Abstract: A semismooth Newton method (refered as DC–SSN) is proposed for the numerical solution of a class of nonconvex optimal control problems governed by linear elliptic partial differential equations. The nonconvex term in the cost functional arises from a Huber-type local regularization of the Lq-quasinorm (q ∈ (0, 1)), therefore it promotes sparsity on the solution. The DC–SSN method solves the optimality system of the regularized problem resulting from the application of difference-of-convex functions programming tools. © 2021, Springer Nature Switzerland AG.

Abstract: We study the stationary Stokes and Navier-Stokes equations with nonhomogeneous Navier boundary conditions in a bounded domain Ω⊂R3 of class C1,1. We prove the existence and uniqueness of weak and strong solutions in W1,p(Ω) and W2,p(Ω) for all 1<p<∞, considering minimal regularity on the friction coefficient α. Moreover, we deduce uniform estimates for the solution with respect to α which enables us to analyze the behavior of the solution when α→∞. © 2021 Elsevier Inc. AUTHOR KEYWORDS: Inf-sup condition; Lp-regularity; Navier-Stokes equations; Nonhomogeneous Navier boundary conditions; Stokes equations; Weak solution.

Abstract: We obtain some Hölder regularity estimates for an Hamilton-Jacobi with fractional time derivative of order-2 (0; 1) cast by a Caputo derivative. The Hölder seminorms are independent of time, which allows to investigate the large time behavior of the solutions. We focus on the Namah-Roquejo re setting whose typical example is the Eikonal equation. Contrary to the classical time derivative case α = 1, the convergence of the solution on the so-called projected Aubry set, which is an important step to catch the large time behavior, is not straightforward. Indeed, a function with nonpositive Caputo derivative for all time does not necessarily converge; we provide such a counterexample. However, we establish partial results of convergence under some geometrical assumptions.

PREDICTIVE ONLINE OPTIMISATION WITH APPLICATIONS TO OPTICAL FLOW

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

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

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

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ERROR ESTIMATES FOR THE FEM APPROXIMATION OF OPTIMAL SPARSE CONTROL OF ELLIPTIC EQUATIONS WITH POINTWISE STATE CONSTRAINTS AND FINITE-DIMENSIONAL CONTROL SPACE

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

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A BDF2-SEMISMOOTH NEWTON ALGORITHM FOR THE NUMERICAL SOLUTION OF THE BINGHAM FLOW WITH TEMPERATURE DEPENDENT PARAMETERS

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

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Circuits and Circulant Minors

Abstract: Circulant contraction minors play a key role for characterizing ideal circular matrices in terms of minimally non ideal structures. In this article we prove necessary and sufficient conditions for a circular matrix A to have circulant contraction minors in terms of circuits in a digraph associated with A. In the particular case when A itself is a circulant matrix, our result provides an alternative characterization to the one previously known from the literature. © 2019 The Author(s).

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Lp theory for Boussinesq system with Dirichlet boundary conditions

Abstract: We consider the stationary Boussinesq system with non-homogeneous Dirichlet boundary conditions in a bounded domain Ω ⊂ R3 of class C1, 1 with a possibly disconnected boundary. We prove the existence of weak solutions in W1,p(Ω), strong solutions in W2,p(Ω) and very weak solutions in Lp(Ω) of the stationary Boussinesq system by assuming that the fluxes of the velocity are sufficiently small. Finally, as it is expected, we obtain the uniqueness of the solution by considering small data. © 2018, © 2018 Informa UK Limited, trading as Taylor & Francis Group.

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Stokes and Navier–Stokes equations with Navier boundary condition [Équations de Stokes et de Navier–Stokes avec la condition de Navier]

Abstract: In this paper, we study the stationary Stokes and Navier–Stokes equations with non-homogeneous Navier boundary condition in a bounded domain Ω⊂R 3 of class C 1,1 from the viewpoint of the behavior of solutions with respect to the friction coefficient α. We first prove the existence of a unique weak solution (and strong) in W 1,p (Ω) (and W 2,p (Ω)) to the linear problem for all 1<p<∞ considering minimal regularity of α using some inf–sup condition concerning the rotational operator. Furthermore, we deduce uniform estimates of the solutions for large α which enables us to obtain the strong convergence of Stokes solutions with Navier slip boundary condition to the one with no-slip boundary condition as α→∞. Finally, we discuss the same questions for the non-linear system. © 2018 Académie des sciences

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A computational model of scientific discovery in a very simple world, aiming at psychological realism

Abstract: We propose a computational model of human scientific discovery and perception of the world. As a prerequisite for such a model, we simulate dynamic microworlds in which physical events take place, as well as an observer that visually perceives and makes interpretations of events in the microworld. Moreover, we give the observer the ability to actively conduct experiments in order to gain evidence about natural regularities in the world. We have broken up the description of our project into two pieces. The first piece deals with the interpreter constructing relatively simple visual descriptions of objects and collisions within a context. The second phase deals with the interpreter positing relationships among the entities, winding up with elaborated construals and conjectures of mathematical laws governing the world. This paper focuses only on the second phase. As is the case with most human scientific observation, observations are subject to interpretation, and the discoveries are influenced by these interpretations. © 2019, © 2019 Informa UK Limited, trading as Taylor & Francis Group.

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First displacement time of a tagged particle in a stochastic cluster in a simple exclusion process with random slow bonds

Abstract: We consider the nearest-neighbour simple exclusion process on the one-dimensional discrete torus , with random rates defined in terms of a homogeneous Poisson process on with intensity . Given a realization of the Poisson process, the jump rate along the edge is 1 if there is not any Poisson mark in ; otherwise, it is . The density profile of this process with initial measure associated to an initial profile , evolves as the solution of a bounded diffusion random equation. This result follows from an appropriate quenched hydrodynamic limit. If then is discontinuous at each Poisson mark with passage through the slow bonds, otherwise the conductance at the slow bonds decreases meaning no passage through the slow bonds in the continuum. The main results are concerned with upper and lower quenched and annealed bounds of , where is the first displacement time of a tagged particle in a stochastic cluster of size j (the cluster is defined via specific macroscopic density profiles). It is possible to observe that when time t grows, then decays quadratically in both the upper and lower bounds, and falls as slow as the presence of more Poisson marks neighbouring the tagged particle, as expected. © Applied Probability Trust 2019.

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PRIMAL-DUAL BLOCK-PROXIMAL SPLITTING FOR A CLASS OF NON-CONVEX PROBLEMS

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

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A BILEVEL LEARNING APPROACH FOR OPTIMAL OBSERVATION PLACEMENT IN VARIATIONAL DATA ASSIMILATION

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

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TOTAL GENERALIZED VARIATION REGULARIZATION IN DATA ASSIMILATION FOR BURGERS' EQUATION

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

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A DIFFERENCE-OF-CONVEX FUNCTIONS APPROACH FOR SPARSE PDE OPTIMAL CONTROL PROBLEMS WITH NONCONVEX COSTS

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

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ACCELERATION AND GLOBAL CONVERGENCE OF A FIRST-ORDER PRIMAL-DUAL METHOD FOR NONCONVEX PROBLEMS

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

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BLOCK-PROXIMAL METHODS WITH SPATIALLY ADAPTED ACCELERATION

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

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

Abstract: In this paper we propose a multigrid optimization algorithm (MG/OPT) for the numerical solution of a class of quasilinear variational inequalities of the second kind. This approach is enabled by the fact that the solution of the variational inequality is given by the minimizer of a nonsmooth energy functional, involving the p-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. © 2017 Elsevier Ltd

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Weakly coupled systems of parabolic Hamilton–Jacobi equations with Caputo time derivative

Abstract: In this paper we prove existence and uniqueness of bounded viscosity solutions of weakly coupled systems of parabolic Hamilton–Jacobi equations with nonlocal ingredients, where the time evolution of each equation is driven by Caputo derivatives of different orders. As an application, we present steady-state large time behavior for the system in the case the stationary equation has uniqueness. © 2018, Springer Nature Switzerland AG.

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TESTING AND NON-LINEAR PRECONDITIONING OF THE PROXIMAL POINT METHOD

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

LEER

ON THE OPTIMAL CONTROL OF SOME NONSMOOTH DISTRIBUTED PARAMETER SYSTEMS ARISING IN MECHANICS

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

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EXISTENCE AND UNIQUENESS OF WEAK SOLUTIONS FOR NONLOCAL PARABOLIC PROBLEMS VIA THE GALERKIN METHOD

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

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AN EXACT APPROACH FOR THE BALANCED K-WAY PARTITIONING PROBLEM WITH WEIGHT CONSTRAINTS AND ITS APPLICATION TO SPORTS TEAM REALIGNMENT

Blow-up for the b-family of equations

Abstract: In this paper, we consider the b-family of equations on the torus ut−utxx+(b + 1)uux=buxuxx+uuxxx, which for appropriate values of b reduces to well-known models, such as the Camassa–Holm equation or the Degasperis–Procesi equation. We establish a local-in-space blow-up criterion. Copyright © 2016 John Wiley & Sons, Ltd. Copyright © 2016 John Wiley & Sons, Ltd.

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Exponential Propagation for Fractional Reaction–Diffusion Cooperative Systems with Fast Decaying Initial Conditions

Abstract: We study the time asymptotic propagation of solutions to the reaction–diffusion cooperative systems with fractional diffusion. We prove that the propagation speed is exponential in time, and we find the precise exponent of propagation. This exponent depends on the smallest index of the fractional laplacians and on the principal eigenvalue of the matrix DF(0) where F is the reaction term. We also note that this speed does not depend on the space direction. © 2015, Springer Science+Business Media New York.

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Exponential Propagation for Fractional Reaction–Diffusion Cooperative Systems with Fast Decaying Initial Conditions

Abstract: We study the time asymptotic propagation of solutions to the reaction–diffusion cooperative systems with fractional diffusion. We prove that the propagation speed is exponential in time, and we find the precise exponent of propagation. This exponent depends on the smallest index of the fractional laplacians and on the principal eigenvalue of the matrix DF(0) where F is the reaction term. We also note that this speed does not depend on the space direction. © 2015, Springer Science+Business Media New York.

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A computer model of context-dependent perception in a very simple world

Abstract: We propose the foundations of a computer model of scientific discovery that takes into account certain psychological aspects of human observation of the world. To this end, we simulate two main components of such a system. The first is a dynamic microworld in which physical events take place, and the second is an observer that visually perceives entities and events in the microworld. For reason of space, this paper focuses only on the starting phase of discovery, which is the relatively simple visual inputs of objects and collisions. © 2017 Informa UK Limited, trading as Taylor & Francis Group.

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A MULTIGRID OPTIMIZATION ALGORITHM FOR THE NUMERICAL SOLUTION OF QUASILINEAR VARIATIONAL INEQUALITIES INVOLVING THE P-LAPLACIAN

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

LEER

FIRST ORDER METHODS FOR HIGH RESOLUTION IMAGE DENOISING

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

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PRECONDITIONED PROXIMAL POINT METHODS AND NOTIONS OF PARTIAL SUBREGULARITY

INTEGRO-DIFFERENTIAL SYSTEMS OF MIXED TYPE INVOLVING HIGHER ORDER FRACTIONAL LAPLACIAN, INTEGRAL TRANSFORMS AND SPECIAL FUNCTIONS

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

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INFIMAL CONVOLUTION OF DATA DISCREPANCIES FOR MIXED NOISE REMOVAL

LEARNING OPTIMAL SPATIALLY-DEPENDENT REGULARIZATION PARAMETERS IN TOTAL VARIATION IMAGE DENOISING

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

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THE JUMP SET UNDER GEOMETRIC REGULARISATION. PART 2: HIGHER-ORDER APPROACHES

Abstract:

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BILEVEL APPROACHES FOR LEARNING OF VARIATIONAL IMAGING MODELS

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

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EXISTENCE AND UNIQUENESS FOR PARABOLIC PROBLEMS WITH CAPUTO TIME DERIVATIVE

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

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SECOND-ORDER ORTHANT-BASED METHODS WITH ENRICHED HESSIAN INFORMATION FOR SPARSE ${\ell }_1$-OPTIMIZATION

A PRECONDITIONED DESCENT ALGORITHM FOR VARIATIONAL INEQUALITIES OF THE SECOND KIND INVOLVING THE P-LAPLACIAN OPERATOR

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

LEER

Balanced partition of a graph for football team realignment in Ecuador

Abstract: In the second category of the Ecuadorian football league, a set of football teams must be grouped into k geographical zones according to some regulations, where the total distance of the road trips that all teams must travel to play a Double Round Robin Tournament in each zone is minimized. This problem can be modeled as a k-clique partitioning problem with constraints on the sizes and weights of the cliques. An integer programming formulation and a heuristic approach were developed to provide a solution to the problem which has been implemented in the 2015 edition of the aforementioned football championship. © Springer International Publishing Switzerland 2016.

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Newton’s method for convex optimization

Abstract: In this chapter we deal with the convex optimization problem (COP). Using the generalized-Newton’s algorithm (GNA) we generate a sequence that converges to a solution of the COP. We use weak-center and weak Lipschitz-type conditions in our semilocal convergence analysis leading to a finer convergence analysis than in earlier studies. Numerical examples where earlier sufficient convergence conditions are not satisfied but our conditions are satisfied are also presented in this chapter. © Springer International Publishing Switzerland 2016.

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

Abstract: We derive a-priori error estimates for the finite-element approximation of a distributed optimal control problem governed by the steady one-dimensional Burgers equation with pointwise box constraints on the control. Here the approximation of the state and the control is done by using piecewise linear functions. With this choice, a superlinear order of convergence for the control is obtained in the (Formula presented.) -norm; moreover, under a further assumption on the regularity structure of the optimal control this error estimate can be improved to (Formula presented.) , extending the results in Rösch (Optim. Methods Softw. 21(1): 121–134, 2006). The theoretical findings are tested experimentally by means of numerical examples. © 2015, Springer Science+Business Media New York.

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Modeling high-dimensional time-varying dependence using dynamic D-vine models

Abstract: We consider the problem of modeling the dependence among many time series. We build high-dimensional time-varying copula models by combining pair-copula constructions with stochastic autoregressive copula and generalized autoregressive score models to capture dependence that changes over time. We show how the estimation of this highly complex model can be broken down into the estimation of a sequence of bivariate models, which can be achieved by using the method of maximum likelihood. Further, by restricting the conditional dependence parameter on higher cascades of the pair copula construction to be constant, we can greatly reduce the number of parameters to be estimated without losing much flexibility. Applications to five MSCI stock market indices and to a large dataset of daily stock returns of all constituents of the Dax 30 illustrate the usefulness of the proposed model class in-sample and for density forecasting. Copyright © 2016 John Wiley & Sons, Ltd. Copyright © 2016 John Wiley & Sons, Ltd.

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Backward estimates for nonnegative solutions to a class of singular parabolic equations

Abstract: In this paper we improve a recent result of the same Authors (Recalde and Vespri, 2015) by using the same approach in a more tricky way. More precisely, we study a class of quasilinear singular parabolic equations with L∞ coefficients, whose prototypes are the p-Laplacian ([formula presented]<p<2) equations. Let us consider nonnegative solutions defined in R+×RN. We recall that in Recalde and Vespri (2015), starting from the value attained in a point (x0,t0) by the solution, we proved sharp estimates in the stripe ((1−ε)t0,∞)×RN. In this note we show that, quite surprisingly, sharp estimates hold also for the remote past i.e. also for the stripe (0,t0)×RN. In the last section we briefly show how these results can be adapted to equations of Porous Medium type in the fast diffusion range i.e. ([formula presented])+<m<1. © 2016 Elsevier Ltd

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OPTIMAL CONTROL OF STATIC ELASTOPLASTICITY IN PRIMAL FORMULATION

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

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PROPAGATION SPEED FOR FRACTIONAL COOPERATIVE SYSTEMS WITH SLOWLY DECAYING INITIAL CONDITIONS

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

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ASYMPTOTIC BEHAVIOUR OF SOLUTIONS TO ONE-DIMENSIONAL REACTION DIFFUSION COOPERATIVE SYSTEMS INVOLVING INFINITESIMAL GENERATORS

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

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BALANCED PARTITION OF A GRAPH FOR FOOTBALL TEAM REALIGNMENT IN ECUADOR

Abstract: In the second category of the Ecuadorian football league, a set of football teams must be grouped into $k$ geographical zones according to some regulations, where the total distance of the road trips that all teams must travel to play a Double Round Robin Tournament in each zone is minimized. This problem can be modeled as a $k$-clique partitioning problem with constraints on the sizes and weights of the cliques. An integer programming formulation and a heuristic approach were developed to provide a solution to the problem which has been implemented in the 2015 edition of the aforementioned football championship.

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BILEVEL PARAMETER LEARNING FOR HIGHER-ORDER TOTAL VARIATION REGULARISATION MODELS

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

LEER

An adaptive numerical method for semi-infinite elliptic control problems based on error estimates

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. © 2014 Taylor & Francis. AUTHOR KEYWORDS: elliptic partial differential equation; FEM; numerical methods; optimal control; semi-infinite programming

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Harnack estimates at large: Sharp pointwise estimates for nonnegative solutions to a class of singular parabolic equations to celebrate the 60th birthday of Enzo Mitidieri

Abstract: In this paper we deal with quasilinear singular parabolic equations with L∞ coefficients, whose prototypes are the p-Laplacian (2NN+1<p<2) equations. In this range of the parameters, we are in the so called fast diffusion case. Extending a recent result (Ragnedda et al. 2013), we are able to prove Harnack estimates at large, i.e. starting from the value attained in a point by the solution, we are able to give explicit and sharp pointwise estimates, from below by using the Barenblatt solutions. In the last section we briefly show how these results can be adapted to equations of porous medium type in the fast diffusion range i.e. (N-2N)+<m

LEER

Local convergence analysis of inexact Gauss–Newton method for singular systems of equations under majorant and center-majorant condition

Abstract: We present a new semi-local convergence analysis of the Gauss–Newton method for solving convex composite optimization problems using the concept of quasi-regularity for an initial point. The convergence analysis is based on a combination of a center-majorant and a majorant function. The results extend the applicability of the Gauss–Newton method under the same computational cost as in earlier studies. In particular, the advantages are: the error estimates on the distances involved are more precise and the convergence ball is at least as large. Numerical examples are also provided in this study. © 2015, Sociedad Española de Matemática Aplicada. AUTHOR KEYWORDS: Center-majorant function; Convergence ball; Gauss–Newton method; Local convergence; Majorant function

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MINOR RELATED ROW FAMILY INEQUALITIES FOR THE SET COVERING POLYHEDRON OF CIRCULANT MATRICES

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

LEER

LIMITING ASPECTS OF NONCONVEX ${TV}^{\varphi }$ MODELS

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

LEER

FINITE ELEMENT ERROR ESTIMATES FOR AN OPTIMAL CONTROL PROBLEM GOVERNED BY THE BURGERS EQUATION

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

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OPTIMAL CONTROL OF ELECTRORHEOLOGICAL FLUIDS THROUGH THE ACTION OF ELECTRIC FIELDS

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

LEER

THE STRUCTURE OF OPTIMAL PARAMETERS FOR IMAGE RESTORATION PROBLEMS

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

LEER

SECOND-ORDER APPROXIMATION AND FAST MULTIGRID SOLUTION OF PARABOLIC BILINEAR OPTIMIZATION PROBLEMS

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

LEER

ERROR ESTIMATES FOR OPTIMAL CONTROL PROBLEMS OF A CLASS OF QUASILINEAR EQUATIONS ARISING IN VARIABLE VISCOSITY FLUID FLOW

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.

LEER

THE DYNAMICS OF DETERMINISTIC SYSTEMS - A SURVEY

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

LEER

AN ADAPTIVE NUMERICAL METHOD FOR SEMI-INFINITE ELLIPTIC

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.

LEER

STRONG STATIONARITY CONDITIONS FOR A CLASS OF OPTIMIZATION PROBLEMS GOVERNED BY VARIATIONAL INEQUALITIES OF THE SECOND KIND

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.

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