arXiv subjects

Alexander Vladimirsky

Publications and source records attributed to Alexander Vladimirsky.

28 records · Page 2Linked to original sources

arXiv2017

A bi-criteria path planning algorithm for robotics applications

Realistic path planning applications often require optimizing with respect to several criteria simultaneously. Here we introduce an efficient algorithm for bi-criteria path planning on graphs. Our approach is based on augmenting the state space to keep track of the "budget" remaining to satisfy the constraints on secondary cost. The resulting augmented graph is acyclic and the primary cost can be then minimized by a simple upward sweep through budget levels. The efficiency and accuracy of our algorithm is tested on Probabilistic Roadmap graphs to minimize the distance of travel subject to a constraint on the overall threat exposure of the robot. We also present the results from field experiments illustrating the use of this approach on realistic robotic systems.

cs.RO↗

arXiv2015

Piecewise-Deterministic Optimal Path Planning

We consider piecewise-deterministic optimal control problems in which the environment randomly switches among several deterministic modes, and the goal is to optimize the expected cost up to the termination while taking the likelihood of future mode-switches into account. The dynamic programming approach yields a weakly-coupled system of Hamilton-Jacobi-Bellman PDEs satisfied by the value functions of individual modes. We derive and implement semi-Lagrangian and Eulerian numerical schemes for that system. We further recover simpler, "asymptotically optimal" controls and test their performance in non-asymptotic regimes. Our approach is illustrated on a simple anisotropic path-planning problem: the time-optimal control for a boat affected by randomly switching winds.

math.OC↗

arXiv2014

Causal Domain Restriction for Eikonal Equations

Many applications require efficient methods for solving continuous shortest path problems. Such paths can be viewed as characteristics of static Hamilton-Jacobi equations. Several fast numerical algorithms have been developed to solve such equations on the whole domain. In this paper we consider a somewhat different problem, where the solution is needed at one specific point, so we restrict the computations to a neighborhood of the characteristic. We explain how heuristic under/over-estimate functions can be used to obtain a causal domain restriction, significantly decreasing the computational work without sacrificing convergence under mesh refinement. The discussed techniques are inspired by an alternative version of the classical A* algorithm on graphs. We illustrate the advantages of our approach on continuous isotropic examples in 2D and 3D. We compare its efficiency and accuracy to previous domain restriction techniques. We also analyze the behavior of errors under the grid refinement and show how Lagrangian (Pontryagin's Maximum Principle-based) computations can be used to enhance our method.

math.NA↗

arXiv2014

A parallel Heap-Cell Method for Eikonal equations

Numerous applications of Eikonal equations prompted the development of many efficient numerical algorithms. The Heap-Cell Method (HCM) is a recent serial two-scale technique that has been shown to have advantages over other serial state-of-the-art solvers for a wide range of problems. This paper presents a parallelization of HCM for a shared memory architecture. The numerical experiments in $R^3$ show that the parallel HCM exhibits good algorithmic behavior and scales well, resulting in a very fast and practical solver. We further explore the influence on performance and scaling of data precision, early termination criteria, and the hardware architecture. A shorter version of this manuscript (omitting these more detailed tests) has been submitted to SIAM Journal on Scientific Computing in 2012.

math.NA↗

arXiv2014

Optimal control with reset-renewable resources

We consider both discrete and continuous control problems constrained by a fixed budget of some resource, which may be renewed upon entering a preferred subset of the state space. In the discrete case, we consider both deterministic and stochastic shortest path problems with full budget resets in all preferred nodes. In the continuous case, we derive augmented PDEs of optimal control, which are then solved numerically on the extended state space with a full/instantaneous budget reset on the preferred subset. We introduce an iterative algorithm for solving these problems efficiently. The method's performance is demonstrated on a range of computational examples, including the optimal path planning with constraints on prolonged visibility by a static enemy observer. In addition, we also develop an algorithm that works on the original state space to solve a related but simpler problem: finding the subsets of the domain "reachable-within-the-budget". This manuscript is an extended version of the paper accepted for publication by SIAM J. on Control and Optimization. In the journal version, Section 3 and the Appendix were omitted due to space limitations.

math.OC↗

arXiv2013

Deterministic control of randomly-terminated processes

We consider both discrete and continuous "uncertain horizon" deterministic control processes, for which the termination time is a random variable. We examine the dynamic programming equations for the value function of such processes, explore their connections to infinite-horizon and optimal-stopping problems, and derive sufficient conditions for the applicability of non-iterative (label-setting) methods. In the continuous case, the resulting PDE has a free boundary, on which all characteristic curves originate. The causal properties of "uncertain horizon" problems can be exploited to design efficient numerical algorithms: we derive causal semi-Lagrangian and Eulerian discretizations for the isotropic randomly-terminated problems, and use them to build a modified version of the Fast Marching Method. We illustrate our approach using numerical examples from optimal idle-time processing and expected response-time minimization.

math.OC↗

arXiv2013

A fast implicit method for time-dependent Hamilton-Jacobi PDEs

We present a new efficient computational approach for time-dependent first-order Hamilton-Jacobi-Bellman PDEs. Since our method is based on a time-implicit Eulerian discretization, the numerical scheme is unconditionally stable, but discretized equations for each time-slice are coupled and non-linear. We show that the same system can be re-interpreted as a discretization of a static Hamilton-Jacobi-Bellman PDE on the same physical domain. The latter was shown to be ``causal' in [Vladimirsky 2006], making fast (non-iterative)methods applicable. The implicit discretization results in higher computational cost per time slice compared to the explicit time marching. However, the latter is subject to a CFL-stability condition, and the implicit approach becomes significantly more efficient whenever the accuracy demands on the time-step are less restrictive than the stability. We also present a hybrid method, which aims to combine the advantages of both the explicit and implicit discretizations. We demonstrate the efficiency of our approach using several examples in optimal control of isotropic fixed-horizon processes.

math.NA↗

arXiv2011

Fast two-scale methods for Eikonal equations

Fast Marching and Fast Sweeping are the two most commonly used methods for solving the Eikonal equation. Each of these methods performs best on a different set of problems. Fast Sweeping, for example, will outperform Fast Marching on problems where the characteristics are largely straight lines. Fast Marching, on the other hand, is usually more efficient than Fast Sweeping on problems where characteristics frequently change their directions and on domains with complicated geometry. In this paper we explore the possibility of combining the best features of both of these approaches, by using marching on a coarser scale and sweeping on a finer scale. We present three new hybrid methods based on this idea and illustrate their properties on several numerical examples with continuous and piecewise-constant speed functions in $R^2$.

math.NA↗

arXiv2009

An efficient method for multiobjective optimal control and optimal control subject to integral constraints

We introduce a new and efficient numerical method for multicriterion optimal control and single criterion optimal control under integral constraints. The approach is based on extending the state space to include information on a "budget" remaining to satisfy each constraint; the augmented Hamilton-Jacobi-Bellman PDE is then solved numerically. The efficiency of our approach hinges on the causality in that PDE, i.e., the monotonicity of characteristic curves in one of the newly added dimensions. A semi-Lagrangian "marching" method is used to approximate the discontinuous viscosity solution efficiently. We compare this to a recently introduced "weighted sum" based algorithm for the same problem. We illustrate our method using examples from flight path planning and robotic navigation in the presence of friendly and adversarial observers.

math.OC↗

arXiv2008

Label-setting methods for Multimode Stochastic Shortest Path problems on graphs

Stochastic shortest path (SSP) problems arise in a variety of discrete stochastic control contexts. An optimal solutions to such a problem is typically computed using the value function, which can be found by solving the corresponding dynamic programming equations. In the deterministic case, these equations can be often solved by the highly efficient label-setting methods (such as Dijkstra's and Dial's algorithms). In this paper we define and study a class of Multimode Stochastic Shortest Path problems and develop sufficient conditions for the applicability of label-setting methods. We illustrate our approach on a number of discrete stochastic control examples. We also discuss the relationship of SSPs with discretizations of static Hamilton-Jacobi equations and provide an alternative derivation for several fast (non-iterative) numerical methods for these PDEs.

math.OC↗