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Unbalanced Optimal TransportMI 02.10.011 (Boltzmannstr. 3, 85748 Garching)

Optimal transport induces a geometrically intuitive metric on the space of probability measures and is a powerful tool for image and data analysis. With the evolution of efficient numerical methods it is becoming increasingly popular. However, in many models the assumption that all measures have unit mass and that mass is exactly preserved locally are too restrictive, for instance in biochemical growth processes. Hence, in recent years, `unbalanced' transport problems, that allow creation or annihilation of mass during transport, have received increased attention. In this talk we present several formulations for such problems, efficient numerical methods and illustrate applications and advantages of unbalanced metrics.

Stochastic systems with time delay – Fokker-Planck approach, Markovian embedding and thermodynamic notionsMI 03.06.011 (Boltzmannstr. 3, 85748 Garching)

A discrete time delay in the Langevin equation naturally leads to an infinite hierarchy of Fokker- Planck (FP) equations for the n-time joint probability distribution functions [1,2]. Finding a probabilistic description is hence challenging, especially for systems subject to nonlinear forces. One major issue is that the higher members of the hierarchy contain unknown functional derivatives between noise and the stochastic state variable. In this talk, I will introduce a new way to derive the Fokker-Planck equation via a Markovian embedding technique. In particular, I will discuss an extended Markovian system with auxiliary variables which generates the same dynamics as the original (delayed) system in the limit of an infinitely large system. This extended system can be studied under a (stochastic) thermodynamical [3] perspective, allowing to find a closed expression for the entropy production, which is a nontrivial problem in the presence of delay [4]. [1] S. Guillouzic et al., PRE 59, 3970 (1999). [2] S. A. M. Loos and S. H. L. Klapp, PRE 96, 012106 (2017). [3] S. A. M. Loos and S. H. L. Klapp, arXiv:1806.04995 (2017). [4] M. L. Rosinberg, T. Munakata, and G. Tarjus, PRE 91, 042114 (2015).

S. A. M. Loos and S. H. L. Klapp

Sharpness of the Percolation Phase Transition for the Contact Process on Z^dB 252 (Theresienstr. 39, 80333 München)

The contact process is a model for the spread of infection on a graph. Each vertex of Z^d is either healthy or infected. Infected vertices become healthy with rate 1, independent of the rest of the process. Healthy sites become infected with a rate of lambda times the number of neighbours that are infected, where lambda is the parameter of the model. There exists a critical value for lambda above which there exists an infinite cluster and below which all clusters are finite. We show that this phase transition is sharp, i.e., below this critical value all clusters are exponentially small. The proof draws on a series of recent papers by Duminil-Copin, Raoufi and Tassion in which they prove sharp phase transitions for a variety of models using the OSSS inequality.

Discrete Schemes for Rate-independent Evolutions with Non-Convex EnergiesGebäude 33, Raum 0231 (Werner-Heisenberg-Weg 39, 85577 Neubiberg)

Rate-independent systems governed by non-convex energies provide several mathematical challenges. Since solutions may in general show discontinuities in time, the design of a suitable, mathematically rigorous notion of solution is all but clear and several different solution concepts exist, such as weak, differential, and global energetic solutions. In the recent past a new promising solution concept was developed, the so-called parametrized solution. The principal idea is to introduce an artificial time, in which the solution is continuous, and to interpret the physical time as a function of the artificial time. A numerical scheme that allows to approximate this class of solutions is the so-called local time-incremental minimization scheme. We investigate this scheme (combined with a standard finite element discretization in space) in detail, provide convergence results in the general case, and prove convergence rates for problems with (locally) convex energies. Numerical tests confirm our theoretical findings.

Microlocal interludeMI HS 3 (Boltzmannstr. 3, 85748 Garching)

Some questions are at the threshold between harmonic analysis and partial differential equations and can be studied in the Euclidean space or in more complicated non-commutative settings like compact Lie groups such as the Heisenberg group. Our goal will be to use one of these questions in order to get familiar with the microlocal approach and to understand how it can be implemented in various frameworks.

Taming non-convex optimization landscapes in data analysisMI 02.08.011 (Boltzmannstr. 3, 85748 Garching)

Nonconvex optimization problems are the bottleneck in many appplications in science and technology. In my talk I will report on two recent breakthroughs in solving some important nonconvex optimization problems. The first example concerns blind deconvolution, a topic that pervades many areas of science and technology, including geophysics, medical imaging, and communications. Here, blind deconvolution refers to the problem of recovering a function f from the convolution of two unknown functions g and h. Blind deconvolution is obviously ill-posed and its optimization landscape is full of undesirable local minima. I will first describe how I once failed to catch a murderer (dubbed the "graveyard murderer" by the media), because I failed in solving a blind deconvolution problem. I will then present a host of new algorithms to solve such nonconvex optimization problems. The proposed methods come with theoretical guarantees, are numerically efficient, robust, and require little or no parameter tuning, thus making them useful for massive datasets. The second example concerns the classical topics of data clustering and graph cuts. I will discuss a convex relaxation approach, which gives rise to a rigorous theoretical analysis of graph cuts. I derive deterministic bounds of finding optimal graph cuts via a natural and intuitive spectral proximity condition. Moreover, our theory provides theoretical guarantees for spectral clustering and for community detection.

Quantum advantage with shallow circuitsMI 03.08.011 (Boltzmannstr. 3, 85748 Garching)

Quantum effects can significantly enhance information-processing capabilities and speed up the solution of certain computational problems. Whether a quantum advantage can be rigorously proved in some setting or demonstrated experimentally using near-term devices is the subject of active debate. Here we show that parallel quantum algorithms running in a constant time are strictly more powerful than their classical counterparts: they are provably better at solving certain linear algebra problems associated with binary quadratic forms. Our work gives the first unconditional proof of a computational quantum advantage and simultaneously pinpoints its origin: it is a consequence of quantum nonlocality. The proposed quantum algorithm is a suitable candidate for near-future experimental realizations as it requires only constant-depth quantum circuits with nearest-neighbor gates on a 2D grid of qubits.

This is joint work with Sergey Bravyi and David Gosset, Science vol. 362, no. 6412 (2018).

Symmetry breaking in the Hartree-Fock jelliumMI 03.08.011 (Boltzmannstr. 3, 85748 Garching)

In this talk, we present two types of symmetry breakings that occur respectively in the Hartree-Fock jellium, and in the free fermionic gas. We first discuss the Overhauser's spatial symmetry breaking in the HF jellium, and give a lower bound on the energy gain due to this symmetry breaking. Then we focus on the free gas, and we present the phase diagram of the spin-polarisation of the gas at different densities and different temperatures. This is joint work with Mathieu Lewin and Christian Hainzl.

Hawkes Processes on Networks B 252 (Theresienstr. 39, 80333 München)

How do neurons in the brain interact with each other? This is a question not just for neuroscientists, but also for mathematicians, as mathematical models can help to elucidate some phenomena of neural networks. We give an example for such a model, namely Hawkes processes on networks. The set of neurons and their connections form a network, and each neuron in this network is assigned a Hawkes process (a special point process) that models its spike train. We outline a proof strategy for existence and uniqueness of Hawkes processes on networks, where we follow the work of Delattre/Fournier/Hoffmann 2016. We then discuss the large-time behaviour of the Hawkes processes, where we extend the results of Delattre et al. to more general networks.

Survival and extinction of the contact process on a symmetric graphB 252 (Theresienstr. 39, 80333 München)

We give a construction of a symmetric graph, on which the contact process survives on the whole graph, but dies out if you remove three privileged edges. For this, we will first introduce the graphical construction of the contact process on graphs, give the construction of our graph and the proof of the above statement.