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Author: Hintermüller, Michael × End date: 2019 × Start date: 2010 × Type: Text × WGL Institute: WIAS ×

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Now showing 1 - 10 of 29
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    A function space framework for structural total variation regularization with applications in inverse problems
    (Bristol [u.a.] : Inst., 2018) Hintermüller, Michael; Holler, Martin; Papafitsoros, Kostas
    In this work, we introduce a function space setting for a wide class of structural/weighted total variation (TV) regularization methods motivated by their applications in inverse problems. In particular, we consider a regularizer that is the appropriate lower semi-continuous envelope (relaxation) of a suitable TV type functional initially defined for sufficiently smooth functions. We study examples where this relaxation can be expressed explicitly, and we also provide refinements for weighted TV for a wide range of weights. Since an integral characterization of the relaxation in function space is, in general, not always available, we show that, for a rather general linear inverse problems setting, instead of the classical Tikhonov regularization problem, one can equivalently solve a saddle-point problem where no a priori knowledge of an explicit formulation of the structural TV functional is needed. In particular, motivated by concrete applications, we deduce corresponding results for linear inverse problems with norm and Poisson log-likelihood data discrepancy terms. Finally, we provide proof-of-concept numerical examples where we solve the saddle-point problem for weighted TV denoising as well as for MR guided PET image reconstruction.
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    A Bayesian approach to parameter identification in gas networks
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Hajian, Soheil; Hintermüller, Michael; Schillings, Claudia; Strogies, Nikolai
    The inverse problem of identifying the friction coefficient in an isothermal semilinear Euler system is considered. Adopting a Bayesian approach, the goal is to identify the distribution of the quantity of interest based on a finite number of noisy measurements of the pressure at the boundaries of the domain. First well-posedness of the underlying non-linear PDE system is shown using semigroup theory, and then Lipschitz continuity of the solution operator with respect to the friction coefficient is established. Based on the Lipschitz property, well-posedness of the resulting Bayesian inverse problem for the identification of the friction coefficient is inferred. Numerical tests for scalar and distributed parameters are performed to validate the theoretical results.
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    A class of second-order geometric quasilinear hyperbolic PDEs and their application in imaging science
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Dong, Guozhi; Hintermüller, Michael; Zhang, Ye
    In this paper, we study damped second-order dynamics, which are quasilinear hyperbolic partial differential equations (PDEs). This is inspired by the recent development of second-order damping systems for accelerating energy decay of gradient flows. We concentrate on two equations: one is a damped second-order total variation flow, which is primarily motivated by the application of image denoising; the other is a damped second-order mean curvature flow for level sets of scalar functions, which is related to a non-convex variational model capable of correcting displacement errors in image data (e.g. dejittering). For the former equation, we prove the existence and uniqueness of the solution. For the latter, we draw a connection between the equation and some second-order geometric PDEs evolving the hypersurfaces which are described by level sets of scalar functions, and show the existence and uniqueness of the solution for a regularized version of the equation. The latter is used in our algorithmic development. A general algorithm for numerical discretization of the two nonlinear PDEs is proposed and analyzed. Its efficiency is demonstrated by various numerical examples, where simulations on the behavior of solutions of the new equations and comparisons with first-order flows are also documented.
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    Analytical 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.
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    Simulation and control of a nonsmooth Cahn--Hilliard Navier--Stokes system with variable fluid densities
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Gräßle, Carmen; Hintermüller, Michael; Hinze, Michael; Keil, Tobias
    We are concerned with the simulation and control of a two phase flow model governed by a coupled Cahn--Hilliard Navier--Stokes system involving a nonsmooth energy potential.We establish the existence of optimal solutions and present two distinct approaches to derive suitable stationarity conditions for the bilevel problem, namely C- and strong stationarity. Moreover, we demonstrate the numerical realization of these concepts at the hands of two adaptive solution algorithms relying on a specifically developed goal-oriented error estimator.In addition, we present a model order reduction approach using proper orthogonal decomposition (POD-MOR) in order to replace high-fidelity models by low order surrogates. In particular, we combine POD with space-adapted snapshots and address the challenges which are the consideration of snapshots with different spatial resolutions and the conservation of a solenoidal property.
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    Stability 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.
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    A physically oriented method for quantitative magnetic resonance imaging
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Dong, Guozhi; Hintermüller, Michael; Papafitsoros, Kostas
    Quantitative magnetic resonance imaging (qMRI) denotes the task of estimating the values of magnetic and tissue parameters, e.g., relaxation times T1, T2, proton density p and others. Recently in [Ma et al., Nature, 2013], an approach named Magnetic Resonance Fingerprinting (MRF) was introduced, being capable of simultaneously recovering these parameters by using a two step procedure: (i) a series of magnetization maps are created and then (ii) these are matched to parameters with the help of a pre-computed dictionary (Bloch manifold). In this paper, we initially put MRF and its variants in the perspective of optimization and inverse problems, providing some mathematical insights into these methods. Motivated by the fact that the Bloch manifold is non-convex, and the accuracy of the MRF type algorithms is limited by the discretization size of the dictionary, we propose here a novel physically oriented method for qMRI. In contrast to the conventional two step models, our model is dictionary-free and it is described by a single nonlinear equation, governed by an operator for which we prove differentiability and other properties. This non-linear equation is efficiently solved via robust Newton type methods. The effectiveness of our method for noisy and undersampled data is shown both analytically and via numerical examples where also improvement over MRF and its variants is observed.
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    On 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.
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    Optimization of a multiphysics problem in semiconductor laser design
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Adam, Lukáš; Hintermüller, Michael; Peschka, Dirk; Surowiec, Thomas M.
    A @multimaterial topology optimization framework is suggested for the simultaneous optimization of mechanical and optical properties to be used in the development of optoelectronic devices. Based on the physical aspects of the underlying device, a nonlinear multiphysics model for the elastic and optical properties is proposed. Rigorous proofs are provided for the sensitivity of the fundamental mode of the device with respect to the changes in the underlying topology. 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 highly favorable electronic properties of this design are demonstrated by steady-state simulations of the corresponding van Roosbroeck (drift-diffusion) system.
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    Variable step mollifiers and applications
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) 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.