Essential m-dissipativity and hypocoercivity of Langevin dynamics with multiplicative noise

We provide a complete elaboration of the $L^2$-Hilbert space hypocoercivity theorem for the degenerate Langevin dynamics with multiplicative noise, studying the longtime behaviour of the strongly continuous contraction semigroup solving the abstract Cauchy problem for the associated backward Kolmogorov operator. Hypocoercivity for the Langevin dynamics with constant diffusion matrix was proven previously by Dolbeault, Mouhot and Schmeiser in the corresponding Fokker-Planck framework, and made rigorous in the Kolmogorov backwards setting by Grothaus and Stilgenbauer. We extend these results to weakly differentiable diffusion coefficient matrices, introducing multiplicative noise for the corresponding stochastic differential equation. The rate of convergence is explicitly computed depending on the choice of these coefficients and the potential giving the outer force. In order to obtain a solution to the abstract Cauchy problem, we first prove essential self-adjointness of non-degenerate elliptic Dirichlet operators on Hilbert spaces, using prior elliptic regularity results and techniques from Bogachev, Krylov and R\"ockner. We apply operator perturbation theory to obtain essential m-dissipativity of the Kolmogorov operator, extending the m-dissipativity results from Conrad and Grothaus. We emphasize that the chosen Kolmogorov approach is natural, as the theory of generalized Dirichlet forms implies a stochastic representation of the Langevin semigroup as the transition kernel of a diffusion process which provides a martingale solution to the Langevin equation with multiplicative noise. Moreover, we show that even a weak solution is obtained this way.Comment: 30 page

Wavelet representation of contour sets

Journal ArticleWe present a new wavelet compression and multiresolution modeling approach for sets of contours (level sets). In contrast to previous wavelet schemes, our algorithm creates a parametrization of a scalar field induced by its contours and compactly stores this parametrization rather than function values sampled on a regular grid. Our representation is based on hierarchical polygon meshes with subdivision connectivity whose vertices are transformed into wavelet coefficients. From this sparse set of coefficients, every set of contours can be efficiently reconstructed at multiple levels of resolution. When applying lossy compression, introducing high quantization errors, our method preserves contour topology, in contrast to compression methods applied to the corresponding field function. We provide numerical results for scalar fields defined on planar domains. Our approach generalizes to volumetric domains, time-varying contours, and level sets of vector fields

Infinite-dimensional supermanifolds over arbitrary base fields

In his recent investigation of a super Teichm\"uller space, Sachse (2007), based on work of Molotkov (1984), has proposed a theory of Banach supermanifolds using the `functor of points' approach of Bernstein and Schwarz. We prove that the the category of Berezin-Kostant-Leites supermanifolds is equivalent to the category of finite-dimensional Molotkov-Sachse supermanifolds. Simultaneously, using the differential calculus of Bertram-Gl\"ockner-Neeb (2004), we extend Molotkov-Sachse's approach to supermanifolds modeled on Hausdorff topological super-vector spaces over an arbitrary non-discrete Hausdorff topological base field of characteristic zero. We also extend to locally k-omega base fields the `DeWitt' supermanifolds considered by Tuynman in his monograph (2004), and prove that this leads to a category which is isomorphic to the full subcategory of Molokov-Sachse supermanifolds modeled on locally k-omega spaces.Comment: 36 pages; minor corrections, expanded introductio

Hypocoercivity for infinite-dimensional non-linear degenerate stochastic differential equations with multiplicative noise

We analyze infinite-dimensional non-linear degenerate stochastic differential equations with multiplicative noise. First, essential m-dissipativity of their associated Kolmogorov backward generators on $L^2(\mu^{\Phi})$ defined on smooth finitely based functions is established. Here $\Phi$ is an appropriate potential and $\mu^{\Phi}$ is the invariant measure with density $e^{-\Phi}$ w.r.t. an infinite-dimensional non-degenerate Gaussian measure. Second, we use resolvent methods to construct corresponding right processes with infinite lifetime, solving the martingale problem for the Kolmogorov backward generators. They provide weak solutions, with weakly continuous paths, to the non-linear degenerate stochastic partial differential equations. Moreover, we identify the transition semigroup of such a process with the strongly continuous contraction semigroup $(T_t)_{t\geq 0}$ generated by the corresponding Kolmogorov backwards generator. Afterwards, we apply a refinement of the abstract Hilbert space hypocoercivity method, developed by Dolbeaut, Mouhot and Schmeiser, to derive hypocoercivity of $(T_t)_{t\geq 0}$. I.e. we take domain issues into account and use the formulation in the Kolmogorov backwards setting worked out by Grothaus and Stilgenbauer. The method enables us to explicitly compute the constants determining the exponential convergence rate to equilibrium of $(T_t)_{t\geq 0}$. The identification between $(T_t)_{t\geq 0}$ and the transition semigroup of the process yields exponential ergodicity of the latter. Finally, we apply our results to second order in time stochastic reaction-diffusion and Cahn-Hilliard type equations with multiplicative noise. More generally, we analyze corresponding couplings of infinite-dimensional deterministic with stochastic differential equations

Supervised and Reinforcement Learning from Observations in Reconnaissance Blind Chess

In this work, we adapt a training approach inspired by the original AlphaGo system to play the imperfect information game of Reconnaissance Blind Chess. Using only the observations instead of a full description of the game state, we first train a supervised agent on publicly available game records. Next, we increase the performance of the agent through self-play with the on-policy reinforcement learning algorithm Proximal Policy Optimization. We do not use any search to avoid problems caused by the partial observability of game states and only use the policy network to generate moves when playing. With this approach, we achieve an ELO of 1330 on the RBC leaderboard, which places our agent at position 27 at the time of this writing. We see that self-play significantly improves performance and that the agent plays acceptably well without search and without making assumptions about the true game state.Comment: 4 Pages, IEEE Conference on Games 2022 short pape

On Moving Least Squares Based Flow Visualization

Modern simulation and measurement methods tend to produce meshfree data sets if modeling of processes or objects with free surfaces or boundaries is desired. In Computational Fluid Dynamics (CFD), such data sets are described by particle-based vector fields. This paper presents a summary of a selection of methods for the extraction of geometric features of such point-based vector fields while pointing out its challenges, limitations, and applications

Generation of Adaptive Streak Surfaces Using Moving Least Squares

We introduce a novel method for the generation of fully adaptive streak surfaces in time-varying flow fields based on particle advection and adaptive mesh refinement. Moving least squares approximation plays an important role in multiple stages of the proposed algorithm, which adaptively refines the surface based on curvature approximation and circumradius properties of the underlying Delaunay mesh. We utilize the grid-less Moving Least Squares approximation method for both curvature and surface estimation as well as vector field evaluation during particle advection. Delaunay properties of the surface triangulation are guaranteed by edge flipping operations on the progressive surface mesh. The results of this work illustrate the benefit of adaptivity techniques to streak surface generation and provide the means for a qualitative analysis of the presented approach