Fast methods for nonsmooth nonconvex problems using variable projectionMI 02.06.011 (Boltzmannstr. 3, 85748 Garching)

Classic inverse problems are formulated using smooth penalties and regularizations. However, nonsmooth and nonconvex penalties/regularizers have proved to be extremely useful in underdetermined and noisy settings. Problems with these features also arise naturally when modeling complex physical and chemical phenomena; including PDE-constrained optimization, phase retrieval, and structural resolution of bio-molecular models. We propose a new technique for solving a broad range of nonsmooth, nonconvex problems. The technique is based on a relaxed reformulation, and can be implemented on a range of problems in a simple and scalable way. In particular, we typically need only solve least squares problems, as well as implement custom separable operators. We discuss the problem class, reformulation and algorithms, and give numerous examples of very promising numerical results in different applications.

Loss of memory and continuity properties for the supercritical contact processB 252 (Theresienstr. 39, 80333 München)

The contact process is an interacting particle system which models the spread of an infection in a population. In this talk I will focus on the evolution of this process in the supercritcal regime within a partial (and finite) subspace of the population. In particular, I will present some recent results on the loss of memory property for such partially observed processes and discuss their continuity properties with respect to conditioning (in the sense of g-measures). Several motivations for studying such projections, and some open questions, will be discussed during the talk.

Material Design for Optimal Excitation Induced Charge Transfer in Photovoltaic DevicesMI 03.08.011 (Boltzmannstr. 3, 85748 Garching)

This talk is concerned with a novel approach in the context of material design for (organic) photovoltaics. We consider a quantum-mechanical model for a prescribed distribution of positive charges (atomic cores) and a corresponding set of orbitals describing the negative charges (electrons). In the ground state, all orbitals up to HOMO are occupied and all higher orbitals starting from LUMO are unoccupied. By a light-induced excitation, the electronic system may end up in the first excited state where HOMO is unoccupied but LUMO is occupied. Our aim is to maximize this light-induced spatial charge transfer from HOMO to LUMO as a function of the underlying nuclear charge distribution. Concerning optimal charge transfer, we will review a general existence proof for the corresponding mathematical optimization problem. Numerical simulations are carried out for a 1D chain of atoms and illustrate the applicability of this approach. This work is part of a joint project with Gero Friesecke.

TBA2.01.10 (Parkring 11, 85748 Garching-Hochbrück)

TBA

A Weierstrass representation for 2D elasticity MI 03.08.011 (Boltzmannstr. 3, 85748 Garching)

We study a class of elastic energy functionals for maps between planar domains (among them the so-called squared distance functional) whose critical points (elastic maps) allow a far more complete theory than one would expect from general elasticity theory. For some of these functionals elastic maps even admit a "Weierstrass representation" in terms of holomorphic functions, reminiscent of the one for minimal surfaces. We also prove a global uniqueness theorem that does not seem to be known in other situations.

Women in Probability 2018MI 00.07.014 (Boltzmannstr. 3, 85748 Garching)

Sprecherinnen sind: Fabienne Castell, Karen Habermann, Irene Marcovici, Sara Merino-Aceituno, Francesca Nardi, Ellen Powell, Ellen Saada, Kristina Schubert

Women in Probability 2018MI 00.07.014 (Boltzmannstr. 3, 85748 Garching)

Integral functionals of cadlag processes and partial superhedging of American optionsB349 (Theresienstr. 39, 80333 München)

In this talk we present advances in convex analysis and obtain a novel interchange rule for convex functionals defined over cadlag processes. This interchange rule allows to develop convex duality for a rich class of convex problems in general stochastic settings and requires a careful analysis of set valued mappings and its cadlag selections. As an application, we develop the dual problem of American option's partial hedging.

Exponential attractors for infinite dimensional dynamical systemsMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

Exponential attractors of infinite dimensional dynamical systems are compact, semi-invariant sets of finite fractal dimension that attract all bounded subsets at an exponential rate. They contain the global attractor and, due to the exponential rate of convergence, are generally more stable under perturbations than global attractors. In the autonomous setting, exponential attractors have been studied for several decades and their existence has been shown for a large variety of dissipative equations. More recently, the theory has been extended to non-autonomous and random problems. We discuss general existence results for exponential attractors for nonautonomous and random dynamical systems in Banach spaces and derive explicit estimates on their fractal dimension. As an application semilinear heat and semilinear damped wave equations are considered. This is joint work with Tomas Caraballo (University of Sevilla) and Alexandre Carvalho (University of São Paulo).

Optimizing over convex functions: motivations and challengesMI 02.06.011 (Boltzmannstr. 3, 85748 Garching)

Optimization problems subject to a convexity constraint on the admissible profiles arise in a variety of contexts in aerodynamics, economics, geometry, shape optimization, mass transport.... They are challenging because discretizing convex functions or shapes is in general very costly and naive algorithms maybe inconsistent. In this survey talk, after describing some applications and theoretical results I will review some tractable numerical approaches to these problems.

For the full programme of the SFB Colloquium see: http://www.discretization.de/en/events/14/

A non-local Fokker-Planck equation related to nucleation and coarseningRoom 2004, 1st floor, Building L1 (Universitätsstr. 14, 86159 Augsburg)

In this talk we consider a Fokker-Planck equation modeling nucleation of clusters very similar to the classical Becker-Döring equation. The main feature of the equation is the dependence of the driving vector field and boundary condition on a non-local order parameter related to the excess mass of the system. In this way the equation has formally the structure of a McKean-Vlasov equation, but with a non-local boundary condition. We briefly discuss the well-posedness and regularity properties of the Cauchy-Problem. Here the main difficulty is to improve basic a priori regularity properties of the non-local order parameter. The main part of the talk focuses on the longtime behavior of the system. The system possesses a free energy, which strictly decreases along the evolution and leads to a gradient structure involving boundary conditions. We generalize the standard entropy method based on suitable weighted logarithmic Sobolev inequalities and interpolation estimates. In this way, we obtain an explicit characterization of the convergence to equilibrium with algebraic or even exponential rates depending on the particular assumptions on the vector fields, diffusivity and initial data. (joint work with J. Conlon)

Regularity theory for functionals on BD - from convexity to symmetric-rank-one convexityRoom 2004, 1st floor, Building L1 (Universitätsstr. 14, 86159 Augsburg)

Linear growth functionals (such as the minimal surface functional) are usually minimised over the space BV of functions of bounded variation. However, if we pass to the vectorial case and replace the full by the symmetric gradient, then Ornstein's Non-Inequality rules out coerciveness of linear growth functionals on BV. We are thereby lead to examine generalised minima over the space BD of functions of bounded deformation: For such maps the symmetric distributional gradients are finite Radon measures. In this talk I aim to give a comprehensive regularity analysis for generalised minimisers both in the convex and the symmetric-semiconvex framework. The results presented therein are partially obtained in collaboration with J. Kristensen (Oxford).

Near-optimal data-driven l^1-regularization (Master Thesis presentation)MI 03.10.011 (Boltzmannstr. 3, 85748 Garching)

Solving underdetermined linear systems of equations by l^1-regularization is a common approach. The amount of regularization is controlled by a regularization parameter. Even though there exist various techniques for choosing this parameter, it is still a relevant challenge. In this thesis, a data-driven approach is proposed, which is neither relying on knowledge of the solution nor on the noise level. The idea is to estimate the regularization parameter of LASSO by a greedy solution computed by OMP. We give explicit error bounds for the vectors reconstructed by the different algorithms and show theoretically that by an optimal choice of the regularization parameter LASSO and OMP can achieve the same error. The numerical results are even more promising: we find scenarios where LASSO, with an estimated parameter, outperforms OMP. Furthermore, we implement an image reconstruction from noisy and undersampled data in MATLAB. Here, LASSO using a regularization parameter chosen by our proposed approach, reconstructs with a smaller error than OMP.

Learning a two layer neural network by fewest samples (Master Thesis presentation)MI 03.10.011 (Boltzmannstr. 3, 85748 Garching)

We consider compositions of weighted sums of ridge functions, which are closely related to two layer feed-forward neural networks. The goal is the recovery of the ridge directions under mild smoothness assumptions of the function and for quasi-orthogonal ridge directions. The reconstruction can be divided into two steps. First, the identification of a matrix space spanned by the symmetric tensor products of the ridge directions. This space is approximated by evaluating the Hessian of the function on random points of the sphere and applying a dimension reduction on the span of the Hessians. Secondly, the recovery of the ridge directions expressed as rank-1 matrices by solving a non-linear program on the intersection of the reduced matrix space with the unit Frobenius sphere. The primary focus of my presentation is the analysis of the involved matrix spaces. We give bounds on the concentration of the span of the Hessians, that hold with high probability. This concentration is essential and enables us to apply the dimension reduction. This is complemented by numerical results for two-layer neural networks with sigmoidal activation functions.

Geometric Singular Perturbation Analysis of a Model for Micro-Electro Mechanical Systems (MEMS)MI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

Many technological devices commonly used today rely on Micro-Electro Mechanical Systems (MEMS). These are deﬁned as very small structures combining electrical and mechanical components on a common substrate to perform several tasks. Their application in various ﬁelds such as medicine, transport industry and communications has raised considerable scientiﬁc interest.

The aim of this talk is to give a general introduction on the electrostatic-elastic case, where an elastic membrane is allowed to deﬂect above a ground plate under the action of an electric potential. This situation can be mathematically described by a parabolic PDE with a particular nonlinear source term that can lead to the “touchdown phenomenon”. Mathematically, touchdown causes non existence of steady states and/or ﬁnite time blow-up of solutions. A recently proposed model depending on a small “regularization” parameter ε is introduced, where considering additional insulating effects allows to avoid singularities. We use tools from geometric singular perturbation theory and blow-up methods to study the bifurcation of steady-state solutions, emphasizing the interplay between the parameters appearing in the model. In particular, we focus our attention on the singular limit as these small parameters tend to zero.

Max-linear models on infinite graphs generated by Bernoulli bond percolation2.01.10 (Parkring 11, 85748 Garching-Hochbrück)

We extend finite-dimensional max-linear models to models on infinite graphs, and investigate their relations to classical percolation theory, more precisely to nearest neighbour bond percolation. We focus on the square lattice $\mathbb{Z}^2$ with edges to the nearest neighbours, where we direct all edges in a natural way (north-east) resulting in a directed acyclic graph (DAG) on $\mathbb{Z}^2$. On this infinite DAG a random sub-DAG may be constructed by choosing vertices and edges between them at random. In a Bernoulli bond percolation DAG edges are independently declared open with probability $p\in [0,1]$ and closed otherwise. The random DAG consists then of the vertices and the open directed edges. We find for the subcritical case where $p\le 1/2$ that two random variables of the max-linear model become independent with probability 1, whenever their distance tends to infinity. In contrast, for the supercritical case where $p>1/2$ two random variables are dependent with positive probability, even when their node distance tends to infinity. We also consider changes in the dependence properties of random variables on a sub-DAG $H$ of a finite or infinite graph in $\mathbb{Z}^2$, when enlarging this subgraph. The method of enlargement consists of adding nodes and edges of Bernoulli percolation clusters. Here we start with $X_i$ and $X_j$ independent in $H$, and answer the question, whether they can become dependent in the enlarged graph. As a possible application we discuss extreme opinions in social networks The talk is based on joint work with Ercan Sönmez and the following paper. [1] Klüppelberg, C. and Sönmez, E. (2018) Max-linear models on infinite graphs generated by Bernoulli bond percolation. In preparation.

Optimal design problems for energies with non standard growthMI 03.10.011 (Boltzmannstr. 3, 85748 Garching)

Some recent results dealing with optimal design problems for energies which describe composite materials, mixed materials and Ogden ones will be presented.

Breaking the curse of dimension in multi-marginal Kantorovich optimal transport on finite state spacesMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

In this talk, I will present a new ansatz space for the general symmetric multimarginal Kantorovich optimal transport problem on finite state spaces which reduces the number of unknowns from combinatorial in both N and `l to `l · (N + 1), where `l is the number of marginal states and N the number of marginals.

The new ansatz space is a careful low-dimensional enlargement of the Monge class, which corresponds to `l · (N - 1) unknowns, and cures the insuffciency of the Monge ansatz, i.e. it is shown that the Kantorovich problem always admits a minimizer in the enlarged class, for arbitrary cost functions.

Our results apply, in particular, to the discretization of multi-marginal optimal transport with Coulomb cost in three dimensions, which has received much recent interest due to its emergence as the strongly correlated limit of Hohenberg-Kohn density functional theory. In this context N corresponds to the number of particles, motivating the interest in large N.

These results were established in collaboration with Gero Friesecke. The corresponding paper can be found under arXiv:1801.00341.

TBAB 252 (Theresienstr. 39, 80333 München)

TBA

Finite element methods for interface problemsMI 02.06.011 (Boltzmannstr. 3, 85748 Garching)

Interface problems arise in many applications involving heterogenous materials, such as contact problems and immiscible two-phase flows. In some situations, non-matching meshes provide an interesting technique. In others, like moving interfaces, avoiding meshing of the interface is desirable. In this talk, we describe some methods for these situations, and investigate relations between the mortar and Niitsch-based methods.

Modelling 100 percent renewable electricityMI 02.06.011 (Boltzmannstr. 3, 85748 Garching)

In the past few decades, power grids across the world have become dependent on markets that aim to efficiently match supply with demand at all times via a variety of pricing and auction mechanisms. These markets are based on models that capture interactions between producers, transmission and consumers. Energy producers typically maximize profits by optimally allocating and scheduling resources over time. A dynamic equilibrium aims to determine prices and dispatches that can be transmitted over the electricity grid to satisfy evolving consumer requirements for energy at different locations and times. Computation allows large-scale practical implementations of socially optimal models to be solved as part of the market operation, and regulations can be imposed that aim to ensure competitive behaviour of market participants. The recent explosion in the use of renewable supply such as wind, solar and hydro has led to increased volatility in this system. We develop models that aim to ensure enough generation capacity for the long term under various constraints related to environmental concerns, and consider the recovery of costs for this enhanced infrastructure. We demonstrate how risk can impose significant costs on the system that are not modeled in the context of socially optimal power system markets and highlight the use of contracts to reduce or recover these costs. We also outline how battery storage can be used as an effective hedging instrument. This is joint work with Andy Philpott, University of Auckland.

Differential Inclusions and A-quasiconvexityMI 02.08.011 (Boltzmannstr. 3, 85748 Garching)

We extend to the abstract A-framework some existence theorems for differential inclusion problems with Dirichlet boundary conditions. Joint work with Ana Cristina Barroso (FCT/UL) and Pedro Miguel Santos (IST/UL).

Rigorous derivation of density functionals for classical systemsMI 03.08.011 (Boltzmannstr. 3, 85748 Garching)

In density functional theory one postulates a free energy functional on a space of functions representing the space/orientation/etc-dependent densities of the system. In this talk we present a rigorous derivation of such functionals starting from a partition function of a system of interacting classical particles. For homogeneous (constant density) systems, this is the standard virial expansion, but the challenge here is to implement it for inhomogeneous densities. These can be realized by considering particles with space dependent activities. The main technical difficulty lies on the need of an inversion formula in an appropriate functional space. We resolve this issue exploiting the combinatorial structure of such inversions. As a byproduct of our method, if we apply it to the homogeneous case, we improve the existing value for the radius of convergence of the virial expansion. Applications include inhomogeneous systems such as multi-species gases (inhomogeneous sizes of the spheres) and liquid crystals (inhomogeneous position and orientation). This is joint work with Tobias Kuna and Dimitrios Tsagkarogiannis.

A posteriori error estimates for eigenvalue problems arising from electronic structure calculationsMI 03.08.011 (Boltzmannstr. 3, 85748 Garching)

(joint work with Eric Cances, Yvon Maday, Benjamin Stamm and Martin Vohralík) The determination of the electronic structure of a molecular system usually requires the resolution of a nonlinear eigenvalue problem. To be solved numerically, this problem is first discretized, using for example finite elements or planewaves. Then the discrete nonlinear equations are solved using an iterative algorithm. At this point, a natural question arising is the size of the error between the computed solution and the exact solution of the given model. In this talk, I will first present an a posteriori error estimation for the eigenvectors and eigenvalues of a Schrödinger-type linear eigenvalue problem. The a posteriori bounds are guaranteed, fully computable, and converge with optimal speed to the given exact eigenvalues and eigenvectors of the problem [1]. These bounds are valid under very few assumptions that can be checked numerically. Also, they can be generalized for clusters of eigenvalues. Numerical simulations conirm the efficiency of the bounds. In a second part, I will illustrate how these estimations on linear eigenvalue problems can be used for the error estimation of nonlinear eigenvalue problems appearing in electronic structure calculations [2].

References [1] Eric Cances, Geneviève Dusson, Yvon Maday, Benjamin Stamm, and Martin Vohralík, Guaranteed and robust a posteriori bounds for Laplace eigenvalues and eigenvectors: conforming approximations, SIAM Journal on Numerical Analysis 55 (5), 2228-2254. [2] Eric Cances, Geneviève Dusson, Yvon Maday, Benjamin Stamm, and Martin Vohralí, A perturbation-method-based a posteriori estimator for the planewave discretization of nonlinear Schrödinger equations, Comptes Rendus Mathematique 352 (11), 941-946.