#### Prof. Dr. Christian Clason

Inverse Problems

University of Duisburg-Essen

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- christian[dot]clason[at]uni-due[dot]de
- +49 201 183 6382
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**Convex Relaxation of Discrete Vector-Valued Optimization Problems**

Clason, C. and Tameling, C. and Wirth, B.*SIAM Review*63 (2021)view abstract 10.1137/21M1426237 **Entropic regularization of continuous optimal transport problems**

Clason, C. and Lorenz, D.A. and Mahler, H. and Wirth, B.*Journal of Mathematical Analysis and Applications*494 (2021)We analyze continuous optimal transport problems in the so-called Kantorovich form, where we seek a transport plan between two marginals that are probability measures on compact subsets of Euclidean space. We consider the case of regularization with the negative entropy with respect to the Lebesgue measure, which has attracted attention because it can be solved by the very simple Sinkhorn algorithm. We first analyze the regularized problem in the context of classical Fenchel duality and derive a strong duality result for a predual problem in the space of continuous functions. However, this problem may not admit a minimizer, which prevents obtaining primal-dual optimality conditions. We then show that the primal problem is naturally analyzed in the Orlicz space of functions with finite entropy in the sense that the entropically regularized problem admits a minimizer if and only if the marginals have finite entropy. We then derive a dual problem in the corresponding dual space, for which existence can be shown by purely variational arguments and primal-dual optimality conditions can be derived. For marginals that do not have finite entropy, we finally show Gamma-convergence of the regularized problem with smoothed marginals to the original Kantorovich problem. © 2020 Elsevier Inc.view abstract 10.1016/j.jmaa.2020.124432 **No-gap second-order optimality conditions for optimal control of a non-smooth quasilinear elliptic equation**

Clason, C. and Nhu, V.H. and Rösch, A.*ESAIM - Control, Optimisation and Calculus of Variations*27 (2021)This paper deals with second-order optimality conditions for a quasilinear elliptic control problem with a nonlinear coefficient in the principal part that is finitely PC2 (continuous and C2 apart from finitely many points). We prove that the control-To-state operator is continuously differentiable even though the nonlinear coefficient is non-smooth. This enables us to establish "no-gap"second-order necessary and sufficient optimality conditions in terms of an abstract curvature functional, i.e., for which the sufficient condition only differs from the necessary one in the fact that the inequality is strict. A condition that is equivalent to the second-order sufficient optimality condition and could be useful for error estimates in, e.g., finite element discretizations is also provided. © EDP Sciences, SMAI 2021.view abstract 10.1051/cocv/2020092 **Optimal control of a non-smooth quasilinear elliptic equation**

Clason, C. and Nhu, V.H. and Rösch, A.*Mathematical Control and Related Fields*11 (2021)This work is concerned with an optimal control problem governed by a non-smooth quasilinear elliptic equation with a nonlinear coefficient in the principal part that is locally Lipschitz continuous and directionally but not Gâteaux differentiable. This leads to a control-to-state operator that is directionally but not Gâteaux differentiable as well. Based on a suitable regu-larization scheme, we derive C-and strong stationarity conditions. Under the additional assumption that the nonlinearity is a P C1 function with countably many points of nondifferentiability, we show that both conditions are equiva-lent. Furthermore, under this assumption we derive a relaxed optimality system that is amenable to numerical solution using a semi-smooth Newton method. This is illustrated by numerical examples. © 2021, American Institute of Mathematical Sciences. All rights reserved.view abstract 10.3934/MCRF.2020052 **Optimal control and inverse problems**

Clason, C. and Kaltenbacher, B.*Inverse Problems*36 (2020)view abstract 10.1088/1361-6420/ab8485 **Optimal Control of the Principal Coefficient in a Scalar Wave Equation**

Clason, C. and Kunisch, K. and Trautmann, P.*Applied Mathematics and Optimization*(2020)We consider optimal control of the scalar wave equation where the control enters as a coefficient in the principal part. Adding a total variation penalty allows showing existence of optimal controls, which requires continuity results for the coefficient-to-solution mapping for discontinuous coefficients. We additionally consider a so-called multi-bang penalty that promotes controls taking on values pointwise almost everywhere from a specified discrete set. Under additional assumptions on the data, we derive an improved regularity result for the state, leading to optimality conditions that can be interpreted in an appropriate pointwise fashion. The numerical solution makes use of a stabilized finite element method and a nonlinear primal–dual proximal splitting algorithm. © 2020, The Author(s).view abstract 10.1007/s00245-020-09733-9 **Optimal control problems with control complementarity constraints: existence results, optimality conditions, and a penalty method**

Clason, C. and Deng, Y. and Mehlitz, P. and Prüfert, U.*Optimization Methods and Software*35 (2020)A special class of optimal control problems with complementarity constraints on the control functions is studied. It is shown that such problems possess optimal solutions whenever the underlying control space is a first-order Sobolev space. After deriving necessary optimality conditions of strong stationarity-type, a penalty method based on the Fischer–Burmeister function is suggested and its theoretical properties are analyzed. Finally, the numerical treatment of the problem is discussed and results of computational experiments are presented. © 2019, © 2019 Informa UK Limited, trading as Taylor & Francis Group.view abstract 10.1080/10556788.2019.1604705 **Primal–Dual Proximal Splitting and Generalized Conjugation in Non-smooth Non-convex Optimization**

Clason, C. and Mazurenko, S. and Valkonen, T.*Applied Mathematics and Optimization*(2020)We demonstrate that difficult non-convex non-smooth optimization problems, such as Nash equilibrium problems and anisotropic as well as isotropic Potts segmentation models, can be written in terms of generalized conjugates of convex functionals. These, in turn, can be formulated as saddle-point problems involving convex non-smooth functionals and a general smooth but non-bilinear coupling term. We then show through detailed convergence analysis that a conceptually straightforward extension of the primal–dual proximal splitting method of Chambolle and Pock is applicable to the solution of such problems. Under sufficient local strong convexity assumptions on the functionals—but still with a non-bilinear coupling term—we even demonstrate local linear convergence of the method. We illustrate these theoretical results numerically on the aforementioned example problems. © 2020, Springer Science+Business Media, LLC, part of Springer Nature.view abstract 10.1007/s00245-020-09676-1 **Acceleration and global convergence of a first-order primal-dual method for nonconvex problems**

Clason, C. and Mazurenko, S. and Valkonen, T.*SIAM Journal on Optimization*29 (2019)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. © 2019 Society for Industrial and Applied Mathematics.view abstract 10.1137/18M1170194 **Bouligand-levenberg-marquardt iteration for a non-smooth ill-posed inverse problem**

Clason, C. and Nhu, V.H.*Electronic Transactions on Numerical Analysis*51 (2019)In this paper, we consider a modified Levenberg-Marquardt method for solving an ill-posed inverse problem where the forward mapping is not Gâteaux differentiable. By relaxing the standard assumptions for the classical smooth setting, we derive asymptotic stability estimates which are then used to prove convergence of the proposed method. This method can be applied to an inverse source problem for a non-smooth semilinear elliptic PDE where a Bouligand subdifferential can be used in place of the non-existing Fréchet derivative, and we show that the corresponding Bouligand-Levenberg-Marquardt iteration is an iterative regularization scheme. Numerical examples illustrate the advantage over the corresponding Bouligand-Landweber iteration. Copyright © 2019, Kent State University.view abstract 10.1553/etna_vol51s274 **Bouligand–Landweber iteration for a non-smooth ill-posed problem**

Clason, C. and Nhu, V.H.*Numerische Mathematik*142 (2019)This work is concerned with the iterative regularization of a non-smooth nonlinear ill-posed problem where the forward mapping is merely directionally but not Gâteaux differentiable. Using a Bouligand subderivative of the forward mapping, a modified Landweber method can be applied; however, the standard analysis is not applicable since the Bouligand subderivative mapping is not continuous unless the forward mapping is Gâteaux differentiable. We therefore provide a novel convergence analysis of the modified Landweber method that is based on the concept of asymptotic stability and merely requires a generalized tangential cone condition. These conditions are verified for an inverse source problem for an elliptic PDE with a non-smooth Lipschitz continuous nonlinearity, showing that the corresponding Bouligand–Landweber iteration converges strongly for exact data as well as in the limit of vanishing data if the iteration is stopped according to the discrepancy principle. This is illustrated with a numerical example. © 2019, Springer-Verlag GmbH Germany, part of Springer Nature.view abstract 10.1007/s00211-019-01038-6 **Contingent derivatives and regularization for noncoercive inverse problems**

Clason, C. and Khan, A.A. and Sama, M. and Tammer, C.*Optimization*68 (2019)We study the inverse problem of parameter identification in noncoercive variational problems that commonly appear in applied models. We examine the differentiability of the set-valued parameter-to-solution map using the first-order and the second-order contingent derivatives. We explore the inverse problem using the output least-squares and the modified output least-squares objectives. By regularizing the noncoercive variational problem, we obtain a single-valued regularized parameter-to-solution map and investigate its smoothness and boundedness. We also consider optimization problems using the output least-squares and the modified output least-squares objectives for the regularized variational problem. We give a complete convergence analysis showing that for the output least-squares and the modified output least-squares, the regularized minimization problems approximate the original optimization problems suitably. We also provide the first-order and the second-order adjoint method for the computation of the first-order and the second-order derivatives of the output least-squares objective. We provide discrete formulas for the gradient and the Hessian calculation and present numerical results. © 2018, © 2018 Informa UK Limited, trading as Taylor & Francis Group.view abstract 10.1080/02331934.2018.1442448 **Generalized modes in Bayesian inverse problems**

Clason, C. and Helin, T. and Kretschmann, R. and Piiroinen, P.*SIAM-ASA Journal on Uncertainty Quantification*7 (2019)Uncertainty quantification requires efficient summarization of high- or even infinite-dimensional (i.e., nonparametric) distributions based on, e.g., suitable point estimates (modes) for posterior distributions arising from model-specific prior distributions. In this work, we consider nonparametric modes and maximum a posteriori (MAP) estimates for priors that do not admit continuous densities, for which previous approaches based on small ball probabilities fail. We propose a novel definition of generalized modes based on the concept of approximating sequences, which reduce to the classical mode in certain situations that include Gaussian priors but also exist for a more general class of priors. The latter includes the case of priors that impose strict bounds on the admissible parameters and in particular of uniform priors. For uniform priors defined by random series with uniformly distributed coefficients, we show that generalized MAP estimates-but not classical MAP estimates-can be characterized as minimizers of a suitable functional that plays the role of a generalized Onsager- Machlup functional. This is then used to show consistency of nonlinear Bayesian inverse problems with uniform priors and Gaussian noise. Copyright © by SIAM and ASA.view abstract 10.1137/18M1191804 **Convex Regularization of Discrete-Valued Inverse Problems**

Clason, C. and Do, T.B.T.*Trends in Mathematics*(2018)This work is concerned with linear inverse problems where a distributed parameter is known a priori to only take on values from a given discrete set. This property can be promoted in Tikhonov regularization with the aid of a suitable convex but nondifferentiable regularization term. This allows applying standard approaches to show well-posedness and convergence rates in Bregman distance. Using the specific properties of the regularization term, it can be shown that convergence (albeit without rates) actually holds pointwise. Furthermore, the resulting Tikhonov functional can be minimized efficiently using a semi-smooth Newton method. Numerical examples illustrate the properties of the regularization term and the numerical solution. © 2018, Springer International Publishing AG.view abstract 10.1007/978-3-319-70824-9_2 **Error estimates for the approximation of multibang control problems**

Clason, C. and Do, T.B.T. and Pörner, F.*Computational Optimization and Applications*71 (2018)This work is concerned with optimal control problems where the objective functional consists of a tracking-type functional and an additional “multibang” regularization functional that promotes optimal control taking values from a given discrete set pointwise almost everywhere. Under a regularity condition on the set where these discrete values are attained, error estimates for the Moreau–Yosida approximation (which allows its solution by a semismooth Newton method) and the discretization of the problem are derived. Numerical results support the theoretical findings. © 2018, Springer Science+Business Media, LLC, part of Springer Nature.view abstract 10.1007/s10589-018-0026-y **Optimal control of a non-smooth semilinear elliptic equation**

Christof, C. and Meyer, C. and Walther, S. and Clason, C.*Mathematical Control and Related Fields*8 (2018)This paper is concerned with an optimal control problem governed by a non-smooth semilinear elliptic equation. We show that the control-to-state mapping is directionally differentiable and precisely characterize its Bouligand subdifferential. By means of a suitable regularization, first-order optimality conditions including an adjoint equation are derived and afterwards interpreted in light of the previously obtained characterization. In addition, the directional derivative of the control-to-state mapping is used to establish strong stationarity conditions. While the latter conditions are shown to be stronger, we demonstrate by numerical examples that the former conditions are amenable to numerical solution using a semi-smooth Newton method. © 2018, American Institute of Mathematical Sciences. All rights reserved.view abstract 10.3934/mcrf.2018011 **Total variation regularization of multi-material topology optimization**

Clason, C. and Kruse, F. and Kunisch, K.*ESAIM: Mathematical Modelling and Numerical Analysis*52 (2018)This work is concerned with the determination of the diffusion coefficient from distributed data of the state. This problem is related to homogenization theory on the one hand and to regularization theory on the other hand. An approach is proposed which involves total variation regularization combined with a suitably chosen cost functional that promotes the diffusion coefficient assuming prespecified values at each point of the domain. The main difficulty lies in the delicate functional-Analytic structure of the resulting nondifferentiable optimization problem with pointwise constraints for functions of bounded variation, which makes the derivation of useful pointwise optimality conditions challenging. To cope with this difficulty, a novel reparametrization technique is introduced. Numerical examples using a regularized semismooth Newton method illustrate the structure of the obtained diffusion coefficient. © EDP Sciences, SMAI 2018.view abstract 10.1051/m2an/2017061 **Vector-valued multibang control of differential equations**

Clason, C. and Tameling, C. and Wirth, B.*SIAM Journal on Control and Optimization*56 (2018)We consider a class of (ill-posed) optimal control problems in which a distributed vector-valued control is enforced to take values pointwise in a finite set M ? Rm. After convex relaxation, one obtains a well-posed optimization problem, which still promotes control values in M. We state the corresponding well-posedness and stability analysis and exemplify the results for two specific cases of quite general interest, optimal control of the Bloch equation and optimal control of an elastic deformation. We finally formulate a semismooth Newton method to numerically solve a regularized version of the optimal control problem and illustrate the behavior of the approach for our example cases. © 2018 Society for Industrial and Applied Mathematics.view abstract 10.1137/16M1104998 **L1 penalization of volumetric dose objectives in optimal control of PDEs**

Barnard, R.C. and Clason, C.*Computational Optimization and Applications*67 (2017)This work is concerned with a class of PDE-constrained optimization problems that are motivated by an application in radiotherapy treatment planning. Here the primary design objective is to minimize the volume where a functional of the state violates a prescribed level, but prescribing these levels in the form of pointwise state constraints leads to infeasible problems. We therefore propose an alternative approach based on (Formula presented.) penalization of the violation that is also applicable when state constraints are infeasible. We establish well-posedness of the corresponding optimal control problem, derive first-order optimality conditions, discuss convergence of minimizers as the penalty parameter tends to infinity, and present a semismooth Newton method for their efficient numerical solution. The performance of this method for a model problem is illustrated and contrasted with an alternative approach based on (regularized) state constraints. © 2017 Springer Science+Business Media New York (outside the USA)view abstract 10.1007/s10589-017-9897-6 **Nonconvex penalization of switching control of partial differential equations**

Clason, C. and Rund, A. and Kunisch, K.*Systems and Control Letters*106 (2017)This paper is concerned with optimal control problems for parabolic partial differential equations with pointwise in time switching constraints on the control. A standard approach to treat constraints in nonlinear optimization is penalization, in particular using L1-type norms. Applying this approach to the switching constraint leads to a nonsmooth and nonconvex infinite-dimensional minimization problem which is challenging both analytically and numerically. Adding H1 regularization or restricting to a finite-dimensional control space allows showing existence of optimal controls. First-order necessary optimality conditions are then derived using tools of nonsmooth analysis. Their solution can be computed using a combination of Moreau–Yosida regularization and a semismooth Newton method. Numerical examples illustrate the properties of this approach. © 2017 Elsevier B.V.view abstract 10.1016/j.sysconle.2017.05.006 **Optimal control of elliptic equations with positive measures**

Clason, C. and Schiela, A.*ESAIM - Control, Optimisation and Calculus of Variations*23 (2017)Optimal control problems without control costs in general do not possess solutions due to the lack of coercivity. However, unilateral constraints together with the assumption of existence of strictly positive solutions of a pre-Adjoint state equation, are sufficient to obtain existence of optimal solutions in the space of Radon measures. Optimality conditions for these generalized minimizers can be obtained using Fenchel duality, which requires a non-standard perturbation approach if the controlto- observation mapping is not continuous (e.g., for Neumann boundary control in three dimensions). Combining a conforming discretization of the measure space with a semismooth Newton method allows the numerical solution of the optimal control problem. © EDP Sciences, SMAI 2016.view abstract 10.1051/cocv/2015046 **Primal-dual extragradient methods for nonlinear nonsmooth PDE-constrained optimization**

Clason, C. and Valkonen, T.*SIAM Journal on Optimization*27 (2017)We study the extension of the Chambolle-Pock primal-dual algorithm to nonsmooth optimization problems involving nonlinear operators between function spaces. Local convergence is shown under technical conditions including metric regularity of the corresponding primal-dual optimality conditions. We also show convergence for a Nesterov-type accelerated variant, provided one part of the functional is strongly convex. We show the applicability of the accelerated algorithm to examples of inverse problems with L1 and L∞ fitting terms as well as of state-constrained optimal control problems, where convergence can be guaranteed after introducing an (arbitrarily small, still nonsmooth) Moreau-Yosida regularization. This is verified in numerical examples. © 2017 SIAMview abstract 10.1137/16M1080859 **Stability of Saddle Points Via Explicit Coderivatives of Pointwise Subdifferentials**

Clason, C. and Valkonen, T.*Set-Valued and Variational Analysis*25 (2017)We derive stability criteria for saddle points of a class of nonsmooth optimization problems in Hilbert spaces arising in PDE-constrained optimization, using metric regularity of infinite-dimensional set-valued mappings. A main ingredient is an explicit pointwise characterization of the regular coderivative of the subdifferential of convex integral functionals. This is applied to several stability properties for parameter identification problems for an elliptic partial differential equation with non-differentiable data fitting terms. © 2016, The Author(s).view abstract 10.1007/s11228-016-0366-7 **A convex analysis approach to multi-material topology optimization**

Clason, C. and Kunisch, K.*ESAIM: Mathematical Modelling and Numerical Analysis*50 (2016)This work is concerned with optimal control of partial differential equations where the control enters the state equation as a coefficient and should take on values only from a given discrete set of values corresponding to available materials. A "multi-bang" framework based on convex analysis is proposed where the desired piecewise constant structure is incorporated using a convex penalty term. Together with a suitable tracking term, this allows formulating the problem of optimizing the topology of the distribution of material parameters as minimizing a convex functional subject to a (nonlinear) equality constraint. The applicability of this approach is validated for two model problems where the control enters as a potential and a diffusion coefficient, respectively. This is illustrated in both cases by numerical results based on a semi-smooth Newton method. © 2016 EDP Sciences, SMAI.view abstract 10.1051/m2an/2016012 **A convex analysis approach to optimal controls with switching structure for partial differential equations**

Clason, C. and Ito, K. and Kunisch, K.*ESAIM - Control, Optimisation and Calculus of Variations*22 (2016)Optimal control problems involving hybrid binary-continuous control costs are challenging due to their lack of convexity and weak lower semicontinuity. Replacing such costs with their convex relaxation leads to a primal-dual optimality system that allows an explicit pointwise characterization and whose Moreau-Yosida regularization is amenable to a semismooth Newton method in function space. This approach is especially suited for computing switching controls for partial differential equations. In this case, the optimality gap between the original functional and its relaxation can be estimated and shown to be zero for controls with switching structure. Numerical examples illustrate the effectiveness of this approach. © EDP Sciences, SMAI 2016.view abstract 10.1051/cocv/2015017 **A convex penalty for switching control of partial differential equations**

Clason, C. and Rund, A. and Kunisch, K. and Barnard, R.C.*Systems and Control Letters*89 (2016)A convex penalty for promoting switching controls for partial differential equations is introduced; such controls consist of an arbitrary number of components of which at most one should be simultaneously active. Using a Moreau-Yosida approximation, a family of approximating problems is obtained that is amenable to solution by a semismooth Newton method. The efficiency of this approach and the structure of the obtained controls are demonstrated by numerical examples. © 2015 Elsevier B.V. All rights reserved.view abstract 10.1016/j.sysconle.2015.12.013 **Efficient high-resolution RF pulse design applied to simultaneous multi-slice excitation**

Aigner, C.S. and Clason, C. and Rund, A. and Stollberger, R.*Journal of Magnetic Resonance*263 (2016)RF pulse design via optimal control is typically based on gradient and quasi-Newton approaches and therefore suffers from slow convergence. We present a flexible and highly efficient method that uses exact second-order information within a globally convergent trust-region CG-Newton method to yield an improved convergence rate. The approach is applied to the design of RF pulses for single- and simultaneous multi-slice (SMS) excitation and validated using phantom and in vivo experiments on a 3 T scanner using a modified gradient echo sequence. © 2015 Elsevier Inc. All rights reserved.view abstract 10.1016/j.jmr.2015.11.013 **Functional error estimators for the adaptive discretization of inverse problems**

Clason, C. and Kaltenbacher, B. and Wachsmuth, D.*Inverse Problems*32 (2016)So-called functional error estimators provide a valuable tool for reliably estimating the discretization error for a sum of two convex functions. We apply this concept to Tikhonov regularization for the solution of inverse problems for partial differential equations, not only for quadratic Hilbert space regularization terms but also for nonsmooth Banach space penalties. Examples include the measure-space norm (i.e., sparsity regularization) or the indicator function of an L∞ ball (i.e., Ivanov regularization). The error estimators can be written in terms of residuals in the optimality system that can then be estimated by conventional techniques, thus leading to explicit estimators. This is illustrated by means of an elliptic inverse source problem with the above-mentioned penalties, and numerical results are provided for the case of sparsity regularization. © 2016 IOP Publishing Ltd.view abstract 10.1088/0266-5611/32/10/104004 **Multi-bang control of elliptic systems**

Clason, C. and Kunisch, K.*Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire*31 (2014)Multi-bang control refers to optimal control problems for partial differential equations where a distributed control should only take on values from a discrete set of allowed states. This property can be promoted by a combination of L2 and L0-type control costs. Although the resulting functional is nonconvex and lacks weak lower-semicontinuity, application of Fenchel duality yields a formal primal-dual optimality system that admits a unique solution. This solution is in general only suboptimal, but the optimality gap can be characterized and shown to be zero under appropriate conditions. Furthermore, in certain situations it is possible to derive a generalized multi-bang principle, i.e., to prove that the control almost everywhere takes on allowed values except on sets where the corresponding state reaches the target. A regularized semismooth Newton method allows the numerical computation of (sub)optimal controls. Numerical examples illustrate the effectiveness of the proposed approach as well as the structural properties of multi-bang controls. © 2013 Elsevier Masson SAS. All rights reserved.view abstract 10.1016/j.anihpc.2013.08.005 **Optimal control of a singular PDE modeling transient MEMS with control or state constraints**

Clason, C. and Kaltenbacher, B.*Journal of Mathematical Analysis and Applications*410 (2014)A particular feature of certain microelectromechanical systems (MEMS) is the appearance of a so-called "pull-in" instability, corresponding to a singularity in the underlying PDE model. We here consider a transient MEMS model and its optimal control via the dielectric properties of the membrane and/or the applied voltage. In contrast to the static case, the control problem suffers from low dimensionality of the control compared to the state and hence requires different techniques for establishing first order optimality conditions. For this purpose, we here use a relaxation approach combined with a localization technique. © 2013 Elsevier Inc.view abstract 10.1016/j.jmaa.2013.08.058 **Avoiding degeneracy in the westervelt equation by state constrained optimal control**

Clason, C. and Kaltenbacher, B.*Evolution Equations and Control Theory*2 (2013)The Westervelt equation, which describes nonlinear acoustic wave propagation in high intensity ultrasound applications, exhibits potential degeneracy for large acoustic pressure values. While well-posedness results on this PDE have so far been based on smallness of the solution in a higher order spatial norm, non-degeneracy can be enforced explicitly by a pointwise state constraint in a minimization problem, thus allowing for pressures with large gradients and higher-order derivatives, as is required in the mentioned applications. Using regularity results on the linearized state equation, well-posedness and necessary optimality conditions for the PDE constrained optimization problem can be shown via a relaxation approach by Alibert and Raymond [2]. © 2013, American Institute of Mathematical Sciences. All rights reserved.view abstract 10.3934/eect.2013.2.281 **On the use of state constraints in optimal control of singular PDEs**

Clason, C. and Kaltenbacher, B.*Systems and Control Letters*62 (2013)We consider optimal control of nonlinear partial differential equations involving potentially singular solution-dependent terms. Singularity can be prevented by either restricting controls to a closed admissible set for which well-posedness of the equation can be guaranteed, or by explicitly enforcing pointwise bounds on the state. By means of an elliptic model problem, we contrast the requirements for deriving the existence of solutions and first order optimality conditions for both the control-constrained and the state-constrained formulation. Our analysis as well as numerical tests illustrate that control constraints lead to severe restrictions on the attainable states, which is not the case for state constraints. © 2012 Elsevier B.V. All rights reserved.view abstract 10.1016/j.sysconle.2012.10.006 **Parabolic control problems in measure spaces with sparse solutions**

Casas, E. and Clason, C. and Kunisch, K.*SIAM Journal on Control and Optimization*51 (2013)Optimal control problems in measure spaces lead to controls that have small support, which is desirable, e.g., in the context of optimal actuator placement. For problems governed by parabolic partial differential equations, well-posedness is guaranteed in the space of square-integrable measure-valued functions, which leads to controls with a spatial sparsity structure. A conforming approximation framework allows one to derive numerically accessible optimality conditions as well as convergence rates. In particular, although the state is discretized, the control problem can still be formulated and solved in the measure space. Numerical examples illustrate the structural features of the optimal controls. © 2013 Society for Industrial and Applied Mathematics.view abstract 10.1137/120872395 **A deterministic approach to the adapted optode placement for illumination of highly scattering tissue**

Brunner, P. and Clason, C. and Freiberger, M. and Scharfetter, H.*Biomedical Optics Express*3 (2012)A novel approach is presented for computing optode placements that are adapted to specific geometries and tissue characteristics, e.g., in optical tomography and photodynamic cancer therapy. The method is based on optimal control techniques together with a sparsity- promoting penalty that favors pointwise solutions, yielding both locations and magnitudes of light sources. In contrast to current discrete approaches, the need for specifying an initial set of candidate configurations as well as the exponential increase in complexity with the number of optodes are avoided. This is demonstrated with computational examples from photodynamic therapy. © 2012 Optical Society of America.view abstract 10.1364/BOE.3.001732 **A general spectral method for the numerical simulation of one-dimensional interacting fermions**

Clason, C. and von Winckel, G.*Computer Physics Communications*183 (2012)This work introduces a general framework for the direct numerical simulation of systems of interacting fermions in one spatial dimension. The approach is based on a specially adapted nodal spectral Galerkin method, where the basis functions are constructed to obey the antisymmetry relations of fermionic wave functions. An efficient MATLAB program for the assembly of the stiffness and potential matrices is presented, which exploits the combinatorial structure of the sparsity pattern arising from this discretization to achieve optimal run-time complexity. This program allows the accurate cliscretization of systems with multiple fermions subject to arbitrary potentials, e.g., for verifying the accuracy of multi-particle approximations such as Hartree-Fock in the few-particle limit. It can be used for eigenvalue computations or numerical solutions of the time-dependent Schrodinger equation. Program summary Program title: assembleFermiMatrix Catalogue identifier: AEKO_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEKO_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 102 No. of bytes in distributed program, including test data, etc.: 2294 Distribution format: tar.gz Programming language: MATLAB Computer: Any architecture supported by MATLAB Operating system: Any supported by MATLAB; tested under Linux (x86-64) and Mac OS X (10.6) RAM: Depends on the data Classification: 4.3, 2.2 Nature of problem: The direct numerical solution of the multi-particle one-dimensional Schrodinger equation in a quantum well is challenging due to the exponential growth in the number of degrees of freedom with increasing particles. Solution method: A nodal spectral Galerkin scheme is used where the basis functions are constructed to obey the antisymmetry relations of the fermionic wave function. The assembly of these matrices is performed efficiently by exploiting the combinatorial structure of the sparsity patterns. Restrictions: Only one-dimensional computational domains with homogeneous Dirichlet or periodic boundary conditions are supported. Running time: Seconds to minutes (C) 2011 Elsevier B.V. All rights reserved.view abstract 10.1016/j.cpc.2011.10.005 **A measure space approach to optimal source placement**

Clason, C. and Kunisch, K.*Computational Optimization and Applications*53 (2012)The problem of optimal placement of point sources is formulated as a distributed optimal control problem with sparsity constraints. For practical relevance, partial observations as well as partial and non-negative controls need to be considered. Although well-posedness of this problem requires a non-reflexive Banach space setting, a primal-predual formulation of the optimality system can be approximated well by a family of semi-smooth equations, which can be solved by a superlinearly convergent semi-smooth Newton method. Numerical examples indicate the feasibility for optimal light source placement problems in diffusive photochemotherapy. © 2011 Springer Science+Business Media, LLC.view abstract 10.1007/s10589-011-9444-9 **A minimum effort optimal control problem for elliptic PDEs**

Clason, C. and Ito, K. and Kunisch, K.*ESAIM: Mathematical Modelling and Numerical Analysis*46 (2012)This work is concerned with a class of minimum effort problems for partial differential equations, where the control cost is of L ∞-type. Since this problem is non-differentiable, a regularized functional is introduced that can be minimized by a superlinearly convergent semi-smooth Newton method. Uniqueness and convergence for the solutions to the regularized problem are addressed, and a continuation strategy based on a model function is proposed. Numerical examples for a convection-diffusion equation illustrate the behavior of minimum effort controls. © EDP Sciences, SMAI, 2012.view abstract 10.1051/m2an/2011074 **A Semismooth Newton Method for Nonlinear Parameter Identification Problems with Impulsive Noise**

Clason, C. and Jin, B.*SIAM Journal on Imaging Sciences*5 (2012)This work is concerned with nonlinear parameter identification in partial differential equations subject to impulsive noise. To cope with the non-Gaussian nature of the noise, we consider a model with L 1 fitting. However, the nonsmoothness of the problem makes its efficient numerical solution challenging. By approximating this problem using a family of smoothed functionals, a semismooth Newton method becomes applicable. In particular, its superlinear convergence is proved under a second-order condition. The convergence of the solution to the approximating problem as the smoothing parameter goes to zero is shown. A strategy for adaptively selecting the regularization parameter based on a balancing principle is suggested. The efficiency of the method is illustrated on several benchmark inverse problems of recovering coefficients in elliptic differential equations, for which one- and two-dimensional numerical examples are presented. © by SIAM.view abstract 10.1137/110826187 **An image space approach to Cartesian based parallel MR imaging with total variation regularization**

Keeling, S.L. and Clason, C. and Hintermüller, M. and Knoll, F. and Laurain, A. and von Winckel, G.*Medical Image Analysis*16 (2012)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. © 2011 Elsevier B.V.view abstract 10.1016/j.media.2011.07.002 **Approximation of elliptic control problems in measure spaces with sparse solutions**

Casas, E. and Clason, C. and Kunisch, K.*SIAM Journal on Control and Optimization*50 (2012)Optimal control problems in measure spaces governed by elliptic equations are considered for distributed and Neumann boundary control, which are known to promote sparse solutions. Optimality conditions are derived and some of the structural properties of their solutions, in particular sparsity, are discussed. A framework for their approximation is proposed which is efficient for numerical computations and for which we prove convergence and provide error estimates. © 2012 Society for Industrial and Applied Mathematics.view abstract 10.1137/110843216 **L ∞ fitting for inverse problems with uniform noise**

Clason, C.*Inverse Problems*28 (2012)For inverse problems where the data are corrupted by uniform noise such as arising from quantization errors, the L norm is a more robust data-fitting term than the standard L 2 norm. Well-posedness and regularization properties for linear inverse problems with L data fitting are shown, and the automatic choice of the regularization parameter is discussed. After introducing an equivalent reformulation of the problem and a Moreau-Yosida approximation, a superlinearly convergent semi-smooth Newton method becomes applicable for the numerical solution of L fitting problems. Numerical examples illustrate the performance of the proposed approach as well as the qualitative behavior of L fitting. © 2012 IOP Publishing Ltd.view abstract 10.1088/0266-5611/28/10/104007 **Parallel imaging with nonlinear reconstruction using variational penalties**

Knoll, F. and Clason, C. and Bredies, K. and Uecker, M. and Stollberger, R.*Magnetic Resonance in Medicine*67 (2012)A new approach based on nonlinear inversion for autocalibrated parallel imaging with arbitrary sampling patterns is presented. By extending the iteratively regularized Gauss-Newton method with variational penalties, the improved reconstruction quality obtained from joint estimation of image and coil sensitivities is combined with the superior noise suppression of total variation and total generalized variation regularization. In addition, the proposed approach can lead to enhanced removal of sampling artifacts arising from pseudorandom and radial sampling patterns. This is demonstrated for phantom and in vivo measurements. Copyright © 2011 Wiley Periodicals, Inc.view abstract 10.1002/mrm.22964 **A duality-based approach to elliptic control problems in non-reflexive Banach spaces ***

Clason, C. and Kunisch, K.*ESAIM - Control, Optimisation and Calculus of Variations*17 (2011)Convex duality is a powerful framework for solving non-smooth optimal control problems. However, for problems set in non-reflexive Banach spaces such as L1(Ω) or BV(Ω), the dual problem is formulated in a space which has difficult measure theoretic structure. The predual problem, on the other hand, can be formulated in a Hilbert space and entails the minimization of a smooth functional with box constraints, for which efficient numerical methods exist. In this work, elliptic control problems with measures and functions of bounded variation as controls are considered. Existence and uniqueness of the corresponding predual problems are discussed, as is the solution of the optimality systems by a semismooth Newton method. Numerical examples illustrate the structural differences in the optimal controls in these Banach spaces, compared to those obtained in corresponding Hilbert space settings. © 2009 EDP Sciences, SMAI.view abstract 10.1051/cocv/2010003 **Adapted random sampling patterns for accelerated MRI**

Knoll, F. and Clason, C. and Diwoky, C. and Stollberger, R.*Magnetic Resonance Materials in Physics, Biology and Medicine*24 (2011)Objective: Variable density random sampling patterns have recently become increasingly popular for accelerated imaging strategies, as they lead to incoherent aliasing artifacts. However, the design of these sampling patterns is still an open problem. Current strategies use model assumptions like polynomials of different order to generate a probability density function that is then used to generate the sampling pattern. This approach relies on the optimization of design parameters which is very time consuming and therefore impractical for daily clinical use. Materials and methods: This work presents a new approach that generates sampling patterns by making use of power spectra of existing reference data sets and hence requires neither parameter tuning nor an a priori mathematical model of the density of sampling points. Results: The approach is validated with downsampling experiments, as well as with accelerated in vivo measurements. The proposed approach is compared with established sampling patterns, and the generalization potential is tested by using a range of reference images. Quantitative evaluation is performed for the downsampling experiments using RMS differences to the original, fully sampled data set. Conclusion: Our results demonstrate that the image quality of the method presented in this paper is comparable to that of an established model-based strategy when optimization of the model parameter is carried out and yields superior results to non-optimized model parameters. However, no random sampling pattern showed superior performance when compared to conventional Cartesian subsampling for the considered reconstruction strategy. © 2010 ESMRMB.view abstract 10.1007/s10334-010-0234-7 **Minimal invasion: An optimal L∞ state constraint problem**

Clason, C. and Ito, K. and Kunisch, K.*ESAIM: Mathematical Modelling and Numerical Analysis*45 (2011)In this work, the least pointwise upper and/or lower bounds on the state variable on a specified subdomain of a control system under piecewise constant control action are sought. This results in a non-smooth optimization problem in function spaces. Introducing a Moreau-Yosida regularization of the state constraints, the problem can be solved using a superlinearly convergent semi-smooth Newton method. Optimality conditions are derived, convergence of the Moreau-Yosida regularization is proved, and well-posedness and superlinear convergence of the Newton method is shown. Numerical examples illustrate the features of this problem and the proposed approach. © EDP Sciences, SMAI, 2010.view abstract 10.1051/m2an/2010064 **Recent advances in analytical and numerical methods in inverse problems for PDEs (minisymposium report)**

Clason, C. and Shishlenin, M.*Journal of Inverse and Ill-Posed Problems*18 (2011)This minisymposium, held at the 5th International Conference "Inverse Problems: Modeling and Simulation", May 24-29, 2010 in Antalya, Turkey, was dedicated to the 60th birthday of Michael V. Klibanov. © de Gruyter 2011.view abstract 10.1515/JIIP.2011.014 **A duality-based splitting method for ℓ1-tv image restoration with automatic regularization parameter choice**

Clason, C. and Jin, B. and Kunisch, K.*SIAM Journal on Scientific Computing*32 (2010)A novel splitting method is presented for the ℓ1-TV restoration of degraded images subject to impulsive noise. The functional is split into an ℓ2-TV denoising part and an ℓ1-ℓ2 deblurring part. The dual problem of the relaxed functional is smooth with convex constraints and can be solved efficiently by applying an Arrow-Hurwicz-type algorithm to the augmented Lagrangian formulation. The regularization parameter is chosen automatically based on a balancing principle. The accuracy, the fast convergence, and the robustness of the algorithm and the use of the parameter choice rule are illustrated on some benchmark images and compared with an existing method. © 2010 Society for Industrial and Applied Mathematics.view abstract 10.1137/090768217 **A semismooth Newton method for L1 data fitting with automatic choice of regularization parameters and noise calibration**

Clason, C. and Jin, B. and Kunisch, K.*SIAM Journal on Imaging Sciences*3 (2010)This paper considers the numerical solution of inverse problems with an L1 data fitting term, which is challenging due to the lack of differentiability of the objective functional. Utilizing convex duality, the problem is reformulated as minimizing a smooth functional with pointwise constraints, which can be efficiently solved using a semismooth Newton method. In order to achieve superlinear convergence, the dual problem requires additional regularization. For both the primal and the dual problems, the choice of the regularization parameters is crucial. We propose adaptive strategies for choosing these parameters. The regularization parameter in the primal formulation is chosen according to a balancing principle derived from the model function approach, whereas the one in the dual formulation is determined by a path-following strategy based on the structure of the optimality conditions. Several numerical experiments confirm the efficiency and robustness of the proposed method and adaptive strategy. © 2010 Society for Industrial and Applied Mathematics.view abstract 10.1137/090758003 **Adaptation and focusing of optode configurations for fluorescence optical tomography by experimental design methods**

Freiberger, M. and Clason, C. and Scharfetter, H.*Journal of Biomedical Optics*15 (2010)Fluorescence tomography excites a fluorophore inside a sample by light sources on the surface. From boundary measurements of the fluorescent light, the distribution of the fluorophore is reconstructed. The optode placement determines the quality of the reconstructions in terms of, e.g., resolution and contrast-to-noise ratio. We address the adaptation of the measurement setup. The redundancy of the measurements is chosen as a quality criterion for the optodes and is computed from the Jacobian of the mathematical formulation of light propagation. The algorithm finds a subset with minimum redundancy in the measurements from a feasible pool of optodes. This allows biasing the design in order to favor reconstruction results inside a given region. Two different variations of the algorithm, based on geometric and arithmetic averaging, are compared. Both deliver similar optode configurations. The arithmetic averaging is slightly more stable, whereas the geometric averaging approach shows a better conditioning of the sensitivity matrix and mathematically corresponds more closely with entropy optimization. Adapted illumination and detector patterns are presented for an initial set of 96 optodes placed on a cylinder with focusing on different regions. Examples for the attenuation of fluorophore signals from regions outside the focus are given. © 2010 Society of Photo-Optical Instrumentation Engineers.view abstract 10.1117/1.3316405 **Fast reduction of undersampling artifacts in radial MR angiography with 3D total variation on graphics hardware**

Knoll, F. and Unger, M. and Diwoky, C. and Clason, C. and Pock, T. and Stollberger, R.*Magnetic Resonance Materials in Physics, Biology and Medicine*23 (2010)Objective: Subsampling of radially encoded MRI acquisitions in combination with sparsity promoting methods opened a door to significantly increased imaging speed, which is crucial for many important clinical applications. In particular, it has been shown recently that total variation (TV) regularization efficiently reduces undersampling artifacts. The drawback of the method is the long reconstruction time which makes it impossible to use in daily clinical practice, especially if the TV optimization problem has to be solved repeatedly to select a proper regularization parameter. Materials and Methods: The goal of this work was to show that for the case of MR Angiography, TV filtering can be performed as a post-processing step, in contrast to the common approach of integrating TV penalties in the image reconstruction process. With this approach, it is possible to use TV algorithms with data fidelity terms in image space, which can be implemented very efficiently on graphic processing units (GPUs). The combination of a special radial sampling trajectory and a full 3D formulation of the TV minimization problem is crucial for the effectiveness of the artifact elimination process. Results and Conclusion: The computation times of GPU-TV show that interactive elimination of undersampling artifacts is possible even for large volume data sets, in particular allowing the interactive determination of the regularization parameter. Results from phantom measurements and in vivo angiography data sets show that 3D TV, together with the proposed sampling trajectory, leads to pronounced improvements in image quality. However, while artifact removal was very efficient for angiography data sets in this work, it cannot be expected that the proposed method of TV post-processing will work for arbitrary types of scans. © 2010 ESMRMB.view abstract 10.1007/s10334-010-0207-x **On a bilinear optimization problem in parallel magnetic resonance imaging**

Clason, C. and von Winckel, G.*Applied Mathematics and Computation*216 (2010)This work is concerned with the structure of bilinear minimization problems arising in recovering sub-sampled and modulated images in parallel magnetic resonance imaging. By considering a physically reasonable simplified model exhibiting the same fundamental mathematical difficulties, it is shown that such problems suffer from poor gradient scaling and non-convexity, which causes standard optimization methods to perform inadequately. A globalized quasi-Newton method is proposed which is able to reconstruct both image and the unknown modulations without additional a priori information. Thus the present paper serves as a first contribution toward understanding and solving such bilinear optimization problems. © 2010 Elsevier Inc. All rights reserved.view abstract 10.1016/j.amc.2010.02.047 **Optimal Boundary Control for Equations of Nonlinear Acoustics**

Clason, C. and Kaltenbacher, B. and Lasiecka, I. and Veljovic, S.*2010 15th International Conference on Methods and Models in Automation and Robotics (mmar)*(2010)Motivated by a medical application from lithotripsy, we study an optimal boundary control problem given by Westervelt equation -1/c(2) D(t)(2)u + Delta u + b/c(2) Delta(D(t)u) = -beta a/rho c(4) D(t)(2)u(2) in (0,T) x Omega (1) modeling the nonlinear evolution of the acoustic pressure u in a smooth, bounded domain Omega subset of R-d, d is an element of {1, 2, 3}. Here c > 0 is the speed of sound, b > 0 the diffusivity of sound, rho > 0 the mass density and beta(a) > 1 the parameter of nonlinearity. We study the optimization problem for existence of an optimal control and derive the first-order necessary optimality conditions. In addition, all results are extended for the more general Kuznetsov equation D-t(2)psi - c(2)Delta psi = D-t(b Delta psi + 1/c(2) B/2A (D-t psi)(2) + vertical bar del psi vertical bar(2)) (2) given in terms of the acoustic velocity potential psi.view abstract 10.1109/MMAR.2010.5587247 **Total variation regularization for nonlinear fluorescence tomography with an augmented Lagrangian splitting approach**

Freiberger, M. and Clason, C. and Scharfetter, H.*Applied Optics*49 (2010)Fluorescence tomography is an imaging modality that seeks to reconstruct the distribution of fluorescent dyes inside a highly scattering sample from light measurements on the boundary. Using common inversion methods with L 2 penalties typically leads to smooth reconstructions, which degrades the obtainable resolution. The use of total variation (TV) regularization for the inverse model is investigated. To solve the inverse problem efficiently, an augmented Lagrange method is utilized that allows separating the Gauss-Newton minimization from the TV minimization. Results on noisy simulation data provide evidence that the reconstructed inclusions are much better localized and that their half-width measure decreases by at least 25% compared to ordinary L p2 reconstructions. © 2010 Optical Society of America.view abstract 10.1364/AO.49.003741

#### inverse problems

#### mathematical image processing

#### medical imaging

#### numerical methods

#### optimal control systems

#### semismooth newton method