method for the efficient solution of the semi-discretized adjoint system, while the nonlinear state equations are fully solved. Our strategy is analysed as a descent method in function spaces and global convergence results are presented. In combination with a Broyden-Fletcher-Goldfarb-Shanno update also superlinear convergence is verified. Numerical examples are given to illustrate the behaviour of the proposed method for different problems.

Optimal control problems of Bingham fluid flow in pipes are considered. After introducing a family of regularized problems, convergence of the regularized solutions towards the orignal one is verified. An optimality condition for the original problem is obtained as limit of the regularized optimality systems. For the solution of each regularized system a semismooth Newton algorithm is proposed.

We investigate a class of nonlinear optimal control problems with pointwise affine control constraints. Necessary optimality conditions of first order and sufficient second-order conditions are obtained. For the numerical solution of the optimal control problems a semismooth Newton method is proposed. Local superlinear convergence of the infinite-dimensional method is proved. Finally, the properties of the method are tested numerically by controlling the Navier-Stokes equations with affine constraints.

Second-order sufficient optimality conditions are established for the optimal control of semilinear elliptic and parabolic equations with pointwise constraints on the control and the state. In contrast to former publications on this subject, the cone of critical directions is the smallest possible in the sense that the second-order sufficient conditions are the closest to the associated necessary ones. The theory is developed for elliptic distributed controls in domains up to dimension three. Moreover, problems of elliptic boundary control and parabolic distributed control are discussed in spatial domains of dimension two and one, respectively.

The concept of active flow control is applied to the steady flow around a NACA4412 and to the unsteady flow around a generic high-lift configuration in order to delay separation. To the former steady suction upstream of the detachment position is applied. In a series of computations the suction angle β is varied and the main flow features are analyzed. A gradient descent method and an adjoint-based method are successfully used to optimize β. For the unsteady case periodic blowing and suction is employed to control the separation. Various calculations are conducted to obtain the dependency of the lift on the amplitude and frequency of the perturbation and the amplitude is optimized with the gradient descent method.

El 60% de los fracasos endodóncicos se deben a una obturación incompleta, por esta razón es importante realizar un sellado apical correcto en los tratamientos de conductos radiculares. El objetivo de este estudio fue comparar el grado de filtración apical entre la técnica de compactación lateral en frío y la técnica de obturación con System B®. Se utilizaron 30 dientes extraídos, se instrumentaron con Protaper®. Se dividieron en cuatro grupos: Grupo 1: dientes obturados con System B®. Grupo 2: dientes obturados con técnica de compactación lateral en frío. Grupo 3: control negativo. Grupo 4: control positivo. Se les dejó en tinta china y se les diafanizó. Los dientes fueron observados en un fotomicroscopio a un aumentó de 5X. Se encontró diferencia estadísticamente significativa en el grado de filtración apical entre la técnica de compactación lateral en frío y la técnica de obturación con System B®. La técnica de condensación lateral en frío filtró en mayor cantidad que la técnica de compactación con System B®. Se concluye que la técnica de compactación vertical con System B® produce un mejor sellado apical, ya que ésta contiene una mayor cantidad de gutapercha dentro del conducto.

The numerical solution of the Dirichlet boundary optimal control problem of the Navier-Stokes equations in presence of pointwise state constraints is investigated. A Moreau-Yosida regularization of the problem is proposed to obtain regular multipliers. Optimality conditions are derived and the convergence of the regularized solutions towards the original one is presented. The paper ends with a numerical experiment. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {TGV}^2$$\end{document}TGV2 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {ICTV}$$\end{document}ICTV is carried out, showing the advantages and shortcomings of both regularisers, depending on the structure of the processed images and their noise level.

This paper is devoted to the numerical simulation of two-dimensional stationary Bingham fluid flow by semismooth Newton methods. We analyze the modeling variational inequality of the second kind, considering both Dirichlet and stress-free boundary conditions. A family of Tikhonov regularized problems is proposed and the convergence of the regularized solutions to the original one is verified. By using Fenchel’s duality, optimality systems which characterize the original and regularized solutions are obtained. The regularized optimality systems are discretized using a finite element method with (cross-grid P1)–Q0 elements for the velocity and pressure, respectively. A semismooth Newton algorithm is proposed in order to solve the discretized optimality systems. Using an additional relaxation, a descent direction is constructed from each semismooth Newton iteration. Local superlinear convergence of the method is also proved. Finally, we perform numerical experiments in order to investigate the behavior and efficiency of the method.