In this paper we solve the problem of finding a trajectory that shows that a given hybrid dynamical system with deterministic evolution leaves a given set of states considered to be safe. The algorithm combines local with global search for achieving both efficiency and global convergence. In local search, it exploits derivatives for efficient computation. Unlike other methods for falsification of hybrid systems with deterministic evolution, we do not restrict our search to trajectories of a certain bounded length but search for error trajectories of arbitrary length.
The falsification problem can also be viewed as a boundary value problem which is a classical topic in numerical mathematics @cite_20 . However, classical numerical methods assume a fixed final time, whereas we search for error trajectories of arbitrary length. Moreover, classical methods for boundary values problems are restricted to purely continuous systems and the formulation of boundary conditions as equalities. Note that, unlike the continuous case, the trajectory of a hybrid system can reach a point @math for which @math holds every reaching a point @math for which @math holds.
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We present an algorithm for falsifying safety properties of hybrid systems, i.e., for finding a trajectory to an unsafe state. The approach is to approximate how close a point is to being an initial point of an error trajectory using a real-valued quality function, and then to use numerical optimisation to search for an optimum of this function. The function is computed by running simulations, where information coming from abstractions computed by a verification algorithm is exploited to determine whether a simulation looks promising and should be continued or cancelled. This information becomes more reliable as the abstraction becomes more refined. We thus interleave falsification and verification attempts. From a random point O in an infinite strip of width 1, we move in a randomly chosen direction along a curve Г. What shape of Г gives the minimum value to the expectation of the length of the path that reaches the boundary of the strip? After certain arguments suggesting that the desired curve belongs to one of four classes, it is proved that the best curve in those classes consists of four parts: an interval OA of length a, its smooth continuation, an arc AB of radius 1 and small length φ, an interval BD that is smooth continuation of the arc, and an interval DF (with a corner at the point D). If O is the origin and OA is the x axis of a coordinate system, then the coordinates of the above-mentioned points are as follows: A(a, 0), B(a + sin φ, 1 − cos φ), F(a, 1), and @math For the best curve, we have: a ≈ 0.814 and φ ≈ 0.032. Related questions are discussed. Bibliography: 9 titles. We consider the following fundamental Mutual Visibility problem: Given a set of n identical autonomous point robots in arbitrary distinct positions in the Euclidean plane, find a schedule to move them such that within finite time they reach, without collisions, a configuration in which they all see each other. The robots operate following Look-Compute-Move cycles and a robot @math can not see other robot @math if there lies a third robot @math in the line segment connecting the positions of @math and @math . Moreover, n is not assumed to be known to the robots. We study this problem in the robots with lights model, where each robot has an externally visible persistent light which can assume colors from a fixed set of colors and the color set is identical to all the robots. This model corresponds to the classical model of oblivious robots when the number of colors @math in the color set. Therefore, we focus here on the objective of minimizing the number of colors required to successfully solve Mutual Visibility. Di [16] presented two algorithms and showed that Mutual Visibility can always be solved without collisions with @math colors for both semi-synchronous and asynchronous robots under both rigid and non-rigid moves. In this paper, we present and analyze an improved algorithm which requires only @math colors and works for both semi-synchronous and asynchronous robots under both rigid and non-rigid moves; this is optimal since any algorithm for Mutual Visibility needs at least 2 colors in the robots with lights model, when n is not known. We employ a non-trivial technique which moves all the interior robots first to the boundary of the initial convex hull and then the robots in the boundary of the hull except the corners to outside of the hull until all the robots eventually become corners, without the need of any third color. Our result is interesting in the sense that asynchronicity and non-rigidity of robot movements have no effect on Mutual Visibility with respect to the number of colors needed to successfully solve it. We also provide an improved solution to the Circle Formation problem. This paper concerns linear first-order hyperbolic systems in one space dimension of the type @math with periodicity conditions in time and reflection boundary conditions in space. We state a non-resonance condition (depending on the coefficients @math and @math and the boundary reflection coefficients), which implies Fredholm solvability of the problem in the space of continuous functions. Further, we state one more non-resonance condition (depending also on @math ), which implies @math -solution regularity. Moreover, we give examples showing that both non-resonance conditions cannot be dropped, in general. Those conditions are robust under small perturbations of the problem data. Our results work for many non-strictly hyperbolic systems, but they are new even in the case of strict hyperbolicity. We show that randomization can lead to significant improvements for a few fundamental problems in distributed tracking. Our basis is the count-tracking problem, where there are @math players, each holding a counter @math that gets incremented over time, and the goal is to track an @math -approximation of their sum @math continuously at all times, using minimum communication. While the deterministic communication complexity of the problem is @math , where @math is the final value of @math when the tracking finishes, we show that with randomization, the communication cost can be reduced to @math . Our algorithm is simple and uses only O(1) space at each player, while the lower bound holds even assuming each player has infinite computing power. Then, we extend our techniques to two related distributed tracking problems: frequency-tracking and rank-tracking , and obtain similar improvements over previous deterministic algorithms. Both problems are of central importance in large data monitoring and analysis, and have been extensively studied in the literature. Finding the so-called characteristic numbers of the complex projective plane @math is a classical problem of enumerative geometry posed by Zeuthen more than a century ago. For a given @math and @math one has to find the number of degree @math genus @math curves that pass through a certain generic configuration of points and in the same time are tangent to certain generic configuration of lines. The total number of points and lines in these two configurations is @math so that the answer is a finite integer number. In this paper we translate this classical problem to the corresponding enumerative problem of tropical geometry in the case when @math . Namely, we show that the tropical problem is well-posed and establish a special case of the correspondence theorem that ensures that the corresponding tropical and classical numbers coincide. Then we use the floor diagram calculus to reduce the problem to pure combinatorics. As a consequence, we compute genus 0 characteristic numbers of @math in terms of open Hurwitz numbers. This paper considers a trapped characteristic initial value problem for the spherically symmetric Einstein-Maxwell-scalar field equations. For an open set of initial data whose closure contains in particular Reissner-Nordstrdata, the future boundary of the maximal domain of development is found to be a light-like surface along which the curvature blows up, and yet the metric can be continuously extended beyond it. This result is related to the strong cosmic censorship conjecture of Roger Penrose. The principle of determinism in classical physics is expressed mathemat- ically by the uniqueness of solutions to the initial value problem for certain equations of evolution. Indeed, in the context of the Einstein equations of general relativity, where the unknown is the very structure of space and time, uniqueness is equivalent on a fundamental level to the validity of this principle. The question of uniqueness may thus be termed the issue of the predictability of the equation. The present paper explores the issue of predictability in general relativity. Since the work of Leray, it has been known that for the Einstein equations, contrary to common experience, uniqueness for the Cauchy problem in the large does not generally hold even within the class of smooth solutions. In other words, uniqueness may fail without any loss in regularity; such failure is thus a global phenomenon. The central question is whether this violation of predictability may occur in solutions representing actual physical processes. Physical phenomena and concepts related to the general theory of relativity, namely gravitational collapse, black holes, angular momentum, etc., must cer- tainly come into play in the study of this problem. Unfortunately, the math- ematical analysis of this exciting problem is very difficult, at present beyond reach for the vacuum Einstein equations in the physical dimension. Conse- We consider a discrete elliptic equation with random coefficients @math , which (to fix ideas) are identically distributed and independent from grid point to grid point @math . On scales large w. r. t. the grid size (i. e. unity), the solution operator is known to behave like the solution operator of a (continuous) elliptic equation with constant deterministic coefficients. These symmetric ''homogenized'' coefficients @math are characterized by @math where the random field @math is the unique stationary solution of the ''corrector problem'' @math and @math denotes the ensemble average. It is known (''by ergodicity'') that the above ensemble average of the energy density @math , which is a stationary random field, can be recovered by a system average. We quantify this by proving that the variance of a spatial average of @math on length scales @math is estimated as follows: @math where the averaging function (i. e. @math , @math ) has to be smooth in the sense that @math . In two space dimensions (i. e. @math ), there is a logarithmic correction. In other words, smooth averages of the energy density @math behave like as if @math would be independent from grid point to grid point (which it is not for @math ). This result is of practical significance, since it allows to estimate the error when numerically computing @math . We introduce a Gibbs measure on nearest-neighbour paths of length @math in the Euclidean @math -dimensional lattice, where each path is penalised by a factor proportional to the size of its boundary and an inverse temperature @math . We prove that, for all @math , the random walk condensates to a set of diameter @math in dimension @math , up to a multiplicative constant. In all dimensions @math , we also prove that the volume is bounded above by @math and the diameter is bounded below by @math . Similar results hold for a random walk conditioned to have local time greater than @math everywhere in its range when @math is larger than some explicit constant, which in dimension two is the logarithm of the connective constant. We study the cover time @math by (continuous-time) random walk on the 2D box of side length @math with wired boundary or on the 2D torus,and show that in both cases with probability approaching @math as @math increases, @math . This improves a result of Dembo, Peres, Rosen, and Zeitouni (2004) and makes progresstowards a conjecture of Bramson and Zeitouni (2009). Assume that two robots are located at the centre of a unit disk. Their goal is to evacuate from the disk through an exit at an unknown location on the boundary of the disk. At any time the robots can move anywhere they choose on the disk, independently of each other, with maximum speed @math . The robots can cooperate by exchanging information whenever they meet. We study algorithms for the two robots to minimize the evacuation time: the time when both robots reach the exit. In [CGGKMP14] the authors gave an algorithm defining trajectories for the two robots yielding evacuation time at most @math and also proved that any algorithm has evacuation time at least @math . We improve both the upper and lower bounds on the evacuation time of a unit disk. Namely, we present a new non-trivial algorithm whose evacuation time is at most @math and show that any algorithm has evacuation time at least @math . To achieve the upper bound, we designed an algorithm which non-intuitively proposes a forced meeting between the two robots, even if the exit has not been found by either of them. Assume that two robots are located at the centre of a unit disk. Their goal is to evacuate from the disk through an exit at an unknown location on the boundary of the disk. At any time the robots can move anywhere they choose on the disk, independently of each other, with maximum speed @math . The robots can cooperate by exchanging information whenever they meet. We study algorithms for the two robots to minimize the evacuation time: the time when both robots reach the exit. In [9] the authors gave an algorithm defining trajectories for the two robots yielding evacuation time at most @math and also proved that any algorithm has evacuation time at least @math . We improve both the upper and lower bounds on the evacuation time of a unit disk. Namely, we present a new non-trivial algorithm whose evacuation time is at most @math and show that any algorithm has evacuation time at least @math . To achieve the upper bound, we designed an algorithm which non-intuitively proposes a forced meeting between the two robots, even if the exit has not been found by either of them. We study the cover time @math by (continuous-time) random walk on the 2D box of side length @math with wired boundary or on the 2D torus, and show that in both cases with probability approaching 1 as @math increases, @math . This improves a result of Dembo, Peres, Rosen, and Zeitouni (2004) and makes progress towards a conjecture of Bramson and Zeitouni (2009). Classical stochastic gradient methods are well suited for minimizing expected-value objective functions. However, they do not apply to the minimization of a nonlinear function involving expected values or a composition of two expected-value functions, i.e., the problem @math minxEvfv(Ew[gw(x)]). In order to solve this stochastic composition problem, we propose a class of stochastic compositional gradient descent (SCGD) algorithms that can be viewed as stochastic versions of quasi-gradient method. SCGD update the solutions based on noisy sample gradients of @math fv,gw and use an auxiliary variable to track the unknown quantity @math Ewgw(x). We prove that the SCGD converge almost surely to an optimal solution for convex optimization problems, as long as such a solution exists. The convergence involves the interplay of two iterations with different time scales. For nonsmooth convex problems, the SCGD achieves a convergence rate of @math O(k-1 4) in the general case and @math O(k-2 3) in the strongly convex case, after taking k samples. For smooth convex problems, the SCGD can be accelerated to converge at a rate of @math O(k-2 7) in the general case and @math O(k-4 5) in the strongly convex case. For nonconvex problems, we prove that any limit point generated by SCGD is a stationary point, for which we also provide the convergence rate analysis. Indeed, the stochastic setting where one wants to optimize compositions of expected-value functions is very common in practice. The proposed SCGD methods find wide applications in learning, estimation, dynamic programming, etc. Low-rank matrix approximations, such as the truncated singular value decomposition and the rank-revealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low-rank matrix approximation. These techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets. This paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions. These methods use random sampling to identify a subspace that captures most of the action of a matrix. The input matrix is then compressed—either explicitly or implicitly—to this subspace, and the reduced matrix is manipulated deterministically to obtain the desired low-rank factorization. In many cases, this approach beats its classical competitors in terms of accuracy, robustness, and or speed. These claims are supported by extensive numerical experiments and a detailed error analysis. The specific benefits of randomized techniques depend on the computational environment. Consider the model problem of finding the @math dominant components of the singular value decomposition of an @math matrix. (i) For a dense input matrix, randomized algorithms require @math floating-point operations (flops) in contrast to @math for classical algorithms. (ii) For a sparse input matrix, the flop count matches classical Krylov subspace methods, but the randomized approach is more robust and can easily be reorganized to exploit multiprocessor architectures. (iii) For a matrix that is too large to fit in fast memory, the randomized techniques require only a constant number of passes over the data, as opposed to @math passes for classical algorithms. In fact, it is sometimes possible to perform matrix approximation with a single pass over the data. The planted partition model (also known as the stochastic blockmodel) is a classical cluster-exhibiting random graph model that has been extensively studied in statistics, physics, and computer science. In its simplest form, the planted partition model is a model for random graphs on @math nodes with two equal-sized clusters, with an between-class edge probability of @math and a within-class edge probability of @math . Although most of the literature on this model has focused on the case of increasing degrees (ie. @math as @math ), the sparse case @math is interesting both from a mathematical and an applied point of view. A striking conjecture of Decelle, Krzkala, Moore and Zdeborov 'a based on deep, non-rigorous ideas from statistical physics gave a precise prediction for the algorithmic threshold of clustering in the sparse planted partition model. In particular, if @math and @math , then conjectured that it is possible to cluster in a way correlated with the true partition if @math , and impossible if @math for some sufficiently large @math . We prove half of their prediction, showing that it is indeed impossible to cluster if @math . Following , our work establishes a rigorous connection between the clustering problem, spin-glass models on the Bethe lattice and the so called reconstruction problem. This connection points to fascinating applications and open problems. Multiple objects in the plane are called pseudodisks if the boundaries of any two of them intersect transversely at most twice. Given a set of @math (possibly intersecting) convex pseudodisks of @math complexity each and two points @math and @math in the plane, we present an efficient algorithm for computing a shortest @math -to- @math path avoiding the pseudodisks. After the union of the pseudodisks is computed, which can be done in @math randomized time or @math deterministic time, our algorithm runs in @math deterministic time, where @math is the size of the extended visibility graph of the union of the pseudodisks. Note that @math in the worst case. In over two decades, the previously best algorithms for this problem have not improved on the bound of @math time, even when all the pseudodisks are pairwise disjoint disks. Our technique is also applicable to a motion planning problem of finding a shortest path to translate a convex object in the plane from one location to anot... In this paper, we study the problem of maximizing continuous submodular functions that naturally arise in many learning applications such as those involving utility functions in active learning and sensing, matrix approximations and network inference. Despite the apparent lack of convexity in such functions, we prove that stochastic projected gradient methods can provide strong approximation guarantees for maximizing continuous submodular functions with convex constraints. More specifically, we prove that for monotone continuous DR-submodular functions, all fixed points of projected gradient ascent provide a factor @math approximation to the global maxima. We also study stochastic gradient methods and show that after @math iterations these methods reach solutions which achieve in expectation objective values exceeding @math . An immediate application of our results is to maximize submodular functions that are defined stochastically, i.e. the submodular function is defined as an expectation over a family of submodular functions with an unknown distribution. We will show how stochastic gradient methods are naturally well-suited for this setting, leading to a factor @math approximation when the function is monotone. In particular, it allows us to approximately maximize discrete, monotone submodular optimization problems via projected gradient ascent on a continuous relaxation, directly connecting the discrete and continuous domains. Finally, experiments on real data demonstrate that our projected gradient methods consistently achieve the best utility compared to other continuous baselines while remaining competitive in terms of computational effort. We investigate the problem of determining visible regions given a set of (moving) obstacles and a (moving) vantage point. Our approach to this problem is through an implicit framework, where the obstacles are represented by a level set function. The visibility problem is formally formulated as a boundary value problem (BVP) of a first order partial differential equation. It is based on the continuation of values along the given ray field. We propose a one-pass, multi-level algorithm for the construction of the solution on a grid. Furthermore, we study the dynamics of shadow boundaries on the surfaces of the obstacles when the vantage point moves along a given trajectory. In all of these situations, topological changes such as merging and breaking occur in the regions of interest. These are automatically handled by the level set framework proposed here. Finally, we obtain additional useful information through simple operations in the level set framework. Given a simplicial complex with weights on its simplices, and a nontrivial cycle on it, we are interested in finding the cycle with minimal weight which is homologous to the given one. Assuming that the homology is defined with integer ( @math ) coefficients, we show the following (Theorem 5.2): For a finite simplicial complex @math of dimension greater than @math , the boundary matrix @math is totally unimodular if and only if @math is torsion-free for all pure subcomplexes @math in @math of dimensions @math and @math , respectively, where @math . Because of the total unimodularity of the boundary matrix, we can solve the optimization problem, which is inherently an integer programming problem, as a linear program and obtain an integer solution. Thus, the problem of finding optimal cycles in a given homology class can be solved in polynomial time. This result is surprising in the backdrop of a recent result which says that the problem is NP-hard under @math coefficients which, being a field, is in general easier to deal with. Our result implies, among other things, that one can compute in polynomial time an optimal @math -cycle in a given homology class for any triangulation of an orientable compact @math -manifold or for any finite simplicial complex embedded in @math . Our optimization approach can also be used for various related problems, such as finding an optimal chain homologous to a given one when these are not cycles. Our result can also be viewed as providing a topological characterization of total unimodularity.