36,473 research outputs found

    Multi-Symplectic Integrators for Nonlinear Wave Equations

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    Symplectic (area-preserving) integrators for Hamiltonian ordinary differential equations have shown to be robust, efficient and accurate in long-term calculations. In this thesis, we show how symplectic integrators have a natural generalization to Hamiltonian PDEs by introducing the concept of multi-symplectic partial differential equations (PDEs). In particular, we show that multi-symplectic PDEs have an underlying spatio-temporal multi-symplectic structure characterized by a multi-symplectic conservation law MSCL). Then multi-symplectic integrators (MSIs) are numerical schemes that preserve exactly the MSCL. Remarkably, we demonstrate that, although not designed to do so, MSIs preserve very well other associated local conservation laws and global invariants, such as the energy and the momentum, for very long periods of time. We develop two types of MSIs, based on finite differences and Fourier spectral approximations, and illustrate their superior performance over traditional integrators by deriving new numerical schemes to the well known 1D nonlinear Schrödinger and sine-Gordon equations and the 2D Gross-Pitaevskii equation. In sensitive regimes, the spectral MSIs are not only more accurate but are better at capturing the spatial features of the solutions. In particular, for the sine-Gordon equation we show that its phase space, as measured by the nonlinear spectrum associated with it, is better preserved by spectral MSIs than by spectral non-symplectic Runge-Kutta integrators. Finally, to further understand the improved performance of MSIs, we develop a backward error analysis of the multi-symplectic centered-cell discretization for the nonlinear Schrödinger equation. We verify that the numerical solution satisfies to higher order a nearby modified multi-symplectic PDE and its modified multi-symplectic energy conservation law. This implies, that although the numerical solution is an approximation, it retains the key feature of the original PDE, namely its multi-symplectic structure

    Comparative Experiments on Disambiguating Word Senses: An Illustration of the Role of Bias in Machine Learning

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    This paper describes an experimental comparison of seven different learning algorithms on the problem of learning to disambiguate the meaning of a word from context. The algorithms tested include statistical, neural-network, decision-tree, rule-based, and case-based classification techniques. The specific problem tested involves disambiguating six senses of the word ``line'' using the words in the current and proceeding sentence as context. The statistical and neural-network methods perform the best on this particular problem and we discuss a potential reason for this observed difference. We also discuss the role of bias in machine learning and its importance in explaining performance differences observed on specific problems.Comment: 10 page

    A Stochastic Interpretation of Stochastic Mirror Descent: Risk-Sensitive Optimality

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    Stochastic mirror descent (SMD) is a fairly new family of algorithms that has recently found a wide range of applications in optimization, machine learning, and control. It can be considered a generalization of the classical stochastic gradient algorithm (SGD), where instead of updating the weight vector along the negative direction of the stochastic gradient, the update is performed in a "mirror domain" defined by the gradient of a (strictly convex) potential function. This potential function, and the mirror domain it yields, provides considerable flexibility in the algorithm compared to SGD. While many properties of SMD have already been obtained in the literature, in this paper we exhibit a new interpretation of SMD, namely that it is a risk-sensitive optimal estimator when the unknown weight vector and additive noise are non-Gaussian and belong to the exponential family of distributions. The analysis also suggests a modified version of SMD, which we refer to as symmetric SMD (SSMD). The proofs rely on some simple properties of Bregman divergence, which allow us to extend results from quadratics and Gaussians to certain convex functions and exponential families in a rather seamless way

    Level Set Methods for Stochastic Discontinuity Detection in Nonlinear Problems

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    Stochastic physical problems governed by nonlinear conservation laws are challenging due to solution discontinuities in stochastic and physical space. In this paper, we present a level set method to track discontinuities in stochastic space by solving a Hamilton-Jacobi equation. By introducing a speed function that vanishes at discontinuities, the iso-zero of the level set problem coincide with the discontinuities of the conservation law. The level set problem is solved on a sequence of successively finer grids in stochastic space. The method is adaptive in the sense that costly evaluations of the conservation law of interest are only performed in the vicinity of the discontinuities during the refinement stage. In regions of stochastic space where the solution is smooth, a surrogate method replaces expensive evaluations of the conservation law. The proposed method is tested in conjunction with different sets of localized orthogonal basis functions on simplex elements, as well as frames based on piecewise polynomials conforming to the level set function. The performance of the proposed method is compared to existing adaptive multi-element generalized polynomial chaos methods

    Decomposing the queue length distribution of processor-sharing models into queue lengths of permanent customer queues

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    We obtain a decomposition result for the steady state queue length distribution in egalitarian processor-sharing (PS) models. In particular, for an egalitarian PS queue with KK customer classes, we show that the marginal queue length distribution for class kk factorizes over the number of other customer types. The factorizing coefficients equal the queue length probabilities of a PS queue for type kk in isolation, in which the customers of the other types reside \textit{ permanently} in the system. Similarly, the (conditional) mean sojourn time for class kk can be obtained by conditioning on the number of permanent customers of the other types. The decomposition result implies linear relations between the marginal queue length probabilities, which also hold for other PS models such as the egalitarian processor-sharing models with state-dependent system capacity that only depends on the total number of customers in the system. Based on the exact decomposition result for egalitarian PS queues, we propose a similar decomposition for discriminatory processor-sharing (DPS) models, and numerically show that the approximation is accurate for moderate differences in service weights. \u

    Solving One Dimensional Scalar Conservation Laws by Particle Management

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    We present a meshfree numerical solver for scalar conservation laws in one space dimension. Points representing the solution are moved according to their characteristic velocities. Particle interaction is resolved by purely local particle management. Since no global remeshing is required, shocks stay sharp and propagate at the correct speed, while rarefaction waves are created where appropriate. The method is TVD, entropy decreasing, exactly conservative, and has no numerical dissipation. Difficulties involving transonic points do not occur, however inflection points of the flux function pose a slight challenge, which can be overcome by a special treatment. Away from shocks the method is second order accurate, while shocks are resolved with first order accuracy. A postprocessing step can recover the second order accuracy. The method is compared to CLAWPACK in test cases and is found to yield an increase in accuracy for comparable resolutions.Comment: 15 pages, 6 figures. Submitted to proceedings of the Fourth International Workshop Meshfree Methods for Partial Differential Equation

    Solving 1D Conservation Laws Using Pontryagin's Minimum Principle

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    This paper discusses a connection between scalar convex conservation laws and Pontryagin's minimum principle. For flux functions for which an associated optimal control problem can be found, a minimum value solution of the conservation law is proposed. For scalar space-independent convex conservation laws such a control problem exists and the minimum value solution of the conservation law is equivalent to the entropy solution. This can be seen as a generalization of the Lax--Oleinik formula to convex (not necessarily uniformly convex) flux functions. Using Pontryagin's minimum principle, an algorithm for finding the minimum value solution pointwise of scalar convex conservation laws is given. Numerical examples of approximating the solution of both space-dependent and space-independent conservation laws are provided to demonstrate the accuracy and applicability of the proposed algorithm. Furthermore, a MATLAB routine using Chebfun is provided (along with demonstration code on how to use it) to approximately solve scalar convex conservation laws with space-independent flux functions
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