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A novel generalization of Γ -convergence applicable to a class of equilibrium problems is studied. After the introduction of the latter, a variety of its applications is discussed. The existence of equilibria with emphasis on Nash equilibrium problems is investigated. Subsequently, our Γ -convergence notion for equilibrium problems is introduced and discussed as well as applied to a class of penalized generalized Nash equilibrium problems and quasi-variational inequalities. The work ends with a comparison of our results to previous generalizations in the literature. [ABSTRACT FROM AUTHOR]

We consider a class of N-player linear quadratic differential generalized Nash equilibrium problems (GNEPs) with bound constraints on the individual control and state variables. In addition, we assume the individual players' optimal control problems are coupled through their dynamics and objectives via a time-dependent mean field interaction term. This assumption allows us to model the realistic setting that strategic players in large games cannot observe the individual states of their competitors. We observe that the GNEPs require a constraint qualification, which necessitates sufficient robustness of the individuals, in order to prove the existence of an open-loop pure strategy Nash equilibrium and to derive optimality conditions. In order to gain qualitative insight into the N-player game, we assume that players are identical and pass to the limit in N to derive a type of first-order constrained mean field game (MFG). We prove that the mean field interaction terms converge to an absolutely continuous curve of probability measures on the set of possible state trajectories. Using variational convergence methods, we show that the optimal control problems converge to a representative agent problem. Under additional regularity assumptions, we provide an explicit form for the mean field term as the solution of a continuity equation and demonstrate the link back to the N-player GNEP. [ABSTRACT FROM AUTHOR]

We propose and analyze a numerical algorithm for solving a class of optimal control problems for learning-informed semilinear partial differential equations (PDEs). Such PDEs contain constituents that are in principle unknown and are approximated by nonsmooth ReLU neural networks. We first show that direct smoothing of the ReLU network with the aim of using classical numerical solvers can have disadvantages, such as potentially introducing multiple solutions for the corresponding PDE. This motivates us to devise a numerical algorithm that treats directly the nonsmooth optimal control problem, by employing a descent algorithm inspired by a bundle-free method. Several numerical examples are provided and the efficiency of the algorithm is shown. [ABSTRACT FROM AUTHOR]

The analysis and boundary optimal control of the nonlinear transport of gas on a network of pipelines is considered. The evolution of the gas distribution on a given pipe is modeled by an isothermal semilinear compressible Euler system in one space dimension. On the network, solutions satisfying (at nodes) the Kirchhoff flux continuity conditions are shown to exist in a neighborhood of an equilibrium state. The associated nonlinear optimization problem then aims at steering such dynamics to a given target distribution by means of suitable (network) boundary controls while keeping the distribution within given (state) constraints. The existence of local optimal controls is established and a corresponding Karush–Kuhn–Tucker (KKT) stationarity system with an almost surely non-singular Lagrange multiplier is derived. [ABSTRACT FROM AUTHOR]

This paper is concerned with the distributed optimal control of a time-discrete Cahn–Hilliard–Navier–Stokes system with variable densities. It focuses on the double-obstacle potential which yields an optimal control problem for a variational inequality of fourth order and the Navier–Stokes equation. The existence of solutions to the primal system and of optimal controls is established. The Lipschitz continuity of the constraint mapping is derived and used to characterize the directional derivative of the constraint mapping via a system of variational inequalities and partial differential equations. Finally, strong stationarity conditions are presented following an approach from Mignot and Puel. [ABSTRACT FROM AUTHOR]

In this paper we consider a fluid-structure interaction problem given by the steady Navier Stokes equations coupled with linear elasticity taken from (Lasiecka et al. in Nonlinear Anal 44:54–85, 2018). An elastic body surrounded by a liquid in a rectangular domain is deformed by the flow which can be controlled by the Dirichlet boundary condition at the inlet. On the walls along the channel homogeneous Dirichlet boundary conditions and on the outflow boundary do-nothing conditions are prescribed. We recall existence results for the nonlinear system from that reference and analyze the control to state mapping generalizing the results of (Wollner and Wick in J Math Fluid Mech 21:34, 2019) to the setting of the nonlinear Navier-Stokes equation for the fluid and the situation of mixed boundary conditions in a domain with corners. [ABSTRACT FROM AUTHOR]

A class of risk-neutral generalized Nash equilibrium problems is introduced in which the feasible strategy set of each player is subject to a common linear elliptic partial differential equation with random inputs. In addition, each player's actions are taken from a bounded, closed, and convex set on the individual strategies and a bound constraint on the common state variable. Existence of Nash equilibria and first-order optimality conditions are derived by exploiting higher integrability and regularity of the random field state variables and a specially tailored constraint qualification for GNEPs with the assumed structure. A relaxation scheme based on the Moreau-Yosida approximation of the bound constraint is proposed, which ultimately leads to numerical algorithms for the individual player problems as well as the GNEP as a whole. The relaxation scheme is related to probability constraints and the viability of the proposed numerical algorithms are demonstrated via several examples. [ABSTRACT FROM AUTHOR]

Total Generalized Variation (TGV) regularization in image reconstruction relies on an infimal convolution type combination of generalized first- and second-order derivatives. This helps to avoid the staircasing effect of Total Variation (TV) regularization, while still preserving sharp contrasts in images. The associated regularization effect crucially hinges on two parameters whose proper adjustment represents a challenging task. In this work, a bilevel optimization framework with a suitable statistics-based upper level objective is proposed in order to automatically select these parameters. The framework allows for spatially varying parameters, thus enabling better recovery in high-detail image areas. A rigorous dualization framework is established, and for the numerical solution, a Newton type method for the solution of the lower level problem, i.e. the image reconstruction problem, and a bilevel TGV algorithm are introduced. Denoising tests confirm that automatically selected distributed regularization parameters lead in general to improved reconstructions when compared to results for scalar parameters. [ABSTRACT FROM AUTHOR]

We analyze a potentially risk-averse convex stochastic optimization problem, where the control is deterministic and the state is a Banach-valued essentially bounded random variable. We obtain strong forms of necessary and sufficient optimality conditions for problems subject to equality and conical constraints. We propose a Moreau-Yosida regularization for the conical constraint and show consistency of the optimality conditions for the regularized problem as the regularization parameter is taken to infinity. [ABSTRACT FROM AUTHOR]

Inspired by applications in optimal control of semilinear elliptic partial differential equations and physics-integrated imaging, differential equation constrained optimization problems with constituents that are only accessible through data-driven techniques are studied. A particular focus is on the analysis and on numerical methods for problems with machine-learned components. For a rather general context, an error analysis is provided, and particular properties resulting from artificial neural network based approximations are addressed. Moreover, for each of the two inspiring applications analytical details are presented and numerical results are provided. [ABSTRACT FROM AUTHOR]

In the field of quantitative imaging, the image information at a pixel or voxel in an underlying domain entails crucial information about the imaged matter. This is particularly important in medical imaging applications, such as quantitative magnetic resonance imaging (qMRI), where quantitative maps of biophysical parameters can characterize the imaged tissue and thus lead to more accurate diagnoses. Such quantitative values can also be useful in subsequent, automatized classification tasks in order to discriminate normal from abnormal tissue, for instance. The accurate reconstruction of these quantitative maps is typically achieved by solving two coupled inverse problems which involve a (forward) measurement operator, typically ill‐posed, and a physical process that links the wanted quantitative parameters to the reconstructed qualitative image, given some underlying measurement data. In this review, by considering qMRI as a prototypical application, we provide a mathematically‐oriented overview on how data‐driven approaches can be employed in these inverse problems eventually improving the reconstruction of the associated quantitative maps. [ABSTRACT FROM AUTHOR]

For a class of quasi-variational inequalities (QVIs) of obstacle type the stability of its solution set and associated optimal control problems are considered. These optimal control problems are nonstandard in the sense that they involve an objective with set-valued arguments. The approach to study the solution stability is based on perturbations of minimal and maximal elements of the solution set of the QVI with respect to monotone perturbations of the forcing term. It is shown that different assumptions are required for studying decreasing and increasing perturbations and that the optimization problem of interest is well-posed. [ABSTRACT FROM AUTHOR]

A distributed elliptic control problem with control constraints is considered, which is formulated as a three field problem and consists of two variational equations for the state and the co-state variables as well as of a variational inequality for the control variable. Two discretization approaches with h p -finite elements are discussed: In the first discretization approach all variables (state, co-state and control) are discretized. A semi smooth Newton method is introduced for solving the resulting algebraic system. In the second discretization approach only the state and the co-state variables are discretized, whereas the control is determined by projection. A simple fixed point scheme is presented for the iterative solution of this approach. The main focus of the paper is on the derivation of reliable and efficient a posteriori error estimates, which enables h p -adaptive mesh refinements. In particular, the estimates can be applied to the iteration solutions, so that they can be used as a stopping criterion of the iterative solution schemes. In several numerical experiments the order of convergence of the (adaptive) discretization approaches and the efficiency as well as the reliability of the a posteriori error estimates are studied. [ABSTRACT FROM AUTHOR]

An identification problem for the friction parameter in a semilinear system of balance laws, describing the transport of gas through a passive network of pipelines, is considered. The existence of broad solutions to the state system is proven and sensitivity results for the corresponding solution operator are obtained. The existence of solutions to the output least squares formulation of the identification problem, based on noisy measurements over time at fixed spatial positions, is established. Finally, numerical experiments validate the theoretical findings. [ABSTRACT FROM AUTHOR]

A multimaterial topology optimization framework using phase fields is suggested for the simultaneous optimization of mechanical and optical properties to be used in the development of optoelectronic devices. The technique provides a means of determining the cross section of the material alignments needed to create a sufficiently large strain profile within an optically active region of a photonic device. Based on the physical aspects of the underlying device, a nonlinear multiphysics model for the elastic and optical properties is proposed in the form of a linear elliptic partial differential equation (elasticity) coupled via the underlying topology to an eigenvalue problem of Helmholtz type (optics). The differential sensitivity of the displacement and eigenfunctions with respect to the changes in the underlying topology is investigated. After proving existence and optimality results, numerical experiments leading to an optimal material distribution for maximizing the strain in a Ge-on-Si microbridge are given. The presence of a net gain at low voltages for the optimal design is demonstrated by solving the steady-state van Roosbroeck (drift-diffusion) system, which proves the viability of the approach for the development of next-generation photonic devices. [ABSTRACT FROM AUTHOR]

We address the problem of optimally placing sensor networks for convection-diffusion processes where the convective part is perturbed. The problem is formulated as an optimal control problem where the integral Riccati equation is a constraint and the design variables are sensor locations. The objective functional involves a term associated to the trace of the solution to the Riccati equation and a term given by a constrained optimization problem for the directional derivative of the previous quantity over a set of admissible perturbations. This paper addresses the existence of the derivative with respect to the convective part of the solution to the Riccati equation, the well-posedness of the optimization problem, and finalizes with a range of numerical tests. [ABSTRACT FROM AUTHOR]

In this paper we study the structure of solutions of the one dimensional weighted total variation regularisation problem, motivated by its application in signal recovery tasks. We study in depth the relationship between the weight function and the creation of new discontinuities in the solution. A partial semigroup property relating the weight function and the solution is shown and analytic solutions for simply data functions are computed. We prove that the weighted total variation minimisation problem is well-posed even in the case of vanishing weight function, despite the lack of coercivity. This is based on the fact that the total variation of the solution is bounded by the total variation of the data, a result that it also shown here. Finally the relationship to the corresponding weighted fidelity problem is explored, showing that the two problems can produce completely different solutions even for very simple data functions. [ABSTRACT FROM AUTHOR]

This paper is concerned with the distributed optimal control of a time-discrete Cahn- Hilliard-Navier-Stokes system with variable densities. It focuses on the double-obstacle potential which yields an optimal control problem for a family of coupled systems in each time instant of a variational inequality of fourth order and the Navier-Stokes equation. By proposing a suitable timediscretization, energy estimates are proved, and the existence of solutions to the primal system and of optimal controls is established for the original problem as well as for a family of regularized problems. The latter correspond to Moreau-Yosida-type approximations of the double-obstacle potential. The consistency of these approximations is shown, and first-order optimality conditions for the regularized problems are derived. Through a limit process with respect to the regularization parameter, a stationarity system for the original problem is established. The resulting system corresponds to a function space version of C-stationarity which is a special notion of stationarity for mathematical programs with equilibrium constraints. [ABSTRACT FROM AUTHOR]

While the quasilinear isothermal Euler equations are an excellent model for gas pipeline flow, the operation of the pipeline flow with high pressure and small Mach numbers allows us to obtain approximate solutions by a simpler semilinear model. We provide a derivation of the semilinear model that shows that the semilinear model is valid for sufficiently low Mach numbers and sufficiently high pressures. We prove an existence result for continuous solutions of the semilinear model that takes into account lower and upper bounds for the pressure and an upper bound for the magnitude of the Mach number of the gas flow. These state constraints are important both in the operation of gas pipelines and to guarantee that the solution remains in the set where the model is physically valid. We show the constrained exact boundary controllability of the system with the same pressure and Mach number constraints. [ABSTRACT FROM AUTHOR]

The derivation of multiplier-based optimality conditions for elliptic mathematical programs with equilibrium constraints (MPEC) is essential for the characterization of solutions and development of numerical methods. Though much can be said for broad classes of elliptic MPECs in both polyhedric and non-polyhedric settings, the calculation becomes significantly more complicated when additional constraints are imposed on the control. In this paper we develop three derivation methods for constrained MPEC problems: via concepts from variational analysis, via penalization of the control constraints, and via penalization of the lower-level problem with the subsequent regularization of the resulting nonsmoothness. The developed methods and obtained results are then compared and contrasted. [ABSTRACT FROM AUTHOR]

A priori estimates of the length of the primal-dual path resulting from a Moreau-Yosida approximation of the feasible set for state constrained optimal control problems are derived. These bounds depend on the regularity of the state and the dimension of the problem. Numerical results indicate that the bounds are indeed sharp and are typically attained in cases where the active set consists of isolated active points. Further conditions on the multiplier approximation are identified which guarantee higher convergence rates for the feasibility violation due to the Moreau-Yosida approximation process. Numerical experiments show again that the results are sharp and accurately predict the convergence behavior. [ABSTRACT FROM AUTHOR]

This paper considers a class of nonlinear evolution quasi-variational inequality (QVI) problems with pointwise gradient constraints in vector-valued function spaces. The existence and approximation of solutions is addressed based on a combination of C0-semigroup methods, Mosco convergence, and monotone operator techniques developed by Brézis. An algorithm based on semidiscretization in time is proposed and its numerical implementation based on a penalty approach and semismooth Newton methods is studied. This paper ends with a report on numerical tests which involve the p-Laplacian and several types of nonlinear constraints. [ABSTRACT FROM AUTHOR]

A mathematical framework for optimal bilinear control of nonlinear Schrödinger equations of Gross--Pitaevskii type arising in the description of Bose--Einstein condensates is presented. The obtained results generalize earlier efforts found in the literature in several aspects. In particular, the cost induced by the physical workload over the control process is taken into account rather than the often used L²- or H¹-norms for the cost of the control action. Well-posedness of the problem and existence of an optimal control are proved. In addition, the first order optimality system is rigorously derived. Also a numerical solution method is proposed, which is based on a Newton-type iteration, and used to solve several coherent quantum control problems. [ABSTRACT FROM AUTHOR]

A class of nonlinear elliptic quasi-variational inequality (QVI) problems with gradient constraints in function space is considered. Problems of this type arise, for instance, in the mathematical description of the magnetization of superconductors, in problems in elastoplasticity, or in electrostatics as well as in game theory. The paper addresses the iterative solution of the QVIs by a sequential minimization technique relying on the repeated solution of variational inequality--type problems. A monotone operator theoretic approach is developed which does not resort to Mosco convergence as is often done in connection with existence analysis for QVIs. For the numerical solution of the QVIs a penalty approach combined with a semismooth Newton iteration is proposed. The paper ends with a report on numerical tests involving the p-Laplace operator and various types of nonlinear constraints. [ABSTRACT FROM AUTHOR]

Abstract: Magnetic resonance images which are corrupted by noise and by smooth modulations are corrected using a variational formulation incorporating a total variation like penalty for the image and a high order penalty for the modulation. The optimality system is derived and numerically discretized. The cost functional used is non-convex, but it possesses a bilinear structure which allows the ambiguity among solutions to be resolved technically by regularization and practically by normalizing the maximum value of the modulation. Since the cost is convex in each single argument, convex analysis is used to formulate the optimality condition for the image in terms of a primal–dual system. To solve the optimality system, a nonlinear Gauss–Seidel outer iteration is used in which the cost is minimized with respect to one variable after the other using an inner generalized Newton iteration. Favorable computational results are shown for artificial phantoms as well as for realistic magnetic resonance images. Reported computational times demonstrate the feasibility of the approach in practice. [Copyright &y& Elsevier]

general multi-scale vectorial total variation model with spatially adapted regularization parameter for color image restoration is introduced in this paper. This total variation model contains an L-data fidelity for any τ∈[1,2]. The use of a spatial dependent regularization parameter improves the reconstruction of features in the image as well as an adequate smoothing for the homogeneous parts. The automated adaptation of this regularization parameter is made according to local statistical characteristics of the noise which contaminates the image. The corresponding multiscale vectorial total variation model is solved by Fenchel-duality and inexact semismooth Newton techniques. Numerical results are presented for the cases τ=1 and τ=2 which reconstruct images contaminated with salt-and-pepper noise and Gaussian noise, respectively. [ABSTRACT FROM AUTHOR]

A general Moreau--Yosida-based framework for minimization problems subject to partial differential equations and pointwise constraints on the control, the state, and its derivative is considered. A range space constraint qualification is used to argue existence of Lagrange multipliers and to derive a KKT-type system for characterizing first-order optimality of the unregularized problem. The theoretical framework is then used to develop a semismooth Newton algorithm in function space and to prove its locally superlinear convergence when solving the regularized problems. Further, for maintaining the local superlinear convergence in function space it is demonstrated that in some cases it might be necessary to add a lifting step to the Newton framework in order to bridge an L²-Lr-norm gap, with r > 2. The paper ends by a report on numerical tests. [ABSTRACT FROM AUTHOR]

Optimal control problems for the heat equation with pointwise bilateral control-state constraints are considered. A locally superlinearly convergent numerical solution algorithm is proposed and its mesh independence is established. Further, for the efficient numerical solution reduced space and Schur complement based preconditioners are proposed which take into account the active and inactive set structure of the problem. The paper ends by numerical tests illustrating our theoretical findings and comparing the efficiency of the proposed preconditioners. [ABSTRACT FROM AUTHOR]

An adjustment scheme for the regularization parameter of a Moreau-Yosida-based regularization, or relaxation, approach to the numerical solution of pointwise state constrained elliptic optimal control problems is introduced. The method utilizes error estimates of an associated finite element discretization of the regularized problems for the optimal selection of the regularization parameter in dependence on the mesh size of discretization and error estimates for the approximation error due to regularization. The theoretical results are verified numerically. [ABSTRACT FROM AUTHOR]

An adjustment scheme for the regularization parameter of a Moreau-Yosida-based regularization, or relaxation, approach to the numerical solution of pointwise state constrained elliptic optimal control problems is introduced. The method utilizes error estimates of an associated finite element discretization of the regularized problems for the optimal selection of the regularization parameter in dependence on the mesh size of discretization and error estimates for the approximation error due to regularization. The theoretical results are verified numerically. [ABSTRACT FROM AUTHOR]

We present an a posteriori error analysis of adaptive finite element approximations of distributed control problems for second order elliptic boundary value problems under bound constraints on the control. The error analysis is based on a residual-type a posteriori error estimator that consists of edge and element residuals. Since we do not assume any regularity of the data of the problem, the error analysis further invokes data oscillations. We prove reliability and efficiency of the error estimator and provide a bulk criterion for mesh refinement that also takes into account data oscillations and is realized by a greedy algorithm. A detailed documentation of numerical results for selected test problems illustrates the convergence of the adaptive finite element method. [ABSTRACT FROM AUTHOR]

Path-following methods for primal-dual active set strategies requiring a regularization parameter are introduced. Existence of a primal-dual path and its differentiability properties are analyzed. Monotonicity and convexity of the primal-dual path value function are investigated. Both feasible and infeasible approximations are considered. Numerical path-following strategies are developed and their efficiency is demonstrated by means of examples. [ABSTRACT FROM AUTHOR]

Distributed optimal control problems for the time-dependent and the stationary Navier-Stokes equations subjected to pointwise control constraints are considered. Under a coercivity condition on the Hessian of the Lagrange function, optimal solutions are shown to be directionally differentiable functions of perturbation parameters such as the Reynolds number, the desired trajectory, or the initial conditions. The derivative is characterized as the solution of an auxiliary linear-quadratic optimal control problem. Thus, it can be computed at relatively low cost. Taylor expansions of the minimum value function are provided as well. [ABSTRACT FROM AUTHOR]

Solving optimal control problems for nonlinear partial differential equations represents a significant numerical challenge due to the tremendous size and possible model difficulties (e.g., nonlinearities) of the discretized problems. In this paper, a novel space-mapping technique for solving the aforementioned problem class is introduced, analyzed, and tested. The advantage of the space-mapping approach compared to classical multigrid techniques lies in the flexibility of not only using grid coarsening as a model reduction but also employing (perhaps less nonlinear) surrogates. The space mapping is based on a regularization approach which, in contrast to other space-mapping techniques, results in a smooth mapping and, thus, avoids certain irregular situations at kinks. A new Broyden update formula for the sensitivities of the space map is also introduced. This quasi-Newton update is motivated by the usual secant condition combined with a secant condition resulting from differentiating the space-mapping surrogate. The overall algorithm employs a trust-region framework for global convergence. Issues involved in the computations are highlighted, and a report on a few illustrative numerical tests is given. [ABSTRACT FROM AUTHOR]

For a class of semismooth operator equations a mesh independence result for generalized Newton methods is established. The main result states that the continuous and the discrete Newton process, when initialized properly, converge q-linearly with the same rate. The problem class considered in the paper includes MCP-function based reformulations of first order conditions of a class of control constrained optimal control problems for partial differential equations for which a numerical validation of the theoretical results is given. [ABSTRACT FROM AUTHOR]

A preface to the special issue of "Mathematical Programming" on the 21st International Symposium on Mathematical Programming (ISMP) which took place in Berlin, Germany is presented.

Motivated by applications in economics and engineering, we consider the optimal control of a variational inequality with point evaluations of the state variable in the objective. This problem class constitutes a specific mathematical program with complementarity constraints (MPCC). In our context, the problem is posed in an adequate function space and the variational inequality involves second order linear elliptic partial differential operators. The necessary functional analytic framework complicates the derivation of stationarity conditions whereas the non-convex and non-differentiable nature of the problem challenges the design of an efficient solution algorithm. In this paper, we present a penalization and smoothing technique to derive first order type conditions related to C-stationarity in the associated Sobolev space setting. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

A Fenchel dualization scheme for the one-step time-discretized elasto-plastic contact problem with kinematic or isotropic hardening is considered. The associated path is induced by a combined Moreau-Yosida / Tichonov regularization of the dual problem. The sequence of solutions to the regularized problems is shown to converge strongly to the solution of the original problem. This property relies on the density of the intersection of certain convex sets. The corresponding conditions are worked out and customary regularization approaches are shown to be valid in this context. It is also argued that without higher regularity assumptions on the data the resulting problems possess Newton differentiable optimality systems in infinite dimensions [2]. Consequently, each regularized subsystem can be solved mesh-independently at a local superlinear rate of convergence [6]. Numerically the problems are solved using conforming finite elements. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

In this paper, the automated spatially dependent regularization parameter selection framework for multi-scale image restoration is applied to total generalized variation (TGV) of order 2. Well-posedness of the underlying continuous models is discussed and an algorithm for the numerical solution is developed. Experiments confirm that due to the spatially adapted regularization parameter, the method allows for a faithful and simultaneous recovery of fine structures and smooth regions in images. Moreover, because of the TGV regularization term, the adverse staircasing effect, which is a well-known drawback of the total variation regularization, is avoided. [ABSTRACT FROM PUBLISHER]

Abstract: The Cartesian parallel magnetic imaging problem is formulated variationally using a high-order penalty for coil sensitivities and a total variation like penalty for the reconstructed image. Then the optimality system is derived and numerically discretized. The objective function used is non-convex, but it possesses a bilinear structure that allows the ambiguity among solutions to be resolved technically by regularization and practically by normalizing a pre-estimated norm of the reconstructed image. Since the objective function is convex in each single argument, convex analysis is used to formulate the optimality condition for the image in terms of a primal–dual system. To solve the optimality system, a nonlinear Gauss–Seidel outer iteration is used in which the objective function is minimized with respect to one variable after the other using an inner generalized Newton iteration. Computational results for in vivo MR imaging data show that a significant improvement in reconstruction quality can be obtained by using the proposed regularization methods in relation to alternative approaches. [Copyright &y& Elsevier]

A two-phase image restoration method based upon total variation regularization combined with an L¹-data-fitting term for impulse noise removal and deblurring is proposed. In the first phase, suitable noise detectors are used for identifying image pixels contaminated by noise. Then, in the second phase, based upon the information on the location of noise-free pixels, images are deblurred and denoised simultaneously. For efficiency reasons, in the second phase a superlinearly convergent algorithm based upon Fenchel-duality and inexact semismooth Newton techniques is utilized for solving the associated variational problem. Numerical results prove the new method to be a significantly advance over several state-of-the-art techniques with respect to restoration capability and computational efficiency. [ABSTRACT FROM AUTHOR]

An interview with mathematicians Christof Schütte, Martin Skutella and Michael Hintermüller is presented. They discussed various topics including launch of the Forschungszentrum der Berliner Mathematik MATH+, challenges faced by mathematicians, mathematical research and technological innovations in mathematics sector.