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Finding nearly accurate distance between two or more nearly intersecting three-dimensional (3D) objects is vital especially for collision determination such as in virtual surgeon simulation and real-time car crash simulation. Instead of…
We design an additive approximation scheme for estimating the cost of the min-weight bipartite matching problem: given a bipartite graph with non-negative edge costs and $\varepsilon > 0$, our algorithm estimates the cost of matching all…
This paper proposes a solution for multiple-impulse orbital maneuvers near circular orbits for special cases where orbital observations are not globally available and the spacecraft is being observed through a limited window from a ground…
We present an algorithm for the rapid numerical integration of smooth, time-periodic differential equations with small nonlinearity, particularly suited to problems with small dissipation. The emphasis is on speed without compromising…
Initial orbit determination (IOD) from line-of-sight (i.e., bearing) measurements is a classical problem in astrodynamics. Indeed, there are many well-established methods for performing the IOD task when given three line-of-sight…
We study the complexity of approximating the multimarginal optimal transport (MOT) distance, a generalization of the classical optimal transport distance, considered here between $m$ discrete probability distributions supported each on $n$…
Consider a set $P$ of $n$ points in $\mathbb{R}^d$. In the discrete median line segment problem, the objective is to find a line segment bounded by a pair of points in $P$ such that the sum of the Euclidean distances from $P$ to the line…
Maximum mean discrepancy (MMD) has been widely employed to measure the distance between probability distributions. In this paper, we propose using MMD to solve continuous multi-objective optimization problems (MOPs). For solving MOPs, a…
Fitting concentric geometric objects to digitized data is an important problem in many areas such as iris detection, autonomous navigation, and industrial robotics operations. There are two common approaches to fitting geometric shapes to…
Given $n$ points in a $d$ dimensional Euclidean space, the Minimum Enclosing Ball (MEB) problem is to find the ball with the smallest radius which contains all $n$ points. We give a $O(nd\Qcal/\sqrt{\epsilon})$ approximation algorithm for…
We propose a new algorithm to approximate the Earth Mover's distance (EMD). Our main idea is motivated by the theory of optimal transport, in which EMD can be reformulated as a familiar $L_1$ type minimization. We use a regularization which…
Elliptic partial differential equations are important both from application and analysis points of views. In this paper we apply the Closest Point Method to solving elliptic equations on general curved surfaces. Based on the closest point…
In this paper, we look into the minimum obstacle displacement (MOD) planning problem from a mobile robot motion planning perspective. This problem finds an optimal path to goal by displacing movable obstacles when no path exists due to…
We present a method for guaranteed collision detection with toleranced motions. The basic idea is to consider the motion as a curve in the 12-dimensional space of affine displacements, endowed with an object-oriented Euclidean metric, and…
Computing collision-free trajectories is of prime importance for safe navigation. We present an approach for computing the collision probability under Gaussian distributed motion and sensing uncertainty with the robot and static obstacle…
Euclidean distance matrix optimization with ordinal constraints (EDMOC) has found important applications in sensor network localization and molecular conformation. It can also be viewed as a matrix formulation of multidimensional scaling,…
We give a reduction from $(1+\varepsilon)$-approximate Earth Mover's Distance (EMD) to $(1+\varepsilon)$-approximate Closest Pair (CP). As a consequence, we improve the fastest known approximation algorithm for high-dimensional EMD. Here,…
Minimizing the Euclidean distance to a set arises frequently in applications. When the set is algebraic, a measure of complexity of this optimization problem is its number of critical points. In this paper we provide a general framework to…
A method is developed to calculate collision probability in this paper. Based on the encounter geometric features of space objects, it is reasonable to separate the radial orbital motions from that in the cross section for most encounter…
A small variation of the circular shape of the hodograph theorem states that for every elliptical solution of the two-body problem, it is possible to find an appropriate inertial frame such that the speed of the bodies is constant. We use…