Risk estimators for choosing regularization parameters in ill-posed problems - properties and limitations
Felix Lucka Katharina Proksch Christoph Brune Nicolai Bissantz Martin Burger Holger Dette Frank Wübbeling
Inverse Problems & Imaging 2018, 12(5): 1121-1155 doi: 10.3934/ipi.2018047

This paper discusses the properties of certain risk estimators that recently regained popularity for choosing regularization parameters in ill-posed problems, in particular for sparsity regularization. They apply Stein's unbiased risk estimator (SURE) to estimate the risk in either the space of the unknown variables or in the data space. We will call the latter PSURE in order to distinguish the two different risk functions. It seems intuitive that SURE is more appropriate for ill-posed problems, since the properties in the data space do not tell much about the quality of the reconstruction. We provide theoretical studies of both approaches for linear Tikhonov regularization in a finite dimensional setting and estimate the quality of the risk estimators, which also leads to asymptotic convergence results as the dimension of the problem tends to infinity. Unlike previous works which studied single realizations of image processing problems with a very low degree of ill-posedness, we are interested in the statistical behaviour of the risk estimators for increasing ill-posedness. Interestingly, our theoretical results indicate that the quality of the SURE risk can deteriorate asymptotically for ill-posed problems, which is confirmed by an extensive numerical study. The latter shows that in many cases the SURE estimator leads to extremely small regularization parameters, which obviously cannot stabilize the reconstruction. Similar but less severe issues with respect to robustness also appear for the PSURE estimator, which in comparison to the rather conservative discrepancy principle leads to the conclusion that regularization parameter choice based on unbiased risk estimation is not a reliable procedure for ill-posed problems. A similar numerical study for sparsity regularization demonstrates that the same issue appears in non-linear variational regularization approaches.

keywords: Ill-posed problems regularization parameter choice risk estimators Stein’s method discrepancy principle
Large time behavior of nonlocal aggregation models with nonlinear diffusion
Martin Burger Marco Di Francesco
Networks & Heterogeneous Media 2008, 3(4): 749-785 doi: 10.3934/nhm.2008.3.749
The aim of this paper is to establish rigorous results on the large time behavior of nonlocal models for aggregation, including the possible presence of nonlinear diffusion terms modeling local repulsions. We show that, as expected from the practical motivation as well as from numerical simulations, one obtains concentrated densities (Dirac $\delta$ distributions) as stationary solutions and large time limits in the absence of diffusion. In addition, we provide a comparison for aggregation kernels with infinite respectively finite support. In the first case, there is a unique stationary solution corresponding to concentration at the center of mass, and all solutions of the evolution problem converge to the stationary solution for large time. The speed of convergence in this case is just determined by the behavior of the aggregation kernels at zero, yielding either algebraic or exponential decay or even finite time extinction. For kernels with finite support, we show that an infinite number of stationary solutions exist, and solutions of the evolution problem converge only in a measure-valued sense to the set of stationary solutions, which we characterize in detail.
Moreover, we also consider the behavior in the presence of nonlinear diffusion terms, the most interesting case being the one of small diffusion coefficients. Via the implicit function theorem we give a quite general proof of a rather natural assertion for such models, namely that there exist stationary solutions that have the form of a local peak around the center of mass. Our approach even yields the order of the size of the support in terms of the diffusion coefficients.
All these results are obtained via a reformulation of the equations considered using the Wasserstein metric for probability measures, and are carried out in the case of a single spatial dimension.
keywords: Wasserstein metric asymptotic behavior biological aggregation nonlocal PDEs stationary solutions Nonlinear diffusion
Mean field games with nonlinear mobilities in pedestrian dynamics
Martin Burger Marco Di Francesco Peter A. Markowich Marie-Therese Wolfram
Discrete & Continuous Dynamical Systems - B 2014, 19(5): 1311-1333 doi: 10.3934/dcdsb.2014.19.1311
In this paper we present an optimal control approach modeling fast exit scenarios in pedestrian crowds. In particular we consider the case of a large human crowd trying to exit a room as fast as possible. The motion of every pedestrian is determined by minimizing a cost functional, which depends on his/her position, velocity, exit time and the overall density of people. This microscopic setup leads in the mean-field limit to a parabolic optimal control problem. We discuss the modeling of the macroscopic optimal control approach and show how the optimal conditions relate to the Hughes model for pedestrian flow. Furthermore we provide results on the existence and uniqueness of minimizers and illustrate the behavior of the model with various numerical results.
keywords: mean field limit optimal control numerical simulations. Pedestrian dynamics calculus of variations
Balanced growth path solutions of a Boltzmann mean field game model for knowledge growth
Martin Burger Alexander Lorz Marie-Therese Wolfram
Kinetic & Related Models 2017, 10(1): 117-140 doi: 10.3934/krm.2017005

In this paper we study balanced growth path solutions of a Boltzmann mean field game model proposed by Lucas and Moll [15] to model knowledge growth in an economy.Agents can either increase their knowledge level by exchanging ideas in learning events or by producing goods with the knowledge they already have.The existence of balanced growth path solutions implies exponential growth of the overall production in time. We prove existence of balanced growth path solutions if the initial distribution of individuals with respect to their knowledge level satisfiesa Pareto-tail condition. Furthermore we give first insights into the existence of such solutions if in addition to production and knowledge exchange theknowledge level evolves by geometric Brownian motion.

keywords: Mean-field games Boltzmann-type equations Hamilton-Jacobi equations travelling wave solutions
Identification of nonlinearities in transport-diffusion models of crowded motion
Martin Burger Jan-Frederik Pietschmann Marie-Therese Wolfram
Inverse Problems & Imaging 2013, 7(4): 1157-1182 doi: 10.3934/ipi.2013.7.1157
The aim of this paper is to formulate a class of inverse problems of particular relevance in crowded motion, namely the simultaneous identification of entropies and mobilities. We study a model case of this class, which is the identification from flux-based measurements in a stationary setup. This leads to an inverse problem for a nonlinear transport-diffusion model, where boundary values and possibly an external potential can be varied. In specific settings we provide a detailed theory for the forward map and an adjoint problem useful in the analysis and numerical solution. We further verify the simultaneous identifiability of the nonlinearities and present several numerical tests yielding further insight into the way variations in boundary values and external potential affect the quality of reconstructions.
keywords: nonlinear drift-diffusion equations. identifiability Inverse problems
A mixed finite element method for nonlinear diffusion equations
Martin Burger José A. Carrillo Marie-Therese Wolfram
Kinetic & Related Models 2010, 3(1): 59-83 doi: 10.3934/krm.2010.3.59
We propose a mixed finite element method for a class of nonlinear diffusion equations, which is based on their interpretation as gradient flows in optimal transportation metrics. We introduce an appropriate linearization of the optimal transport problem, which leads to a mixed symmetric formulation. This formulation preserves the maximum principle in case of the semi-discrete scheme as well as the fully discrete scheme for a certain class of problems. In addition solutions of the mixed formulation maintain exponential convergence in the relative entropy towards the steady state in case of a nonlinear Fokker-Planck equation with uniformly convex potential. We demonstrate the behavior of the proposed scheme with 2D simulations of the porous medium equations and blow-up questions in the Patlak-Keller-Segel model.
keywords: porous medium equation Patlak-Keller-Segel model. mixed finite element method optimal transportation problem Nonlinear diffusion problems
Continuous limit of a crowd motion and herding model: Analysis and numerical simulations
Martin Burger Peter Alexander Markowich Jan-Frederik Pietschmann
Kinetic & Related Models 2011, 4(4): 1025-1047 doi: 10.3934/krm.2011.4.1025
In this paper we study the continuum limit of a cellular automaton model used for simulating human crowds with herding behaviour. We derive a system of non-linear partial differential equations resembling the Keller-Segel model for chemotaxis, however with a non-monotone interaction. The latter has interesting consequences on the behaviour of the model's solutions, which we highlight in its analysis. In particular we study the possibility of stationary states, the formation of clusters and explore their connection to congestion.
    We also introduce an efficient numerical simulation approach based on an appropriate hybrid discontinuous Galerkin method, which in particular allows flexible treatment of complicated geometries. Extensive numerical studies also provide a better understanding of the strengths and shortcomings of the herding model, in particular we examine trapping effects of crowds behind non-convex obstacles.
keywords: Crowd motion asymptotic analysis. continuum model herding

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