Abstract. Parti-game is a new algorithm for learning feasible trajectories to goal regions in high dimensional continuous state-spaces. In high dimensions it is essential that learning does not plan uniformly over a state-space. Parti-game maintains a decision-tree partitioning of state-space and applies techniques from game-theory and computational geometry to e ciently and adaptively concentrate high resolution only on critical areas. The currentversion of the algorithm is designed to nd feasible paths or trajectories to goal regions in high dimensional spaces. Future versions will be designed to nd a solution that optimizes a real-valued criterion. Many simulated problems have been tested, ranging from two-dimensional to nine-dimensional state-spaces, including mazes, path planning, non-linear dynamics, and planar snake robots in restricted spaces. In all cases, a good solution is found in less than ten trials and a few minutes.

While 2D and 3D vector fields are ubiquitous in computational sciences, their use in graphics is often limited to regular grids, where computations are easily handled through finite-difference methods. In this paper, we propose a set of simple and accurate tools for the analysis of 3D discrete vector fields on arbitrary tetrahedral grids. We introduce a variational, multiscale decomposition of vector fields into three intuitive components: a divergence-free part, a curl-free part, and a harmonic part. We show how our discrete approach matches its well-known smooth analog, called the HelmotzHodge decomposition, and that the resulting computational tools have very intuitive geometric interpretation. We demonstrate the versatility of these tools in a series of applications, ranging from data visualization to fluid and deformable object simulation.

Decision making usually involves choosing among different courses of action over a broad range of time scales. For instance, a person planning a trip to a distant location makes high-level decisions regarding what means of transportation to use, but also chooses low-level actions, such as the movements for getting into a car. The problem of picking an appropriate time scale for reasoning and learning has been explored in artificial intelligence, control theory and robotics. In this dissertation we develop a framework that allows novel solutions to this problem, in the context of Markov Decision Processes (MDPs) and reinforcement learning. In this dissertation, we present a general framework for prediction, control and learning at multipl...

Scientific and engineering applications often involve structured meshes. These meshes may be nested (for multigrid codes) and/or irregularly coupled (called multiblock or irregularly coupled regular mesh problems). In this paper, we present a combined runtime and compile-time approach for parallelizing these applications on distributed memory parallel machines in an efficient and machine-independent fashion. Wehave designed and implemented a runtime library which can be used to port these applications on distributed memory machines. The library is currently implemented on several different systems. To further ease the task of application programmers, wehave developed methods for integrating this runtime library with compilers for HPF-like parallel programming languages. We discuss howwehaveintegrated this runtime library with the Fortran 90D compiler being developed at Syracuse University. We present experimental results to demonstrate the efficacy of our approach. Wehave exper...

In this paper we present a method for solving the equations governing time-dependent, variable density incompressible flow in two or three dimensions on an adaptive hierarchy of grids. The method is based on a projection formulation in which we first solve advectiondiffusion equations to predict intermediate velocities, and then project these velocities onto a space of approximately divergence-free vector fields. Our treatment of the first step uses a specialized second-order upwind method for differencing the nonlinear convection terms that provides a robust treatment of these terms suitable for inviscid and high Reynolds number flow. Density and other scalars are advected in such a way as to maintain conservation, if appropriate, and free-stream preservation. Our approach to adaptive refinement uses a nested hierarchy of logically-rectangular grids with simultaneous refinement of the grids in both space and time. The integration algorithm on the grid hierarchy is a recursive procedur...

Scientific and engineering applications often involve structured meshes. These meshes may be nested (for multigrid or adaptive codes) and/or irregularly coupled(called Irregularly CoupledRegular Meshes). We have designed and implemented a runtime library for parallelizing this general class of applications on distributed memory parallel machines in an efficient and machine independent manner. In this paper we present how this runtime library can be integrated with compilers for High PerformanceFortran (HPF) style parallel programming languages. We discuss how we have integrated this runtime library with the Fortran 90D compiler being developed at Syracuse University and provide experimental data on a block structured NavierStokes solver template and a small multigrid example parallelized using this compiler and run on an Intel iPSC/860. We show that the compiler parallelizedcode performs within 20% of the code parallelized by inserting calls to the runtime library manually.

. An abstract framework of auxiliary space method is proposed and, as an application, an optimal multigrid technique is developed for general unstructured grids. The auxiliary space method is a (nonnested) two level preconditioning technique based on a simple relaxation scheme (smoother) and an auxiliary space (that may be roughly understood as a nonnested coarser space). An optimal multigrid preconditioner is then obtained for a discretized partial differential operator defined on an unstructured grid by using an auxiliary space defined on a more structured grid in which a further nested multigrid method can be naturally applied. This new technique make it possible to apply multigrid methods to general unstructured grids without too much more programming effort than traditional solution methods. Some simple examples are also given to illustrate the abstract theory and for instance the Morley finite element space is used as an auxiliary space to construct a preconditioner for Argyris ...

Abstract. We analyze the convergence rate of an asynchronous space decomposition method for constrained convex minimization in a reflexive Banach space. This method includes as special cases parallel domain decomposition methods and multigrid methods for solving elliptic partial differential equations. In particular, the method generalizes the additive Schwarz domain decomposition methods to allow for asynchronous updates. It also generalizes the BPX multigrid method to allow for use as solvers instead of as preconditioners, possibly with asynchronous updates, and is applicable to nonlinear problems. Applications to an overlapping domain decomposition for obstacle problems are also studied. The method of this work is also closely related to relaxation methods for nonlinear network flow. Accordingly, we specialize our convergence rate results to the above methods. The asynchronous method is implementable in a multiprocessor system, allowing for communication and computation delays among the processors. 1.

: Variational methods have been employed with considerable success in computer vision, particularly for surface reconstruction problems. Formulations of this type require the solution of computationally complex Euler-Lagrange partial differential equations (PDEs) to obtain the desired reconstructions. Further, the calculation of reconstruction error covariances for such approaches are usually neglected. In this paper we describe a computationally efficient multiscale approach to surface reconstruction which differs fundamentally from other multiresolution methods that are used to solve the Euler-Lagrange PDEs. Instead, we interpret the variational problem as a statistical estimation problem in order to define a nearby, but slightly different, multiscale estimation problem that admits efficient solutions for both surface reconstruction and the calculation of error statistics. In particular, the membrane and thin-plate variational models for surfaces are interpreted as 1=f 2...

This paper presents an adaptive technique to animate deformable bodies in real-time. In contrast to most previous work, we introduce a multi-resolution model that locally refines or simplifies the simulated object over time in order to optimize the computational effort. We use the mixed Finite-Volume/Finite-Element method to derive fast, local discrete differential operators over irregular grids with tight error bounds. The linear elasticity equations can be simulated using an arbitrary non-nested hierarchy of volumetric meshes, allowing the computation load to be automatically concentrated where and when needed. Real-time simulation, with a guaranteed frame rate, can be achieved as demonstrated through a series of examples on our video.