Convergent adaptive hybrid higher-order schemes for convex minimization

This paper proposes two convergent adaptive mesh-refining algorithms for the hybrid high-order method in convex minimization problems with two-sided p-growth. Examples include the p-Laplacian, an optimal design problem in topology optimization, and the convexified double-well problem. The hybrid high-order method utilizes a gradient reconstruction in the space of piecewise Raviart-Thomas finite element functions without stabilization on triangulations into simplices or in the space of piecewise polynomials with stabilization on polytopal meshes. The main results imply the convergence of the energy and, under further convexity properties, of the approximations of the primal resp. dual variable. Numerical experiments illustrate an efficient approximation of singular minimizers and improved convergence rates for higher polynomial degrees. Computer simulations provide striking numerical evidence that an adopted adaptive HHO algorithm can overcome the Lavrentiev gap phenomenon even with empirical higher convergence rates

Iterated regularization methods for solving inverse problems

Typical inverse problems are ill-posed which frequently leads to difficulties in calculatingnumerical solutions. A common approximation method to solve ill-posed inverse problemsis iterated Tikhonov-Lavrentiev regularization.We examine iterated Tikhonov-Lavrentiev regularization and show that, in the casethat regularity properties are not globally satisfied, certain projections of the error converge faster than the theoretical predictions of the global error. We also explore the sensitivity of iterated Tikhonov regularization to the choice of the regularization parameter. We show that by calculating higher order sensitivities we improve the accuracy. We present a simple to implement algorithm that calculates the iterated Tikhonov updates and the sensitivities to the regularization parameter. The cost of this new algorithm is one vector addition and one scalar multiplication per step more than the standard iterated Tikhonov calculation.In considering the inverse problem of inverting the Helmholz-differential filter (with filterradius δ), we propose iterating a modification to Tikhonov-Lavrentiev regularization (withregularization parameter α and J iteration steps). We show that this modification to themethod decreases the theoretical error bounds from O(α(δ^2 +1)) for Tikhonov regularizationto O((αδ^2)^(J+1) ). We apply this modified iterated Tikhonov regularization method to theLeray deconvolution model of fluid flow. We discretize the problem with finite elements inspace and Crank-Nicolson in time and show existence, uniqueness and convergence of thissolution.We examine the combination of iterated Tikhonov regularization, the L-curve method,a new stopping criterion, and a bootstrapping algorithm as a general solution method inbrain mapping. This method is a robust method for handling the difficulties associated withbrain mapping: uncertainty quantification, co-linearity of the data, and data noise. Weuse this method to estimate correlation coefficients between brain regions and a quantified performance as well as identify regions of interest for future analysis

An Overdetermined Problem in Potential Theory

We investigate a problem posed by L. Hauswirth, F. H\'elein, and F. Pacard, namely, to characterize all the domains in the plane that admit a "roof function", i.e., a positive harmonic function which solves simultaneously a Dirichlet problem with null boundary data, and a Neumann problem with constant boundary data. Under some a priori assumptions, we show that the only three examples are the exterior of a disk, a halfplane, and a nontrivial example. We show that in four dimensions the nontrivial simply connected example does not have any axially symmetric analog containing its own axis of symmetry.Comment: updated version. 20 pages, 3 figure

Unstabilized hybrid high-order method for a class of degenerate convex minimization problems

The relaxation procedure in the calculus of variations leads to minimization problems with a quasi-convex energy density. In some problems of nonlinear elasticity, topology optimization, and multiphase models, the energy density is convex with some convexity control plus two-sided $p$-growth. The minimizers may be non-unique in the primal variable, but define a unique stress variable $\sigma$. The approximation by hybrid high-order (HHO) methods utilizes a reconstruction of the gradients in the space of piecewise Raviart-Thomas finite element functions without stabilization on a regular triangulation into simplices. The application of the HHO methodology to this class of degenerate convex minimization problems allows for a unique $H(\div)$ conform stress approximation $\sigma_h$. The a priori estimates for the stress error $\sigma - \sigma_h$ in the Lebesgue norm are established for mixed boundary conditions without additional assumptions on the primal variable and lead to convergence rates for smooth solutions. The a posteriori analysis provides guaranteed error control, including a computable lower energy bound, and a convergent adaptive scheme. Numerical benchmarks display higher convergence rates for higher polynomial degrees and provide empirical evidence for the superlinear convergence of the lower energy bound. Although the focus is on the unstabilized HHO method, a detailed error analysis is provided for the stabilized version with a gradient reconstruction in the space of piecewise polynomials

A priori error analysis for state constrained boundary control problems. Part II: Full discretization

This is the second of two papers concerned with a state-constrained optimal control problems with boundary control, where the state constraints are only imposed in an interior subdomain. We apply the virtual control concept introduced in [26] to regularize the problem. The arising regularized optimal control problem is discretized by finite elements and linear and continuous ansatz functions for the boundary control. In the first part of the work, we investigate the errors induced by the regularization and the discretization of the boundary control. The second part deals with the error arising from discretization of the PDE. Since the state constraints only appear in an inner subdomain, the obtained order of convergence exceeds the known results in the field of a priori analysis for state-constrained problems. The theoretical results are illustrated by numerical computations

A nash game algorithm for the solution of coupled conductivity identification and data completion in cardiac electrophysiology

International audienceWe consider the identification problem of the conductivity coefficient for an elliptic operator using an incomplete over specified measures on the surface. Our purpose is to introduce an original method based on a game theory approach, and design a new algorithm for the simultaneous identification of conductivity coefficient and data completion process. We define three players with three corresponding criteria. The two first players use Dirichlet and Neumann strategies to solve the completion problem, while the third one uses the conductivity coefficient as strategy, and uses a cost which basically relies on an identifiability theorem. In our work, the numerical experiments seek the development of this algorithm for the electrocardiography imaging inverse problem, dealing with in-homogeneities in the torso domain. Furthermore, in our approach, the conductivity coefficients are known only by an approximate values. we conduct numerical experiments on a 2D torso case including noisy measurements. Results illustrate the ability of our computational approach to tackle the difficult problem of joint identification and data completion. Mathematics Subject Classification. 35J25, 35N05, 91A80. The dates will be set by the publisher

Numerische Konzepte und Fehleranalysis zu elliptischen Randsteuerungsproblemen mit punktweisen Zustands- und Kontrollbeschränkungen

Optimization in technical applications described by partial differential equations plays a more and more important role. By means of the control the solution of a partial differential equation called state is influenced. Simultaneously a cost functional has to be minimized. In many technical applications pointwise constraints to the state or the control are reasonable. It is well known that the Lagrange multipliers with respect to pure state constraints are in general only regular Borel measures. This fact implies a lower regularity of the optimal solution of the problem. In this dissertation a linear quadratic optimal control problem governed by an elliptic partial differential equation an Neumann boundary control is investigated. Furthermore, we consider pointwise state constraints in an inner subdomain and bilateral constraints on the boundary control. Despite the above mentioned problems, we benefit from the localization of the Lagrange multiplier in the inner subdomain such that a higher regularity of the optimal control is shown. However, the so called dual variables of the optimal control problem are not unique. Hence, the application of well known and efficient optimization algorithms becomes difficult. Presenting a regularization concept, we will avoid these problems. We introduce an additional distributed control ("virtual control") which appears in the cost functional, the right hand side of the partial differential equation and in the regularized state constraints. The effect of regularization is influenced by several parameter functions. We derive an error estimate for the error between the optimal solution of the original problem and the regularized one. Moreover, under some assumptions on the parameter functions we obtain certain convergence rates of the regularization error. In the following a finite element based approximation of the regularized optimal control problems is established. Based on appropriate feasible test functions, we derive an error estimate between the optimal solution of the unregularized original problem and the regularized and discretized one. Thereby, we consider the regularization and discretization simultaneously and we propose a suitable coupling of the parameter functions and the mesh size. Forthcoming, we present the primal-dual active set strategy as a optimization method for solving the regularized optimal control problems. Moreover, we derive an error estimate between the current iterates of the algorithm and the optimal solution. Based on this, we construct an error estimator, which is reliable as an alternative stopping criterion for the primal-dual active set strategy. Finally, the theoretical results of this work are illustrated by several numerical examples.Physikalische und technische Anwendungen werden häufig durch partielle Differentialgleichungen beschrieben. Die Optimierung solcher Prozesse führt auf sogenannte Optimalsteuerprobleme mit partiellen Differentialgleichungen. Mit Hilfe einer Steuerungsvariable wird die Lösung der Differentialgleichung, welche Zustand genannt wird, beeinflusst. Gleichzeitig soll ein Zielfunktional minimiert werden. Bei vielen technischen Anwendungen sind punktweise Beschränkungen an den Zustand oder die Steuerung sinnvoll. Es ist bekannt, dass die zu den Zustandsbeschränkungen gehörigen Lagrangsche Multiplikatoren im allgemeinen nur reguläre Borel-Maße sind. Dies führt zu einer geringeren Regularität der optimalen Lösung des Problems. In dieser Dissertationsschrift wird ein linear-quadratisches Optimalsteuerproblem mit elliptischer partieller Differentialgleichung und Neumann-Randsteuerung untersucht. Wir betrachten punkteweise Zustandsschranken in einem inneren Teilgebiet und bilaterale Schranken an die Randsteuerung. Die räumliche Trennung der Zustandsbeschränkungen von dem Wirkungsgebiet der Steuerung gestattet an vielen Stellen den Einsatz von speziell konstruierten mathematischen Techniken. Dies betrifft sowohl Regularitätsaussagen als auch Fehlerabschätzungen. Allerdings sind die sogenannten dualen Variablen des Problems nicht eindeutig. Dies macht die Anwendung bekannter effizienter Optimierungsalgorithmen unmöglich. Es wird ein Regularisierungskonzept vorgestellt, um dieses Problem zu vermeiden. Dabei wird eine zusätzliche verteilte Steuerung ("virtuelle Steuerung") eingeführt, welche im Zielfunktional, in der rechten Seite der Differentialgleichungen und in den regularisierten Zustandsbeschränkungen auftaucht. Die Regularisierung wird durch verschiedene Parameterfunktionen beeinflusst. Wir leiten Abschätzungen für den Fehler zwischen der optimalen Lösung des Ausgangsproblems und der des regularisierten Problems her. Bei Verwendung geschickt gewählter Parameterfunktionen ergeben sich aus diesen Abschätzungen direkt Konvergenzraten für die Regularisierung. Im weiteren betrachten wir auch eine Diskretisierung des regularisierten Problems mit Hilfe von finiten Elementen. Basierend auf geeignet konstruierten zulässigen Testfunktionen wird eine Fehlerabschätzung der optimalen Lösung des unregularisierten Problems zur diskretisierten und regularisierten Lösung hergeleitet. Da der Regularisierungs- und der Diskretisierungsfehler gleichzeitig auftreten, wird eine geeignete Kopplung des Regularisierungsparameters mit der Gitterweite angegeben. Eine primal-duale aktive Mengenstrategie wird als Optimierungsalgorithmus zur Lösung der regularisierten Probleme vorgestellt. Weiterhin wird eine Fehlerabschätzung der aktuellen Iterierten dieses Algorithmus zur optimalen Lösung bewiesen. Basierend auf diesem Resultat wird ein Fehlerschätzer konstruiert, welcher als alternatives Abbruchkriterium für die aktive Mengenstrategie benutzt werden kann. Die Resultate der Arbeit werden durch verschiedene numerische Beispiele bestätigt

Implicit iteration methods in Hilbert scales under general smoothness conditions

For solving linear ill-posed problems regularization methods are required when the right hand side is with some noise. In the present paper regularized solutions are obtained by implicit iteration methods in Hilbert scales. % By exploiting operator monotonicity of certain functions and interpolation techniques in variable Hilbert scales, we study these methods under general smoothness conditions. Order optimal error bounds are given in case the regularization parameter is chosen either {\it a priori} or {\it a posteriori} by the discrepancy principle. For realizing the discrepancy principle, some fast algorithm is proposed which is based on Newton's method applied to some properly transformed equations