Trajectory planning tasks for non-holonomic or collaborative systems are naturally modeled by state spaces with non-Euclidean metrics. However, existing proofs of convergence for sample-based motion planners only consider the setting of Euclidean state spaces. We resolve this issue by formulating a flexible framework and set of assumptions for which the widely-used PRM*, RRT, and RRT* algorithms remain asymptotically optimal in the non-Euclidean setting. The framework is compatible with collaborative trajectory planning: given a fleet of robotic systems that individually satisfy our assumptions, we show that the corresponding collaborative system again satisfies the assumptions and therefore has guaranteed convergence for the trajectory-finding methods. Our joint state space construction builds in a coupling parameter $1\leq p\leq \infty$, which interpolates between a preference for minimizing total energy at one extreme and a preference for minimizing the travel time at the opposite extreme. We illustrate our theory with trajectory planning for simple coupled systems, fleets of Reeds-Shepp vehicles, and a highly non-Euclidean fractal space.
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In the logic synthesis stage, structure transformations in the synthesis tool need to be combined into optimization sequences and act on the circuit to meet the specified circuit area and delay. However, logic synthesis optimization sequences are time-consuming to run, and predicting the quality of the results (QoR) against the synthesis optimization sequence for a circuit can help engineers find a better optimization sequence faster. In this work, we propose a deep learning method to predict the QoR of unseen circuit-optimization sequences pairs. Specifically, the structure transformations are translated into vectors by embedding methods and advanced natural language processing (NLP) technology (Transformer) is used to extract the features of the optimization sequences. In addition, to enable the prediction process of the model to be generalized from circuit to circuit, the graph representation of the circuit is represented as an adjacency matrix and a feature matrix. Graph neural networks(GNN) are used to extract the structural features of the circuits. For this problem, the Transformer and three typical GNNs are used. Furthermore, the Transformer and GNNs are adopted as a joint learning policy for the QoR prediction of the unseen circuit-optimization sequences. The methods resulting from the combination of Transformer and GNNs are benchmarked. The experimental results show that the joint learning of Transformer and GraphSage gives the best results. The Mean Absolute Error (MAE) of the predicted result is 0.412.
To minimize the average of a set of log-convex functions, the stochastic Newton method iteratively updates its estimate using subsampled versions of the full objective's gradient and Hessian. We contextualize this optimization problem as sequential Bayesian inference on a latent state-space model with a discriminatively-specified observation process. Applying Bayesian filtering then yields a novel optimization algorithm that considers the entire history of gradients and Hessians when forming an update. We establish matrix-based conditions under which the effect of older observations diminishes over time, in a manner analogous to Polyak's heavy ball momentum. We illustrate various aspects of our approach with an example and review other relevant innovations for the stochastic Newton method.
We consider the solution to the biharmonic equation in mixed form discretized by the Hybrid High-Order (HHO) methods. The two resulting second-order elliptic problems can be decoupled via the introduction of a new unknown, corresponding to the boundary value of the solution of the first Laplacian problem. This technique yields a global linear problem that can be solved iteratively via a Krylov-type method. More precisely, at each iteration of the scheme, two second-order elliptic problems have to be solved, and a normal derivative on the boundary has to be computed. In this work, we specialize this scheme for the HHO discretization. To this aim, an explicit technique to compute the discrete normal derivative of an HHO solution of a Laplacian problem is proposed. Moreover, we show that the resulting discrete scheme is well-posed. Finally, a new preconditioner is designed to speed up the convergence of the Krylov method. Numerical experiments assessing the performance of the proposed iterative algorithm on both two- and three-dimensional test cases are presented.
The distributed task allocation problem, as one of the most interesting distributed optimization challenges, has received considerable research attention recently. Previous works mainly focused on the task allocation problem in a population of individuals, where there are no constraints for affording task amounts. The latter condition, however, cannot always be hold. In this paper, we study the task allocation problem with constraints of task allocation in a game-theoretical framework. We assume that each individual can afford different amounts of task and the cost function is convex. To investigate the problem in the framework of population games, we construct a potential game and calculate the fitness function for each individual. We prove that when the Nash equilibrium point in the potential game is in the feasible solutions for the limited task allocation problem, the Nash equilibrium point is the unique globally optimal solution. Otherwise, we also derive analytically the unique globally optimal solution. In addition, in order to confirm our theoretical results, we consider the exponential and quadratic forms of cost function for each agent. Two algorithms with the mentioned representative cost functions are proposed to numerically seek the optimal solution to the limited task problems. We further perform Monte Carlo simulations which provide agreeing results with our analytical calculations.
Statistical depth functions provide measures of the outlyingness, or centrality, of the elements of a space with respect to a distribution. It is a nonparametric concept applicable to spaces of any dimension, for instance, multivariate and functional. Liu and Singh (1993) presented a multivariate two-sample test based on depth-ranks. We dedicate this paper to improving the power of the associated test statistic and incorporating its applicability to functional data. In doing so, we obtain a more natural test statistic that is symmetric in both samples. We derive the null asymptotic of the proposed test statistic, also proving the validity of the testing procedure for functional data. Finally, the finite sample performance of the test for functional data is illustrated by means of a simulation study and a real data analysis on annual temperature curves of ocean drifters is executed.
A standard approach to solve ordinary differential equations, when they describe dynamical systems, is to adopt a Runge-Kutta or related scheme. Such schemes, however, are not applicable to the large class of equations which do not constitute dynamical systems. In several physical systems, we encounter integro-differential equations with memory terms where the time derivative of a state variable at a given time depends on all past states of the system. Secondly, there are equations whose solutions do not have well-defined Taylor series expansion. The Maxey-Riley-Gatignol equation, which describes the dynamics of an inertial particle in nonuniform and unsteady flow, displays both challenges. We use it as a test bed to address the questions we raise, but our method may be applied to all equations of this class. We show that the Maxey-Riley-Gatignol equation can be embedded into an extended Markovian system which is constructed by introducing a new dynamical co-evolving state variable that encodes memory of past states. We develop a Runge-Kutta algorithm for the resultant Markovian system. The form of the kernels involved in deriving the Runge-Kutta scheme necessitates the use of an expansion in powers of $t^{1/2}$. Our approach naturally inherits the benefits of standard time-integrators, namely a constant memory storage cost, a linear growth of operational effort with simulation time, and the ability to restart a simulation with the final state as the new initial condition.
Fixed-point iteration algorithms like RTA (response time analysis) and QPA (quick processor-demand analysis) are arguably the most popular ways of solving schedulability problems for preemptive uniprocessor FP (fixed-priority) and EDF (earliest-deadline-first) systems. Several IP (integer program) formulations have also been proposed for these problems, but it is unclear whether the algorithms for solving these formulations are related to RTA and QPA. By discovering connections between the problems and the algorithms, we show that RTA and QPA are, in fact, suboptimal cutting-plane algorithms for specific IP formulations of FP and EDF schedulability, where optimality is defined with respect to convergence rate. We propose optimal cutting-plane algorithms for these IP formulations. We compare the new algorithms with RTA and QPA on large collections of synthetic systems to gauge the improvement in convergence rates and running times.
The forecasting and computation of the stability of chaotic systems from partial observations are tasks for which traditional equation-based methods may not be suitable. In this computational paper, we propose data-driven methods to (i) infer the dynamics of unobserved (hidden) chaotic variables (full-state reconstruction); (ii) time forecast the evolution of the full state; and (iii) infer the stability properties of the full state. The tasks are performed with long short-term memory (LSTM) networks, which are trained with observations (data) limited to only part of the state: (i) the low-to-high resolution LSTM (LH-LSTM), which takes partial observations as training input, and requires access to the full system state when computing the loss; and (ii) the physics-informed LSTM (PI-LSTM), which is designed to combine partial observations with the integral formulation of the dynamical system's evolution equations. First, we derive the Jacobian of the LSTMs. Second, we analyse a chaotic partial differential equation, the Kuramoto-Sivashinsky (KS), and the Lorenz-96 system. We show that the proposed networks can forecast the hidden variables, both time-accurately and statistically. The Lyapunov exponents and covariant Lyapunov vectors, which characterize the stability of the chaotic attractors, are correctly inferred from partial observations. Third, the PI-LSTM outperforms the LH-LSTM by successfully reconstructing the hidden chaotic dynamics when the input dimension is smaller or similar to the Kaplan-Yorke dimension of the attractor. This work opens new opportunities for reconstructing the full state, inferring hidden variables, and computing the stability of chaotic systems from partial data.
Barrier terms for Incremental Potential Contact (IPC) energy are crucial for maintaining an intersection and inversion free simulation trajectory. However, existing formulations which directly use distance for measuring the state of contact can restrict implementation design and performance. This is because numerical eigendecompositions are required for guaranteeing positive semi-definiteness of the Hessian of the barrier energy during optimization with Projected-Newton solvers, and alternative Gauss-Newton methods suffer from significantly reduced convergence rates. We rewrite the barrier function of IPC to derive an efficient approximation its Hessian, which can then be used for simultaneous construction and projection to positive semi-definite state. The key idea is to formulate a simplicial geometric measure of contact using mesh boundary elements, where analytic eigensystems are obtained for minimising the proposed barrier energy with superlinear convergence to retain much of the advantage of Newton-type methods. Our approach is suitable for standard second order unconstrained optimization strategies for IPC based collision processing, minimizing nonlinear nonconvex functions where the Hessian may be indefinite. The result is a 3-times speedup over the standard full-space IPC barrier formulation based on direct Euclidean distance measures. We further apply our analytic proxy eigensystems to produce an entirely GPU-based implementation of IPC with significant further acceleration.
This paper develops power series expansions of a general class of moment functions, including transition densities and option prices, of continuous-time Markov processes, including jump--diffusions. The proposed expansions extend the ones in Kristensen and Mele (2011) to cover general Markov processes. We demonstrate that the class of expansions nests the transition density and option price expansions developed in Yang, Chen, and Wan (2019) and Wan and Yang (2021) as special cases, thereby connecting seemingly different ideas in a unified framework. We show how the general expansion can be implemented for fully general jump--diffusion models. We provide a new theory for the validity of the expansions which shows that series expansions are not guaranteed to converge as more terms are added in general. Thus, these methods should be used with caution. At the same time, the numerical studies in this paper demonstrate good performance of the proposed implementation in practice when a small number of terms are included.
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