DID: Distributed Incremental Block Coordinate Descent for Nonnegative Matrix Factorization

Nonnegative matrix factorization (NMF) has attracted much attention in the last decade as a dimension reduction method in many applications. Due to the explosion in the size of data, naturally the samples are collected and stored distributively in local computational nodes. Thus, there is a growing need to develop algorithms in a distributed memory architecture. We propose a novel distributed algorithm, called \textit{distributed incremental block coordinate descent} (DID), to solve the problem. By adapting the block coordinate descent framework, closed-form update rules are obtained in DID. Moreover, DID performs updates incrementally based on the most recently updated residual matrix. As a result, only one communication step per iteration is required. The correctness, efficiency, and scalability of the proposed algorithm are verified in a series of numerical experiments.Comment: Accepted by AAAI 201

A refined numerical investigation of a large equivalent shallow-depth underwater explosion

The large equivalent shallow-depth explosion problem is very significant in the field of naval architecture and ocean engineering, as such explosions can be used to attack and demolish ships and anti-ship missiles. In the current work, a refined numerical study of the flow-field characteristics of a large equivalent shallow-depth explosion is carried out using a self-developed Eulerian finite element solver. Firstly, the numerical model is validated against theoretical results and a small equivalent explosion test in a tank. The numerical results are found to agree well with the theoretical and experimental results. In the next step, the cavitation cut-off effect is added to the underwater explosion model, and the cavitation phenomenon is quantitatively analyzed through the flow-field pressure. In addition, the dynamic characteristics of the bubble and water hump under various initial conditions for different stand-off parameters are analyzed. The effect of gravity on these physical processes is also discussed. The bubble pulsation period, taking into account the free surface effect, is then quantitatively studied and compared with Cole's experimental formula for an underwater explosion. Overall, when the stand-off parameter > 2, the influence of the free surface on the empirical period of the bubble is not significant. Our investigation provides broad insights into shallow-depth underwater explosions from theoretical, experimental, and numerical perspectives

Conflict Equivalence of Branching Processes

International audienceFor concurrent and large systems, specification step is a crucial point. Combinatory explosion is a limit that can be encountered when a state space exploration is driven on large specification modeled with Petri nets. Considering bounded Petri nets, technics like unfolding can be a way to cope with this problem. This paper is a first attempt to present an axiomatic model to produce the set of processes of unfoldings into a canonic form. This canonic form allows to define a conflict equivalence

Quantitative and Algorithmic aspects of Barrier Synchronization in Concurrency

In this paper we address the problem of understanding Concurrency Theory from a combinatorial point of view. We are interested in quantitative results and algorithmic tools to refine our understanding of the classical combinatorial explosion phenomenon arising in concurrency. This paper is essentially focusing on the the notion of synchronization from the point of view of combinatorics. As a first step, we address the quantitative problem of counting the number of executions of simple processes interacting with synchronization barriers. We elaborate a systematic decomposition of processes that produces a symbolic integral formula to solve the problem. Based on this procedure, we develop a generic algorithm to generate process executions uniformly at random. For some interesting sub-classes of processes we propose very efficient counting and random sampling algorithms. All these algorithms have one important characteristic in common: they work on the control graph of processes and thus do not require the explicit construction of the state-space

Time Integration Methods of Fundamental Solutions and Approximate Fundamental Solutions for Nonlinear Elliptic Partial Differential Equations

A time-dependent method is coupled with the Method of Approximate Particular Solutions (MAPS) of Delta-shaped basis functions, the Method of Fundamental Solutions (MFS), and the Method of Approximate Fundamental Solutions (MAFS) to solve a second order nonlinear elliptic partial differential equation (PDE) on regular and irregular shaped domains. The nonlinear PDE boundary value problem is first transformed into a time-dependent quasilinear problem by introducing a fictitious time. Forward Euler integration is then used to ultimately convert the problem into a sequence of time-dependent linear nonhomogeneous modified Helmholtz boundary value problems on which the superposition principle is applied to split the numerical solution at each time step into a homogeneous solution and an approximate particular solution. The Crank-Nicholson method is also examined as an option for the numerical integration as opposed to the forward Euler method. A Delta-shaped basis function, which can handle scattered data in various domains, is used to provide an approximation of the source function at each time step and allows for a derivation of an approximate particular solution of the associated nonhomogeneous equation using the MAPS. The corresponding homogeneous boundary value problem is solved using MFS or MAFS. Numerical results support the accuracy and validity of these computational methods. The proposed numerical methods are additionally applied in nonlinear thermal explosion to determine the steady state critical condition in explosive regimes

Homogeneous explosion and shock initiation for a three-step chain-branching reaction model

The role of chain-branching cross-over temperatures in shock-induced ignition of reactive materials is studied by numerical simulation, using a three-step chainbranching reaction model. In order to provide insight into shock initiation, the simpler problem of a spatially homogeneous explosion is first considered. It is shown that for ratios of the cross-over temperature to the initial temperature, T-B, sufficiently less than unity, the homogeneous explosion can be quantitatively described by a widely used two-step model, while for T-B sufficiently above unity the homogeneous explosion can be effectively described by the standard one-step model. From the matchings between these homogeneous-explosion solutions, the parameters of the reduced models are identified in terms of those of the three-step model. When T-B is close to unity, all the reactions of the three-step model have a leading role, and hence in this case the model cannot be reduced further. In the case of shock initiation, for T-B (which is now the ratio of the cross-over temperature to the initial shock temperature) sufficiently below unity, the three-step solutions are qualitatively described by those of the matched two-step model, but there are quantitative differences due to the assumption in the reduced model that a purely chain-branching explosion occurs instantaneously. For T-B sufficiently above unity, the matched one-step model is found to effectively describe the way in which the heat release and fluid dynamics couple. For T-B close to unity, the competition between chain branching and chain termination is important from the outset. In these cases the speed at which the forward moving explosion wave that emerges from the piston is sensitive to T-B, and changes from supersonic to subsonic for a value of T-B just below unity