A fast iterative PDE-based algorithm for feedback controls of nonsmooth mean-field control problems

A PDE-based accelerated gradient algorithm is proposed to seek optimal feedback controls of McKean-Vlasov dynamics subject to nonsmooth costs, whose coefficients involve mean-field interactions both on the state and action. It exploits a forward-backward splitting approach and iteratively refines the approximate controls based on the gradients of smooth costs, the proximal maps of nonsmooth costs, and dynamically updated momentum parameters. At each step, the state dynamics is realized via a particle approximation, and the required gradient is evaluated through a coupled system of nonlocal linear PDEs. The latter is solved by finite difference approximation or neural network-based residual approximation, depending on the state dimension. Exhaustive numerical experiments for low and high-dimensional mean-field control problems, including sparse stabilization of stochastic Cucker-Smale models, are presented, which reveal that our algorithm captures important structures of the optimal feedback control, and achieves a robust performance with respect to parameter perturbation.Comment: Add Sections 2.3 and 2.4 for theoretical convergence result

Well-posedness and numerical schemes for one-dimensional McKean-Vlasov equations and interacting particle systems with discontinuous drift

In this paper, we first establish well-posedness results for one-dimensional McKean-Vlasov stochastic differential equations (SDEs) and related particle systems with a measure-dependent drift coefficient that is discontinuous in the spatial component, and a diffusion coefficient which is a Lipschitz function of the state only. We only require a fairly mild condition on the diffusion coefficient, namely to be non-zero in a point of discontinuity of the drift, while we need to impose certain structural assumptions on the measure-dependence of the drift. Second, we study fully implementable Euler-Maruyama type schemes for the particle system to approximate the solution of the one-dimensional McKean-Vlasov SDE. Here, we will prove strong convergence results in terms of the number of time-steps and number of particles. Due to the discontinuity of the drift, the convergence analysis is non-standard and the usual strong convergence order $1/2$ known for the Lipschitz case cannot be recovered for all schemes.Comment: 33 pages, 4 figures, revised introduction and Section

Well-posedness and tamed schemes for McKean-Vlasov Equations with Common Noise

In this paper, we first establish well-posedness of McKean-Vlasov stochastic differential equations (McKean-Vlasov SDEs) with common noise, possibly with coefficients having super-linear growth in the state variable. Second, we present stable time-stepping schemes for this class of McKean-Vlasov SDEs. Specifically, we propose an explicit tamed Euler and tamed Milstein scheme for an interacting particle system associated with the McKean-Vlasov equation. We prove stability and strong convergence of order $1/2$ and $1$, respectively. To obtain our main results, we employ techniques from calculus on the Wasserstein space. The proof for the strong convergence of the tamed Milstein scheme only requires the coefficients to be once continuously differentiable in the state and measure component. To demonstrate our theoretical findings, we present several numerical examples, including mean-field versions of the stochastic $3/2$ volatility model and the stochastic double well dynamics with multiplicative noise.Comment: 36 pages, 3 figure

Milstein schemes for delay McKean equations and interacting particle systems

In this paper, we derive fully implementable first order time-stepping schemes for point delay McKean stochastic differential equations (McKean SDEs), possibly with a drift term exhibiting super-linear growth in the state component. Specifically, we propose different tamed Milstein schemes for a time-discretised interacting particle system associated with the McKean equation and prove strong convergence of order 1 and moment stability, making use of techniques from calculus on the space of probability measures with finite second order moments. In addition, we introduce a truncated tamed Milstein scheme based on an antithetic multi-level Monte Carlo approach, which leads to optimal complexity estimators for expected functionals without the need to simulate L\'evy areas.Comment: 33 pages, 4 figure

First order convergence of Milstein schemes for McKean-Vlasov equations and interacting particle systems

In this paper, we derive fully implementable first order time-stepping schemes for McKean--Vlasov stochastic differential equations (McKean--Vlasov SDEs), allowing for a drift term with super-linear growth in the state component. We propose Milstein schemes for a time-discretised interacting particle system associated with the McKean--Vlasov equation and prove strong convergence of order 1 and moment stability, taming the drift if only a one-sided Lipschitz condition holds. To derive our main results on strong convergence rates, we make use of calculus on the space of probability measures with finite second order moments. In addition, numerical examples are presented which support our theoretical findings.Comment: 28 pages, 10 figure

Listener Modeling and Context-aware Music Recommendation Based on Country Archetypes

Music preferences are strongly shaped by the cultural and socio-economic background of the listener, which is reflected, to a considerable extent, in country-specific music listening profiles. Previous work has already identified several country-specific differences in the popularity distribution of music artists listened to. In particular, what constitutes the "music mainstream" strongly varies between countries. To complement and extend these results, the article at hand delivers the following major contributions: First, using state-of-the-art unsupervised learning techniques, we identify and thoroughly investigate (1) country profiles of music preferences on the fine-grained level of music tracks (in contrast to earlier work that relied on music preferences on the artist level) and (2) country archetypes that subsume countries sharing similar patterns of listening preferences. Second, we formulate four user models that leverage the user's country information on music preferences. Among others, we propose a user modeling approach to describe a music listener as a vector of similarities over the identified country clusters or archetypes. Third, we propose a context-aware music recommendation system that leverages implicit user feedback, where context is defined via the four user models. More precisely, it is a multi-layer generative model based on a variational autoencoder, in which contextual features can influence recommendations through a gating mechanism. Fourth, we thoroughly evaluate the proposed recommendation system and user models on a real-world corpus of more than one billion listening records of users around the world (out of which we use 369 million in our experiments) and show its merits vis-a-vis state-of-the-art algorithms that do not exploit this type of context information.Comment: 30 pages, 3 tables, 12 figure

Solving an inverse problem concerning the sheet metal blank geometry in an industrial application to minimize the processing time

This paper concerns the optimization of an industrial sheet metal forming process on an automatic panel bender from Salvagnini Maschinenbau GmbH combined with a corner former from an external company. By combining these two machines and optimizing the sheet metal blank geometry, production is much cheaper and faster because one processing step is eliminated

A Comparison Between Commercial and Open-Source Software for Finite Element Analysis of Elasto-Plastic Bending

Nowadays, simulation is becoming more and more important in industries. Here we consider a typical industrial application in the field of sheet metal bending. A high number of simulations is necessary during the development process to perform parameter studies and optimizations. On the other hand, simulation tools should be also available for the customers of these machines, e.g., to plan the production of very specific profiles. In such cases, the optimal process parameters only can be found by simulation. Very important in this context are the license costs for commercial simulation software. Frequently, the simulations are not limited by computational power but by the number of available licenses, such that the duration for parameter studies is elongated. Also, with license costs it very expensive to provide a simulation platform to the customers. The presented case study has been carried out with the goal of comparing possible open source alternatives to expensive commercial Finite Element software. Exemplarily, we consider the elasto-plastic bending of a cantilever, using the Johnson Cook constitutive law. For this test case, a three dimensional Finite Element analysis is performed, comparing the results of open-source software (Salome-Meca) and a com mercial counterpart (Abaqus). Different element types and mesh sizes are compared, the usability of both tools, and the computational time. Considering the obvious price difference, both platforms show comparable results. Comparing the functionality of both programs, both are capable for modelling highly detailed and complex models for elasto-plastic material processing. However, for under standing the structure of the user interface of Salome-Meca is far more time consuming. Additionally, the performance of Salome-Meca on different operating systems is com pared: Salome-Meca on Linux, Salome-Meca on Linux, installed in a virtual machine on Windows, and finally Salome-Meca on Windows. All in all, it turned out that depending on the specific application Salome-Meca can be a powerful alternative to Abaqus for the considered industrial application

Influence of rate dependent plasticity on a sheet metal bending process

The high demands on precision and quality of industrial sheet metal forming processes are increasing steadily. Therefore, more and more effects concerning the machines but also the material behaviour of the workpiece must be considered. Here, we consider an automatic panel bender of Salvagnini Maschinenbau GmbH. In this application, it turned out that the speed of bending is a relevant influence factor. Goal of this work is to estimate the influence of strain rate on bending forces and the shape of the bent profile