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Dynamical low-rank algorithms are a class of numerical methods that compute lowrank approximations of dynamical systems. This is accomplished by projecting the dynamics onto a low-dimensional manifold and writing the solution directly in terms of the low-rank factors. The approach has been successfully applied to many types of differential equations. Recently, efficient dynamical low-rank algorithms have been proposed in [L. Einkemmer, A Low-Rank Algorithm for Weakly Compressible Flow, arXiv:1804.04561, 2018; L. Einkemmer and C. Lubich, SIAM J. Sci. Comput., 40 (2018), pp. B1330-B1360] to treat kinetic equations, including the Vlasov-Poisson and the Boltzmann equation. There it was demonstrated that the methods are able to capture the lowrank structure of the solution and significantly reduce numerical cost, while often maintaining high accuracy. However, no numerical analysis is currently available. In this paper, we perform an error analysis for a dynamical low-rank algorithm applied to the multiscale linear Boltzmann equation (a classical model in kinetic theory) to showcase the validity of the application of dynamical lowrank algorithms to kinetic theory. The equation, in its parabolic regime, is known to be rank 1 theoretically, and we will prove that the scheme can dynamically and automatically capture this low-rank structure. This work thus serves as the first mathematical error analysis for a dynamical low-rank approximation applied to a kinetic problem. [ABSTRACT FROM AUTHOR]

The Euler genus of a graph is a fundamental and well-studied parameter in graph theory and topology. Computing it has been shown to be NP-hard by Thomassen [J. Algorithms, 10 (1989), pp. 568{576; J. Combin. Theory, Ser. B, 57 (1993), pp. 196{206], and it is known to be fixed-parameter tractable. However, the approximability of the Euler genus is wide open. While the existence of an O(1)-approximation is not ruled out, only an O( p n)-approximation [J. Chen, S. P. Kanchi, and A. Kanevsky, Inform. Process. Lett., 61 (1997), pp. 317{322] is known even in bounded-degree graphs. In this paper we give a polynomial-time algorithm which, given a boundeddegree graph of Euler genus g, computes a drawing in a surface of Euler genus gO(1) · logO(1) n. Combined with the upper bound from [J. Chen, S. P. Kanchi, and A. Kanevsky, Inform. Process. Lett., 61 (1997), pp. 317{322], our result also implies a O(n1/2-α)-approximation for some constant α > 0. Using our algorithm for approximating the Euler genus as a subroutine, we obtain, in a uniform fashion, algorithms with approximation ratios of the form OPTO(1) · logO(1) n for several related problems on bounded-degree graphs. These include the problems of orientable genus, crossing number, and planar edge and vertex deletion. Our algorithm and proof of correctness for the crossing number problem are simpler compared to the long and difficult proof in the recent breakthrough by Chuzhoy [Proceedings of the ACM Symposium on Theory of Computing, 2011, pp. 303{312], while essentially obtaining a qualitatively similar result. For planar edge and vertex deletion problems our results are the first to obtain a bound of the form poly(OPT; log n). We also highlight some further applications of our results in the design of algorithms for graphs with small genus. Many such algorithms require that a drawing of the graph is given as part of the input. Our results imply that in several interesting cases, we can implement such algorithms even when the drawing is unknown. [ABSTRACT FROM AUTHOR]

The RKFIT algorithm outlined in [M. Berljafa and S. Güttel, SIAM J. Matrix Anal. Appl., 36 (2015), pp. 894-916] is a Krylov-based approach for solving nonlinear rational least squares problems. This paper puts RKFIT into a general framework, allowing for its extension to nondiagonal rational approximants and a family of approximants sharing a common denominator. Furthermore, we derive a strategy for the degree reduction of the approximants, as well as methods for their conversion to partial fraction form, for the efficient evaluation, and for root-finding. We also discuss similarities and differences between RKFIT and the popular vector fitting algorithm. A MATLAB implementation of RKFIT is provided, and numerical experiments, including the fitting of a multiple-input/multiple-output (MIMO) dynamical system and an optimization problem related to exponential integration, demonstrate its applicability. [ABSTRACT FROM AUTHOR]

Let G be an interval graph and take one of its vertices x. Can we find in linear time a minimum number of vertex disjoint paths of G which cover the vertex set of G and have x as one of their endpoints? This paper provides a positive answer to this problem. In the course of developing such an algorithm, we explore the possibility of getting insight on the path structure of interval graphs via greedy graph searches. [ABSTRACT FROM AUTHOR]

Systems of differential equations with polynomial right-hand sides are very common in applications. On the other hand, their mathematical analysis is very challenging in general, due to the possibility of complex dynamics: multiple basins of attraction, oscillations, and even chaotic dynamics. Even if we restrict our attention to mass-action systems, all of these complex dynamical behaviors are still possible. On the other hand, if a polynomial dynamical system has a weakly reversible deficiency zero (WR0) realization, then its dynamics is known to be remarkably simple: oscillations and chaotic dynamics are ruled out, and, up to linear conservation laws, there exists a single positive steady state, which is asymptotically stable. Here we describe an algorithm for finding WR0 realizations of polynomial dynamical systems, whenever such realizations exist. [ABSTRACT FROM AUTHOR]

The Kaczmarz and Gauss-Seidel methods aim to solve an m x n linear system Xβ = y by iteratively refining the solution estimate; the former uses random rows of X to update β given the corresponding equations and the latter uses random columns of X to update corresponding coordinates in β. Recent work analyzed these algorithms in a parallel comparison for the overcomplete and undercomplete systems, showing convergence to the ordinary least squares (OLS) solution and the minimum Euclidean norm solution, respectively. This paper considers the natural follow-up to the OLS problem--ridge or Tikhonov regularized regression. By viewing them as variants of randomized coordinate descent, we present variants of the randomized Kaczmarz (RK) and randomized Gauss-Siedel (RGS) for solving this system and derive their convergence rates. We prove that a recent proposal, which can be interpreted as randomizing over both rows and columns, is strictly suboptimal--instead, one should always work with randomly selected columns (RGS) when m > n (#rows > #cols) and with randomly selected rows (RK) when n > m (#cols > #rows). [ABSTRACT FROM AUTHOR]

The well-studied "power of two choices" family of algorithms creates balanced allocations of m balls into n bins by, for each ball, selecting a few bins at random and then placing the item in the least-loaded bin. A natural variation is to create an unbalanced allocation by, for each ball, selecting a few bins at random and then placing the ball in the most-loaded bin. Surprisingly, this variation has not been previously studied. This paper introduces this family of unbalanced allocation processes and begins its analysis. The behavior of the bounded m case is analyzed in detail via differential equations and coupling, and some preliminary results for the general case are presented. [ABSTRACT FROM AUTHOR]

Given a set of t ≥ κ +2 words of length n over a κ-letter alphabet, it is proved that there exists a common subsequence among two of them of length at least n/κ + cn1-1/(t-k-2) for some c > 0 depending on κ and t. This is sharp up to the value of c. [ABSTRACT FROM AUTHOR]

Graphs with bounded highway dimension were introduced by Abraham et al. [Proceedings of SODA 2010, pp. 782{793] as a model of transportation networks. We show that any such graph can be embedded into a distribution over bounded treewidth graphs with arbitrarily small distortion. More concretely, given a weighted graph G = (V;E) of constant highway dimension, we show how to randomly compute a weighted graph H = (V;E0) that distorts shortest path distances of G by at most a 1 + ε factor in expectation, and whose treewidth is polylogarithmic in the aspect ratio of G. Our probabilistic embedding implies quasi-polynomial time approximation schemes for a number of optimization problems that naturally arise in transportation networks, including Travelling Salesman, Steiner Tree, and Facility Location. To construct our embedding for low highway dimension graphs we extend Talwar's [Proceedings of STOC 2004, pp. 281-290] embedding of low doubling dimension metrics into bounded treewidth graphs, which generalizes known results for Euclidean metrics. We add several nontrivial ingredients to Talwar's techniques, and in particular thoroughly analyze the structure of low highway dimension graphs. Thus we demonstrate that the geometric toolkit used for Euclidean metrics extends beyond the class of low doubling metrics. [ABSTRACT FROM AUTHOR]

Let G be an n-node simple directed planar graph with nonnegative edge weights. We study the fundamental problems of computing (1) a global cut of G with minimum weight and (2) a cycle of G with minimum weight. The best previously known algorithm for the former problem, running in O(n log³ n) time, can be obtained from the algorithm of Łącki, Nussbaum, Sankowski, and Wulff-Nilsen for single-source all-sinks maximum flows. The best previously known result for the latter problem is the O(n log³ n)-time algorithm of Wulff-Nilsen. By exploiting duality between the two problems in planar graphs, we solve both problems in O(n log n log log n) time via a divideand-conquer algorithm that finds a shortest nondegenerate cycle. The kernel of our result is an O(n log log n)-time algorithm for computing noncrossing shortest paths among nodes well ordered on a common face of a directed plane graph, which is extended from the algorithm of Italiano, Nussbaum, Sankowski, and Wulff-Nilsen for an undirected plane graph. [ABSTRACT FROM AUTHOR]

A class of stochastic algorithms for the numerical treatment of the Wigner equation is introduced. The algorithms are derived using the theory of pure jump processes with a general state space. The class contains several new algorithms as well as some of the algorithms previously considered in the literature. The approximation error and the efficiency of the algorithms are analyzed. Numerical experiments are performed on a benchmark test case, where certain advantages of the new class of algorithms are demonstrated. [ABSTRACT FROM AUTHOR]

This article details a novel numerical scheme to approximate gradient flows for optimal transport (i.e., Wasserstein) metrics. These flows have proved useful to tackle theoretically and numerically nonlinear diffusion equations that model, for instance, porous media or crowd evolutions. These gradient flows define a suitable notion of weak solutions for these evolutions and they can be approximated in a stable way using discrete flows. These discrete flows are implicit Euler time stepping according to the Wasserstein metric. A bottleneck of these approaches is the high computational load induced by the resolution of each step. Indeed, this corresponds to the resolution of a convex optimization problem involving a Wasserstein distance to the previous iterate. Following several recent works on the approximation of Wasserstein distances, we consider a discrete flow induced by an entropic regularization of the transportation coupling. This entropic regularization allows one to trade the initial Wasserstein fidelity term for a Kullback-Leibler divergence, which is easier to deal with numerically. We show how Kullback-Leibler first order proximal schemes and, in particular, Dykstra's algorithm, can be used to compute each step of the regularized flow. The resulting algorithm is both fast, parallelizable, and versatile, because it only requires multiplications by the Gibbs kernel e -c/γ, where c is the ground cost and γ > 0 the regularization strength. On Euclidean domains discretized on a uniform grid, this corresponds to a linear filtering (for instance, a Gaussian filtering when c is the squared Euclidean distance) which can be computed in nearly linear time. On more general domains, such as (possibly nonconvex) shapes or on manifolds discretized by a triangular mesh, following a recently proposed numerical scheme for optimal transport, this Gibbs kernel multiplication is approximated by a short-time heat diffusion. We show numerical illustrations of this method to approximate crowd motions with congestion on complicated domains as well as extensions to take into account interactions between multiple densities. [ABSTRACT FROM AUTHOR]

We propose a new class of primal-dual Fejér monotone algorithms for solving systems of composite monotone inclusions. Our construction is inspired by a framework used by Eckstein and Svaiter for the basic problem of finding a zero of the sum of two monotone operators. At each iteration, points in the graph of the monotone operators present in the model are used to construct a half-space containing the Kuhn-Tucker set associated with the system. The primal-dual update is then obtained via a relaxed projection of the current iterate onto this half-space. An important feature that distinguishes the resulting splitting algorithms from existing ones is that they do not require prior knowledge of bounds on the linear operators involved or the inversion of linear operators. [ABSTRACT FROM AUTHOR]