36,473 research outputs found
Multi-Symplectic Integrators for Nonlinear Wave Equations
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
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
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
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
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 customer classes, we show that the marginal queue length distribution for class factorizes over the number of other customer types. The factorizing coefficients equal the queue length probabilities of a PS queue for type in isolation, in which the customers of the other types reside \textit{ permanently} in the system. Similarly, the (conditional) mean sojourn time for class 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
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
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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