Browsing by Author "Rautenberg, Carlos N."
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- ItemAdaptive regularization for image reconstruction from subsampled data(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Hintermüller, Michael; Langer, Andreas; Rautenberg, Carlos N.; Wu, TaoChoices of regularization parameters are central to variational methods for image restoration. In this paper, a spatially adaptive (or distributed) regularization scheme is developed based on localized residuals, which properly balances the regularization weight between regions containing image details and homogeneous regions. Surrogate iterative methods are employed to handle given subsampled data in transformed domains, such as Fourier or wavelet data. In this respect, this work extends the spatially variant regularization technique previously established in [15], which depends on the fact that the given data are degraded images only. Numerical experiments for the reconstruction from partial Fourier data and for wavelet inpainting prove the efficiency of the newly proposed approach.
- ItemAnalysis of a quasi-variational contact problem arising in thermoelasticity(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Alphonse, Amal; Rautenberg, Carlos N.; Rodrigues, José FranciscoWe formulate and study two mathematical models of a thermoforming process involving a membrane and a mould as implicit obstacle problems. In particular, the membrane-mould coupling is determined by the thermal displacement of the mould that depends in turn on the membrane through the contact region. The two models considered are a stationary (or elliptic) model and an evolutionary (or quasistatic) one. For the first model, we prove the existence of weak solutions by solving an elliptic quasi-variational inequality coupled to elliptic equations. By exploring the fine properties of the variation of the contact set under non-degenerate data, we give sufficient conditions for the existence of regular solutions, and under certain contraction conditions, also a uniqueness result. We apply these results to a series of semi-discretised problems that arise as approximations of regular solutions for the evolutionary or quasistatic problem. Here, under certain conditions, we are able to prove existence for the evolutionary problem and for a special case, also the uniqueness of time-dependent solutions.
- ItemAnalytical aspects of spatially adapted total variation regularisation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Hintermüller, Michael; Papafitsoros, Konstantinos; Rautenberg, Carlos N.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.
- ItemThe Boussinesq system with mixed non-smooth boundary conditions and do-nothing boundary flow(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Ceretani, Andrea N.; Rautenberg, Carlos N.A @stationary Boussinesq system for an incompressible viscous fluid in a bounded domain with a nontrivial condition at an open boundary is studied. We consider a novel non-smooth boundary condition associated to the heat transfer on the open boundary that involves the temperature at the boundary, the velocity of the fluid, and the outside temperature. We show that this condition is compatible with two approaches at dealing with the do-nothing boundary condition for the fluid: 1) the directional do-nothing condition and 2) the do-nothing condition together with an integral bound for the backflow. Well-posedness of variational formulations is proved for each problem.
- ItemDensity of convex intersections and applications(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Hintermüller, Michael; Rautenberg, Carlos N.; Rösel, SimonIn this paper we address density properties of intersections of convex sets in several function spaces. Using the concept of Gamma-convergence, it is shown in a general framework, how these density issues naturally arise from the regularization, discretization or dualization of constrained optimization problems and from perturbed variational inequalities. A variety of density results (and counterexamples) for pointwise constraints in Sobolev spaces are presented and the corresponding regularity requirements on the upper bound are identified. The results are further discussed in the context of finite element discretizations of sets associated to convex constraints. Finally, two applications are provided, which include elasto-plasticity and image restoration problems.
- ItemDirectional differentiability for elliptic quasi-variational inequalities of obstacle type(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Alphonse, Amal; Hintermüller, Michael; Rautenberg, Carlos N.The directional differentiability of the solution map of obstacle type quasi-variational inequalities (QVIs) with respect to perturbations on the forcing term is studied. The classical result of Mignot is then extended to the quasi-variational case under assumptions that allow multiple solutions of the QVI. The proof involves selection procedures for the solution set and represents the directional derivative as the limit of a monotonic sequence of directional derivatives associated to specific variational inequalities. Additionally, estimates on the coincidence set and several simplifications under higher regularity are studied. The theory is illustrated by a detailed study of an application to thermoforming comprising of modelling, analysis and some numerical experiments.
- ItemDissipative and non-dissipative evolutionary quasi-variational inequalities with gradient constraints(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2017) Hintermüller, Michael; Rautenberg, Carlos N.; Strogies, NikolaiEvolutionary quasi-variational inequality (QVI) problems of dissipative and non-dissipative nature with pointwise constraints on the gradient are studied. A semi-discretization in time is employed for the study of the problems and the derivation of a numerical solution scheme, respectively. Convergence of the discretization procedure is proven and properties of the original infinite dimensional problem, such as existence, extra regularity and non-decrease in time, are derived. The proposed numerical solver reduces to a finite number of gradient-constrained convex optimization problems which can be solved rather efficiently. The paper ends with a report on numerical tests obtained by a variable splitting algorithm involving different nonlinearities and types of constraints.
- ItemDualization and automatic distributed parameter selection of total generalized variation via bilevel optimization(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Hintermüller, Michael; Papafitsoros, Kostas; Rautenberg, Carlos N.; Sun, HongpengTotal 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, two Newton type methods for the solution of the lower level problem, i.e. the image reconstruction problem, and two bilevel TGV algorithms are introduced, respectively. Denoising tests confirm that automatically selected distributed regularization parameters lead in general to improved reconstructions when compared to results for scalar parameters.
- ItemExistence, iteration procedures and directional differentiability for parabolic QVIs(Berlin ; Heidelberg : Springer, 2020) Alphonse, Amal; Hintermüller, Michael; Rautenberg, Carlos N.We study parabolic quasi-variational inequalities (QVIs) of obstacle type. Under appropriate assumptions on the obstacle mapping, we prove the existence of solutions of such QVIs by two methods: one by time discretisation through elliptic QVIs and the second by iteration through parabolic variational inequalities. Using these results, we show the directional differentiability (in a certain sense) of the solution map which takes the source term of a parabolic QVI into the set of solutions, and we relate this result to the contingent derivative of the aforementioned map. We finish with an example where the obstacle mapping is given by the inverse of a parabolic differential operator.
- ItemExistence, iteration procedures and directional differentiability for parabolic QVIs(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Alphonse, Amal; Hintermüller, Michael; Rautenberg, Carlos N.We study parabolic quasi-variational inequalities (QVIs) of obstacle type. Under appropriate assumptions on the obstacle mapping, we prove the existence of solutions of such QVIs by two methods: one by time discretisation through elliptic QVIs and the second by iteration through parabolic variational inequalities (VIs). Using these results, we show the directional differentiability (in a certain sense) of the solution map which takes the source term of a parabolic QVI into the set of solutions, and we relate this result to the contingent derivative of the aforementioned map. We finish with an example where the obstacle mapping is given by the inverse of a parabolic differential operator.
- ItemExistence, uniqueness, and stabilization results for parabolic variational inequalities(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Kröner, Axel; Rautenberg, Carlos N.; Rodrigues, SérgioIn this paper we consider feedback stabilization for parabolic variational inequalities of obstacle type with time and space depending reaction and convection coefficients and show exponential stabilization to nonstationary trajectories. Based on a Moreau--Yosida approximation, a feedback operator is established using a finite (and uniform in the approximation index) number of actuators leading to exponential decay of given rate of the state variable. Several numerical examples are presented addressing smooth and nonsmooth obstacle functions.
- ItemOn the differentiability of the minimal and maximal solution maps of elliptic quasi-variational inequalities(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Alphonse, Amal; Hintermüller, Michael; Rautenberg, Carlos N.In this short note, we prove that the minimal and maximal solution maps associated to elliptic quasi-variational inequalities of obstacle type are directionally differentiable with respect to the forcing term and for directions that are signed. On the way, we show that the minimal and maximal solutions can be seen as monotone limits of solutions of certain variational inequalities and that the aforementioned directional derivatives can also be characterised as the monotone limits of sequences of directional derivatives associated to variational inequalities.
- ItemOn the uniqueness and numerical approximation of solutions to certain parabolic quasi-variational inequalities(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Hintermüller, Michael; Rautenberg, Carlos N.We provide a series of rigidity results for a nonlocal phase transition equation. The results that we obtain are an improvement of flatness theorem and a series of theorems concerning the one-dimensional symmetry for monotone and minimal solutions, in the research line dictaded by a classical conjecture of E. De Giorgi. Here, we collect a series of pivotal results, of geometric type, which are exploited in the proofs of the main results in a companion paper.
- ItemOptimal control and directional differentiability for elliptic quasi-variational inequalities(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Alphonse, Amal; Hintermüller, Michael; Rautenberg, Carlos N.We focus on elliptic quasi-variational inequalities (QVIs) of obstacle type and prove a number of results on the existence of solutions, directional differentiability and optimal control of such QVIs. We give three existence theorems based on an order approach, an iteration scheme and a sequential regularisation through partial differential equations. We show that the solution map taking the source term into the set of solutions of the QVI is directionally differentiable for general unsigned data, thereby extending the results of our previous work which provided a first differentiability result for QVIs in infinite dimensions. Optimal control problems with QVI constraints are also considered and we derive various forms of stationarity conditions for control problems, thus supplying among the first such results in this area.
- ItemOptimal selection of the regularization function in a generalized total variation model. Part I: Modelling and theory(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Hintermüller, Michael; Rautenberg, Carlos N.A generalized total variation model with a spatially varying regularization weight is considered. Existence of a solution is shown, and the associated Fenchel-predual problem is derived. For automatically selecting the regularization function, a bilevel optimization framework is proposed. In this context, the lower-level problem, which is parameterized by the regularization weight, is the Fenchel predual of the generalized total variation model and the upper-level objective penalizes violations of a variance corridor. The latter object relies on a localization of the image residual as well as on lower and upper bounds inspired by the statistics of the extremes.
- ItemOptimal selection of the regularization function in a generalized total variation model. Part II: Algorithm, its analysis and numerical tests(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Hintermüller, Michael; Rautenberg, Carlos N.; Wu, Tao; Langer, AndreasBased on the generalized total variation model and its analysis pursued in part I (WIAS Preprint no. 2235), in this paper a continuous, i.e., infinite dimensional, projected gradient algorithm and its convergence analysis are presented. The method computes a stationary point of a regularized bilevel optimization problem for simultaneously recovering the image as well as determining a spatially distributed regularization weight. Further, its numerical realization is discussed and results obtained for image denoising and deblurring as well as Fourier and wavelet inpainting are reported on.
- ItemOptimal sensor placement: A robust approach(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Hintermüller, Michael; Rautenberg, Carlos N.; Mohammadi, Masoumeh; Kanitsar, MartinWe 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. The 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.
- ItemSobolev spaces with non-Muckenhoupt weights, fractional elliptic operators, and applications(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Antil, Harbir; Rautenberg, Carlos N.We propose a new variational model in weighted Sobolev spaces with non-standard weights and applications to image processing. We show that these weights are, in general, not of Muckenhoupt type and therefore the classical analysis tools may not apply. For special cases of the weights, the resulting variational problem is known to be equivalent to the fractional Poisson problem. The trace space for the weighted Sobolev space is identified to be embedded in a weighted L2 space. We propose a finite element scheme to solve the Euler-Lagrange equations, and for the image denoising application we propose an algorithm to identify the unknown weights. The approach is illustrated on several test problems and it yields better results when compared to the existing total variation techniques.
- ItemStability of the solution set of quasi-variational inequalities and optimal control(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Alphonse, Amal; Hintermüller, Michael; Rautenberg, Carlos N.For a class of quasivariational inequalities (QVIs) of obstacle-type the stability of its solution set and associated optimal control problems are considered. These optimal control problems are non-standard 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 to the solution set of the QVI with respect to monotonic 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.
- ItemVariable Step Mollifiers and Applications(Berlin ; Heidelberg : Springer, 2020) Hintermüller, Michael; Papafitsoros, Kostas; Rautenberg, Carlos N.We consider a mollifying operator with variable step that, in contrast to the standard mollification, is able to preserve the boundary values of functions. We prove boundedness of the operator in all basic Lebesgue, Sobolev and BV spaces as well as corresponding approximation results. The results are then applied to extend recently developed theory concerning the density of convex intersections. © 2020, The Author(s).