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We propose an implicit neural formulation of optimal transport that eliminates adversarial min--max optimization and multi-network architectures commonly used in existing approaches. Our key idea is to parameterize a single potential in the…
Moment optimization techniques have been recently proposed to solve globally various classes of optimal control problems. As those methods return truncated moment sequences of occupation measures, this paper explores a numeric method for…
Motivated, in particular, by the entropy-regularized optimal transport problem, we consider convex optimization problems with linear equality constraints, where the dual objective has Lipschitz $p$-th order derivatives, and develop two…
Mesh refinement in pseudospectral (PS) optimal control is embarrassingly easy --- simply increase the order $N$ of the Lagrange interpolating polynomial and the mathematics of convergence automates the distribution of the grid points.…
The article presents a computationally effective algorithm for calculating the multiresolution discrete Fourier transform (MrDFT). The algorithm is based on the idea of reducing the computational complexity which was introduced by Wen and…
We consider the numerical solution of the scattering of time-harmonic plane waves from an infinite periodic array of reflection or transmission obstacles in a homogeneous background medium, in two dimensions. Boundary integral formulations…
Fast exact algorithms are known for Hamiltonian paths in undirected and directed bipartite graphs through elegant though involved algorithms that are quite different from each other. We devise algorithms that are simple and similar to each…
Unbalanced optimal transport (UOT) extends optimal transport (OT) to take into account mass variations to compare distributions. This is crucial to make OT successful in ML applications, making it robust to data normalization and outliers.…
The optimal mass transport problem gives a geometric framework for optimal allocation, and has recently gained significant interest in application areas such as signal processing, image processing, and computer vision. Even though it can be…
In our previous work [arXiv:2202.10042], the complexity of Sinkhorn iteration is reduced from $O(N^2)$ to the optimal $O(N)$ by leveraging the special structure of the kernel matrix. In this paper, we explore the special structure of kernel…
We consider the problem of generating a time-optimal quadrotor trajectory that attains a set of prescribed waypoints. This problem is challenging since the optimal trajectory is located on the boundary of the set of dynamically feasible…
This paper improves the convergence and robustness of a multigrid-based solver for the cross sections of the driven Schroedinger equation. Adding an Coupled Channel Correction Step (CCCS) after each multigrid (MG) V-cycle efficiently…
In [Q. Liao et al., Commun. Math. Sci., 20(2022)], a linear-time Sinkhorn algorithm is developed based on dynamic programming, which significantly reduces the computational complexity involved in solving optimal transport problems. However,…
We present a fast direct solver for structured linear systems based on multilevel matrix compression. Using the recently developed interpolative decomposition of a low-rank matrix in a recursive manner, we embed an approximation of the…
In this paper, we propose a class of super-schemes for efficiently solving nonlinear unconstrained optimization problems. The proposed approach introduces two novel choices of step-size parameters, leading to efficient descent directions…
We present an efficient transcription method for highly oscillatory optimal control problems. For these problems, the optimal state trajectory consists of fast oscillations that change slowly over the time horizon. Out of a large number of…
In this work we construct multigrid preconditioners to accelerate the solution process of a linear-quadratic optimal control problem constrained by the Stokes system. The first order optimality conditions of the control problem form a…
Motion planning is still an open problem for many disciplines, e.g., robotics, autonomous driving, due to their need for high computational resources that hinder real-time, efficient decision-making. A class of methods striving to provide…
This paper is concerned with the theoretical and computational development of a new class of nonlinear filtering algorithms called the optimal transport particle filters (OTPF). The algorithm is based on a recently introduced variational…
Most common Optimal Transport (OT) solvers are currently based on an approximation of underlying measures by discrete measures. However, it is sometimes relevant to work only with moments of measures instead of the measure itself, and many…