Optimal Local Approximation Spaces for Generalized Finite Element Methods with Application to Multiscale Problems

The paper addresses a numerical method for solving second order elliptic partial differential equations that describe fields inside heterogeneous media. The scope is general and treats the case of rough coefficients, i.e. coefficients with values in $L^\infty(\Omega)$. This class of coefficients includes as examples media with micro-structure as well as media with multiple non-separated length scales. The approach taken here is based on the the generalized finite element method (GFEM) introduced in \cite{107}, and elaborated in \cite{102}, \cite{103} and \cite{104}. The GFEM is constructed by partitioning the computational domain $\Omega$ into to a collection of preselected subsets $\omega_{i},i=1,2,..m$ and constructing finite dimensional approximation spaces $\Psi_{i}$ over each subset using local information. The notion of the Kolmogorov $n$-width is used to identify the optimal local approximation spaces. These spaces deliver local approximations with errors that decay almost exponentially with the degrees of freedom $N_{i}$ in the energy norm over $\omega_i$. The local spaces $$ are used within the GFEM scheme to produce a finite dimensional subspace $S^N$ of $H^{1}(\Omega)$ which is then employed in the Galerkin method. It is shown that the error in the Galerkin approximation decays in the energy norm almost exponentially (i.e., super-algebraicly) with respect to the degrees of freedom $N$. When length scales "`separate" and the microstructure is sufficiently fine with respect to the length scale of the domain $\omega_i$ it is shown that homogenization theory can be used to construct local approximation spaces with exponentially decreasing error in the pre-asymtotic regime.Comment: 30 pages, 6 figures, updated references, sections 3 and 4 typos corrected, minor text revision, results unchange

Sampling-based Approximations with Quantitative Performance for the Probabilistic Reach-Avoid Problem over General Markov Processes

This article deals with stochastic processes endowed with the Markov (memoryless) property and evolving over general (uncountable) state spaces. The models further depend on a non-deterministic quantity in the form of a control input, which can be selected to affect the probabilistic dynamics. We address the computation of maximal reach-avoid specifications, together with the synthesis of the corresponding optimal controllers. The reach-avoid specification deals with assessing the likelihood that any finite-horizon trajectory of the model enters a given goal set, while avoiding a given set of undesired states. This article newly provides an approximate computational scheme for the reach-avoid specification based on the Fitted Value Iteration algorithm, which hinges on random sample extractions, and gives a-priori computable formal probabilistic bounds on the error made by the approximation algorithm: as such, the output of the numerical scheme is quantitatively assessed and thus meaningful for safety-critical applications. Furthermore, we provide tighter probabilistic error bounds that are sample-based. The overall computational scheme is put in relationship with alternative approximation algorithms in the literature, and finally its performance is practically assessed over a benchmark case study

Least-squares methods for policy iteration

Approximate reinforcement learning deals with the essential problem of applying reinforcement learning in large and continuous state-action spaces, by using function approximators to represent the solution. This chapter reviews least-squares methods for policy iteration, an important class of algorithms for approximate reinforcement learning. We discuss three techniques for solving the core, policy evaluation component of policy iteration, called: least-squares temporal difference, least-squares policy evaluation, and Bellman residual minimization. We introduce these techniques starting from their general mathematical principles and detailing them down to fully specified algorithms. We pay attention to online variants of policy iteration, and provide a numerical example highlighting the behavior of representative offline and online methods. For the policy evaluation component as well as for the overall resulting approximate policy iteration, we provide guarantees on the performance obtained asymptotically, as the number of samples processed and iterations executed grows to infinity. We also provide finite-sample results, which apply when a finite number of samples and iterations are considered. Finally, we outline several extensions and improvements to the techniques and methods reviewed

Volumetric real-time particle-based representation of large unstructured tetrahedral polygon meshes

In this paper we propose a particle-based volume rendering approach for unstructured, three-dimensional, tetrahedral polygon meshes. We stochastically generate millions of particles per second and project them on the screen in real-time. In contrast to previous rendering techniques of tetrahedral volume meshes, our method does not need a prior depth sorting of geometry. Instead, the rendered image is generated by choosing particles closest to the camera. Furthermore, we use spatial superimposing. Each pixel is constructed from multiple subpixels. This approach not only increases projection accuracy, but allows also a combination of subpixels into one superpixel that creates the well-known translucency effect of volume rendering. We show that our method is fast enough for the visualization of unstructured three-dimensional grids with hard real-time constraints and that it scales well for a high number of particles

Imitrob: Imitation Learning Dataset for Training and Evaluating 6D Object Pose Estimators

This paper introduces a dataset for training and evaluating methods for 6D pose estimation of hand-held tools in task demonstrations captured by a standard RGB camera. Despite the significant progress of 6D pose estimation methods, their performance is usually limited for heavily occluded objects, which is a common case in imitation learning where the object is typically partially occluded by the manipulating hand. Currently, there is a lack of datasets that would enable the development of robust 6D pose estimation methods for these conditions. To overcome this problem, we collect a new dataset (Imitrob) aimed at 6D pose estimation in imitation learning and other applications where a human holds a tool and performs a task. The dataset contains image sequences of three different tools and six manipulation tasks with two camera viewpoints, four human subjects, and left/right hand. Each image is accompanied by an accurate ground truth measurement of the 6D object pose, obtained by the HTC Vive motion tracking device. The use of the dataset is demonstrated by training and evaluating a recent 6D object pose estimation method (DOPE) in various setups. The dataset and code are publicly available at http://imitrob.ciirc.cvut.cz/imitrobdataset.php